Find a method for your problem
Pick a problem area and set your limits. You'll see the methods whose records meet them, and why. When a record doesn't state something, the result says so instead of guessing.
285 methods match.
- Algorithms
Active-space VQE workflow
Frozen-core and active-orbital choices define the Hamiltonian size before variational optimization.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Find a better parameterized ansatz for the quantum approximate optimization algorithm (QAOA) applied to combinatorial optimization problems such as Max-Cut, where the standard, fixed-form QAOA ansatz is not known to be optimal and no systematic method exists for improving on it.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
ADAPT-VQE
An adaptive ansatz grows one operator at a time using measured energy gradients from a predefined pool.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Basic circuits
Alternating-parity oracle scaffold · 16 qubits
A deterministic parity-oracle circuit that marks alternating input wires through phase kickback.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Alternating-parity oracle scaffold · 3 qubits
A deterministic parity-oracle circuit that marks alternating input wires through phase kickback.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Algorithms
Amplitude amplification
The general framework behind Grover's algorithm: boosting the success probability of any subroutine with a known reflection structure.
- Cost, as recorded
- 1 (toy instance) qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Analytic-gradient VQE
Parameter-shift or analytic derivative measurements supply gradients to a classical optimizer.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Given query access to the entries of a d × d Hermitian matrix A, output a classical description of a good approximation of its top eigenvector, the eigenvector belonging to the largest eigenvalue.
- Cost, as recorded
- Two algorithms for a d × d Hermitian matrix under an assumed constant eigenvalue gap: one of time complexity Õ(d^1.75), one of time complexity d^(1.5+o(1)), against the best-possible classical algorithm, which the abstract states needs Ω(d²) queries to the entries of A and hence Ω(d²) time. The same abstract states time q·d^(1.5+o(1)) for a classical description of the subspace spanned by the top-q eigenvectors, and a nearly-optimal lower bound of Ω̃(d^1.5) on the quantum query complexity of approximating the top eigenvector. The Zoo instead gives a single figure, Õ(d^(3/2)) against a best classical Õ(d²), which matches neither of the abstract's two upper bounds exactly.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Price discretely monitored Asian options over T monitoring points, where the underlying asset is modeled by a geometric Brownian motion.
- Cost, as recorded
- Poly-logarithmic in T and polynomial in 1/ε for both quantum algorithms given, where T is the number of monitoring points and ε is the additive approximation error; the better of the two reaches Õ(1/ε³), with the tilde suppressing factors that are poly-logarithmic in T and 1/ε. One algorithm is built from an O(log T)-qubit semi-digital quantum encoding of the Brownian motion. The abstract states no further constant, no explicit qubit or gate count for the Monte-Carlo-derived algorithm, and no total circuit-resource count for either.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Batched ADAPT-VQE
Several high-gradient operators are appended per adaptive iteration to reduce optimization and measurement rounds.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Basic circuits
Bell state measurement
A two-qubit entanglement example with a return contract and distribution check.
- Cost, as recorded
- 2 qubits, 2 gates
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Textbook reference
- Basic circuits
Bell-pair ladder benchmark · 16 qubits
A bank of disjoint Bell-pair preparations that exposes parallel two-qubit-gate structure.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Bell-pair ladder benchmark · 4 qubits
A bank of disjoint Bell-pair preparations that exposes parallel two-qubit-gate structure.
- Cost, as recorded
- 4 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Bernstein–Vazirani all-ones oracle · 16 qubits
A fixed all-ones Bernstein–Vazirani instance with the last qubit used as the phase-kickback ancilla.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Bernstein–Vazirani all-ones oracle · 3 qubits
A fixed all-ones Bernstein–Vazirani instance with the last qubit used as the phase-kickback ancilla.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Algorithms
Bernstein–Vazirani hidden-string circuit
A query circuit that recovers the hidden bit string in f(x) = s · x.
- Cost, as recorded
- n + 1 qubits, oracle-dependent
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Primary paper
- Algorithms
Classical-shadow VQE estimation
Randomized measurements are reused to estimate many observables from a shared data set.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Given oracle access to k matrices of size n × n, where a query names a matrix index x together with a pair of indices i, j and returns the ij entry of the x-th matrix, decide whether all k of the matrices commute with one another.
- Cost, as recorded
- O(k^(4/5) n^(9/5)) oracle queries to decide whether all k of the n × n matrices commute; the Zoo states that classically the task requires Ω(k n²) queries. The abstract separately states a lower bound of Ω(k^(1/2) n) for the problem, obtained with Ambainis's lower-bounding technique, so the two bounds the abstract gives do not meet.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
A parametrized entangling gate that applies e^{iλ} exactly when both qubits are |1⟩, generalizing CZ and forming the entangling primitive of the quantum Fourier transform.
- Cost, as recorded
- 2 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
Controlled-X gate
The canonical two-qubit controlled operation used to create correlation and entanglement.
- Cost, as recorded
- 2 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
Controlled-Z gate
A diagonal two-qubit entangling gate that applies a -1 phase exactly when both qubits are |1⟩, and is symmetric between the two qubits it acts on.
- Cost, as recorded
- 2 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
CVaR-VQE objective
Conditional value-at-risk averages only a selected low-energy tail of samples for combinatorial objectives.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Deutsch–Jozsa query circuit
A one-query promise-problem circuit that separates constant from balanced Boolean oracles.
- Cost, as recorded
- 3 qubits, oracle-dependent
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Primary paper
- Algorithms
Discrete logarithm on a quantum computer
Given three n-bit numbers a, b and N with the promise that b = a^s mod N for some s, recover the exponent s.
- Cost, as recorded
- Polynomial in the input size: the Zoo states that s can be found in poly(n) time for n-bit inputs while the fastest known classical algorithm takes time superpolynomial in n, and neither source quotes an exponent or constant. For the elliptic-curve variant over an n-bit prime field, Roetteler et al. estimate at most 9n + 2⌈log₂(n)⌉ + 10 **logical** qubits and a circuit of at most 448 n³ log₂(n) + 4090 n³ Toffoli gates. Their abstract states that qubit figure without the qualifier; section 5 is where they call it logical, and the paper states no error-correction assumption at all, so it supports no physical-qubit reading.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Discrete-time quantum walk on a line
A coin-and-shift circuit that spreads a walker's position amplitude ballistically, in contrast to the diffusive spread of a classical random walk.
- Cost, as recorded
- 1 coin + 3 position (8 sites) qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Element distinctness by quantum walk
Given oracle access to N items carrying no promise about the underlying function, find two items that are equal, or establish that all N are distinct.
- Cost, as recorded
- O(N^(2/3)) queries for element distinctness on N items, matching the lower bound of Shi and improving the previous O(N^(3/4))-query quantum algorithm; the same algorithm uses O(N^(k/(k+1))) queries to find k equal items among N. The Zoo gives Θ(N) classical query complexity for element distinctness, and for the two-to-one collision problem O(N^(1/3)) quantum queries against Θ(√N) classical randomized queries, with Brassard, Hoyer and Tapp stating O((N/r)^(1/3)) expected function evaluations for r-to-one functions.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Determine the role quantum dynamical effects play in the efficiency of exciton (energy) transfer within photosynthetic molecular arrays — such as the Fenna-Matthews-Olson (FMO) protein complex, whose long-lived coherence had recently been demonstrated experimentally — that interact with a thermal bath.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Operators
First-order Trotter product
Product-formula approximation to Hamiltonian evolution. Representative form: e^{-itΣH_j} ≈ ∏_j e^{-itH_j}.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Algorithms
Folded-spectrum VQE
Minimizing the squared shifted Hamiltonian targets eigenstates near a chosen energy shift.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Generalized excitation VQE
Generalized singles and doubles relax occupied-to-virtual restrictions to enlarge the variational manifold.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Basic circuits
GHZ chain benchmark · 16 qubits
A width-scaled GHZ preparation circuit with one Hadamard and a nearest-neighbor CNOT chain.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
GHZ chain benchmark · 3 qubits
A width-scaled GHZ preparation circuit with one Hadamard and a nearest-neighbor CNOT chain.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
GHZ state preparation
A reusable GHZ state preparation circuit with simulator-only evidence.
- Cost, as recorded
- 4 qubits, 2 layers
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Framework example
- Algorithms
Grover unstructured search
The canonical search primitive, paired with the linear classical baseline it is meant to improve.
- Cost, as recorded
- 2 toy qubits, Oracle + diffusion
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
- Gates
Hadamard gate
The smallest useful reference for putting a qubit into an equal superposition.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Hardware-efficient VQE ansatz
Alternating native one-qubit rotations and entanglers reduce compilation overhead but can change trainability.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Hidden shift problem
Given oracle access to a function f on the integers mod N together with the promise that f(x) = g(x+s) for a known function g and an unknown shift s, recover s.
- Cost, as recorded
- O(1) quantum queries for the special case in which f is a multiplicative character of a finite ring or field; for hidden shift in general, at least √N queries are necessary, by reduction from Grover's problem.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Basic circuits
Ising Trotter-step benchmark · 16 qubits
A first-order nearest-neighbor ZZ evolution scaffold expressed as CNOT–RZ–CNOT blocks.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Ising Trotter-step benchmark · 3 qubits
A first-order nearest-neighbor ZZ evolution scaffold expressed as CNOT–RZ–CNOT blocks.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Algorithms
Iterative phase estimation
Estimates the eigenphase of a unitary using a single reused ancilla qubit instead of a full phase-estimation register.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Iterative qubit coupled cluster
Iterative QCC repeatedly dresses the Hamiltonian and selects new entanglers instead of fixing one deep circuit.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
k-UpCCGSD ansatz
Repeated paired generalized doubles with generalized singles trade expressivity against shallower chemistry circuits.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Layerwise VQE training
Circuit depth grows in stages so each newly introduced layer can be initialized and optimized locally.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Puts a list of 2ⁿ probabilities into n qubits as amplitudes √p, exactly, with one uniformly controlled RY rotation per qubit.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Estimate a matrix element of a unitary irreducible representation of a group — a single entry of a matrix whose dimension may be exponentially large — to within an additive error, for the symmetric and alternating groups and for the unitary, special unitary and special orthogonal groups of polynomial highest weight.
- Cost, as recorded
- Time polynomial in n and 1/ε to obtain any matrix element of any irreducible representation of the symmetric or alternating group to within ±ε, and likewise for any irreducible representation of polynomial highest weight of the unitary, special unitary and special orthogonal groups. The Hadamard test itself needs O(1/ε²) measurements.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given three n×n matrices A, B and C over a field, decide whether AB = C, rather than computing the product and comparing it entry by entry.
- Cost, as recorded
- Worst-case time n^(5/3) to verify a product of two n×n matrices over any field with bounded error, and expected time n^(5/3) / min(w, √n)^(1/3), where w is the number of wrong entries; the Zoo gives O(n^(7/4)) for the earlier quantum algorithm it credits to Ambainis et al., O(n^2) for the best known randomized classical verification, and O(n^(2.373)) for the best known classical matrix multiplication.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Measurement-grouped VQE
Commuting Pauli terms are partitioned into compatible bases to reduce distinct measurement circuits.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Compute the ground-state energy of an atom or molecule, a calculation whose time the paper states scales exponentially with system size on a classical computer.
- Cost, as recorded
- Polynomial in place of exponential: the abstract states that the calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms, that the number of quantum bits required scales linearly with the number of basis functions, and that the number of gates required grows polynomially with the number of quantum bits. No exponent, constant factor or target accuracy is attached to either polynomial. The one concrete count in the abstract is for readout: the recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Multistate contracted VQE
A contracted reference subspace is jointly entangled before a small effective Hamiltonian is diagonalized.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Option pricing by amplitude estimation
Price an option, or a portfolio of options — vanilla contracts, multi-asset contracts, and path-dependent contracts such as barrier options — on a gate-based quantum computer, in the setting where the paper takes classical Monte Carlo methods as its point of comparison.
- Cost, as recorded
- A quadratic speedup compared to classical Monte Carlo methods, stated in that form and in no other: the abstract attaches the speedup to amplitude estimation itself, and gives no error scaling, no query count, no qubit count and no gate count for the option-pricing circuits, so nothing here says how many samples or oracle calls pricing a given contract to a given accuracy would take, nor at what accuracy the two methods cross over.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Orbital-optimized VQE
Orbital rotations are optimized alongside circuit parameters to improve compact active-space descriptions.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Gates
P phase gate
The generalized phase gate that fixes the |0⟩ amplitude at 1 and applies e^{iλ} to |1⟩, unifying S, T, and Z as special cases of one continuous parameter.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Particle-conserving VQE circuits
Givens-style or excitation-preserving blocks keep evolution inside a fixed-particle-number sector.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Gates
Pauli-Y gate
The Pauli gate that combines a bit flip with a phase rotation around the y axis.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
Pauli-Z gate
A phase-flip gate whose behavior is invisible to a basis measurement until interference is introduced.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Given a positive non-square integer d, Pell's equation is x² − dy² = 1 and the goal is to find all integer solutions. The least solution can have exponentially many bits, so it cannot be written down; the computational problem is instead to compute the integer part of the regulator R = ln(x₁ + y₁√d), from which the solution can be recovered.
- Cost, as recorded
- Polynomial in log Δ and log δ, where Δ is the quadratic discriminant and δ the precision to which the regulator is approximated, with probability exponentially close to one.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Penalty-based excited-state VQE
Orthogonality or symmetry penalties augment the energy objective to exclude previously identified sectors.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Prove that a given integer N is prime, or in most cases prove it composite and produce a witness, rather than merely declaring it probably prime as a randomized primality test does.
- Cost, as recorded
- O((log n)² n³) operations for an n-bit number N, reducible to O(log log n (log n)³ n²) in the asymptotic limit if fast multiplication is used. The paper states that this reduces the asymptotic complexity of the earlier Chau-Lo quantum primality test from O((log n)(log log n) n³) to O((log log n)(log n)³ n²).
- Speedup, as the sources state it
- Polynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
QAOA MaxCut on a 5-node ring
A p=1 QAOA circuit with a classical MaxCut comparison and reproducible simulation evidence.
- Cost, as recorded
- 5 qubits, 14 gates
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Verified run
- Basic circuits
QAOA ring layer benchmark · 16 qubits
A fixed-angle p=1 MaxCut-style QAOA layer on a cycle graph, including cost and mixer blocks.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
QAOA ring layer benchmark · 3 qubits
A fixed-angle p=1 MaxCut-style QAOA layer on a cycle graph, including cost and mixer blocks.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
Given the low autocorrelation binary sequences (LABS) problem, an optimization problem that is classically intractable even for moderately sized instances, determine whether QAOA can act as an algorithmic component that provides an advantage over the best classical exact solvers.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Solve large-scale Maximum Cut (MaxCut) problems on near-term quantum hardware by decomposing the graph into many subgraph problems that can be solved in parallel, and determine how such a decomposition compares to a purely classical alternative for the same task.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
QFold: protein folding by quantum walk
Predict the three-dimensional structure of a protein from its amino-acid sequence, addressed here by parameterizing the protein in terms of the torsion angles of its amino acids rather than by a lattice-model simplification.
- Cost, as recorded
- A polynomial quantum advantage over the classical analog algorithm, found by comparing the two across different annealing schedules. The abstract states no exponent, no big-O expression, and no other resource count for either algorithm.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
QFT resource screen
A resource-focused QFT construction with a readable compile comparison.
- Cost, as recorded
- 4 qubits, 18 gates
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Quantum amplitude estimation
A finance- and Monte-Carlo-facing primitive that makes the oracle and error model visible.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
Price rainbow options — a type of path-independent multi-asset derivative — on a quantum computer.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Quantum equation-of-motion VQE
Commutator matrix elements over a VQE reference produce excitation energies through an equation-of-motion problem.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Quantum Fourier Transform
The unitary that maps computational basis states to Fourier-basis phase patterns, and the phase-estimation primitive behind Shor's and HHL's speedups.
- Cost, as recorded
- 3 shown (n general) qubits, O(n²) gates exact · O(n log n) approximate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Quantum natural-gradient VQE
The Fubini–Study metric preconditions parameter updates according to circuit-state geometry.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Quantum phase estimation
The eigenphase-estimation primitive behind chemistry, simulation, and period-finding workflows.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
Evaluate candidate designs for a simplified cooling system within an engineering design process that normally requires numerous computationally intensive numerical simulations, in a way that avoids the data input/output overhead that otherwise erodes any quantum speedup on such simulation tasks.
- Cost, as recorded
- For a single QuSO iteration, the paper's gate-level complexity analysis shows that the cost function can be coherently computed, over a superposition of exponentially many configurations, using circuits of polynomial complexity — but the abstract states this does not yield a speedup for a single simulation instance; any advantage is left as a potential one arising from the subsequent QAOA-based search over many such iterations, not a speedup proven here. The abstract also states that it expresses the scaling in terms of problem parameters and QAOA depth and iterations, without giving that expression itself, so no explicit formula, exponent, or constant for that scaling is recorded above.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Quantum subspace expansion
Measured response operators around a VQE state define a generalized eigenproblem for excitations and mitigation.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Quantum teleportation
A communication primitive that makes entanglement, measurement, and classical feed-forward explicit.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
- Algorithms
Quantum walk speedup of backtracking
Backtracking is the general classical technique for exploiting problem structure in constraint satisfaction: explore a tree of partial assignments and prune the branches a predicate rules out. The question is whether an arbitrary backtracking algorithm — any predicate and any branching heuristic — can be sped up quantumly, rather than replaced by brute-force search over the whole assignment space.
- Cost, as recorded
- O(√T · n^(3/2) · log n) evaluations of the predicate and the heuristic to find a solution or report none, where T is the number of vertices in the classical backtracking algorithm's tree and n the number of variables. Detecting whether a solution exists is cheaper, at O(√T · n) evaluations, and the promised-unique case costs O(√T · n · log³ n).
- Speedup, as the sources state it
- Polynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Qubit coupled-cluster VQE
Qubit coupled-cluster uses Pauli-word entanglers and a product-state reference directly in qubit space.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Qubit-ADAPT-VQE
Qubit-space Pauli generators replace fermionic excitation operators to seek shorter adaptive circuits.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Readout-mitigated VQE
Calibrated assignment errors are inverted or regularized before Pauli expectations are assembled.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Carry Shor's discrete-logarithm algorithm through concretely for the group of points on an elliptic curve over GF(p), and count the qubits and operations it needs, so that the cost of attacking elliptic-curve cryptography can be compared with the cost of attacking RSA at an equivalent classical security level.
- Cost, as recorded
- About 6n **logical** qubits and order n³ operations for an n-bit prime field, given as roughly 360n³ n-bit additions against about 4kn³ for factoring. Concretely the paper's table pairs a 160-bit elliptic-curve key at around 1000 logical qubits on a machine the authors twice call perfect and noise-free, with the security-equivalent 1024-bit RSA modulus at about 2000, and it gives the same pairing at 512, 2048, 3072 and 15360-bit RSA against 110, 224, 256 and 512-bit curves.
- Speedup, as the sources state it
- Various (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Bound the quantum-computing resources a fault-tolerant quantum computer would need to price derivatives — autocallable and Target Accrual Redemption Forward (TARF) instruments serving as the paper's benchmark use cases — at a scale offering a quantum advantage over classical pricing.
- Cost, as recorded
- 8k logical qubits and a T-depth of 54 million, and execution at the order of a second to reach quantum advantage, reported by the abstract for the benchmark use cases the paper examines — autocallable and Target Accrual Redemption Forward (TARF) derivatives together. The abstract states this figure for the two benchmark use cases jointly and does not disaggregate it between the two derivative types, so it is not stated here as a resource bound specific to autocallable options alone.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Gates
RX rotation gate
A parametrized single-qubit rotation about the Bloch-sphere X axis, one of the two generators (with RZ) used to synthesize arbitrary single-qubit unitaries.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
RY rotation gate
A parametrized single-qubit rotation about the Bloch-sphere Y axis whose matrix entries are always real, making it the standard tool for encoding real-valued amplitudes.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
RZ rotation gate
A parametrized single-qubit rotation about the Z axis that adds a relative phase between |0⟩ and |1⟩ without changing measurement probabilities in the computational basis.
- Cost, as recorded
- 1 qubits, 1 gate (often 0-duration on hardware)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
A parametrized two-qubit gate implementing evolution under the Ising ZZ coupling, the entangling primitive most directly native to superconducting and trapped-ion hardware built from always-on or tunable ZZ interactions.
- Cost, as recorded
- 2 qubits, 1 native / 3-gate decomposition (CX, RZ, CX)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
S phase gate
A quarter-turn phase gate that exposes the difference between global and relative phase.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
The Hermitian conjugate (and inverse) of the S gate, applying a -π/2 phase to |1⟩ and undoing whatever an S gate did earlier in a circuit.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Shor period finding
The cryptography-facing quantum algorithm record, with the assumptions that make its security relevance precise.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- FTQC required
- Source
- Literature reference
- Algorithms
Spin-adapted VQE ansatz
Spin-complemented generators reduce leakage from a target total-spin sector.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
SPSA-optimized VQE
Simultaneous perturbation estimates a stochastic gradient with two objective evaluations per iteration.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Subset finding by quantum walk
In L-subset finding, a black box f maps a domain D of size N into a range R, and a property P picks out some L-element subsets by their arguments and values. The task is to find an L-subset {x₁, …, x_L} of D whose pairs ((x₁, f(x₁)), …, (x_L, f(x_L))) satisfy P, or to reject if none exists, using as few queries as possible. Unstructured search is the case L = 1 and element distinctness the case L = 2.
- Cost, as recorded
- O(N^(L/(L+1))) queries for L-subset finding. The L-clique application is given in two variants: a simple one using O(N^(2L/(L+1))) edge queries and a recursive one using Õ(N^((5L−2)/(2L+4))), which the abstract says is an improvement for L ≤ 5.
- Speedup, as the sources state it
- Polynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given integers x₁, x₂, …, xₙ and s, decide whether some subset I of {1, 2, …, n} satisfies the sum of xᵢ over I equal to s. The authors take as the typical hard case that the xᵢ are independent uniform random integers in {0, 1, …, 2ⁿ}. Subset-sum was one of the first problems shown NP-complete, so the question the paper asks is not whether it can be solved in polynomial time but how far below the 2ⁿ cost of searching every subset an exponential-time algorithm can get.
- Cost, as recorded
- 2^{(0.241…+o(1))n} qubit operations, where n is the number of integers x₁, …, xₙ and the paper lists an algorithm using 2^{(e+o(1))n} operations as exponent e. The classical algorithms the paper names for the same problem: brute force at 2ⁿ, the Horowitz-Sahni left-right split at 2^{n/2}, and Howgrave-Graham and Joux's representation algorithm at 2^{(0.337…+o(1))n}. The best classical exponent, 0.291… from Becker, Coron and Joux, appears in the paper's comparison table and in no sentence of its text.
- Speedup, as the sources state it
- Polynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Subspace-search VQE
One shared unitary transforms several orthogonal inputs while a weighted objective orders multiple eigenstates.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Superdense coding circuit
An entanglement-assisted communication protocol that encodes two classical bits into one transmitted qubit.
- Cost, as recorded
- 2 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Primary paper
- Gates
SWAP gate
A routing primitive that exchanges two qubit states and makes hardware connectivity explicit.
- Cost, as recorded
- 2 qubits, 1 native / 3 CX
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Symmetry-preserving VQE ansatz
The ansatz is constrained to preserve selected particle-number, parity, or spin symmetries.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Symmetry-tapered VQE
Known Z2 symmetries remove qubits and constrain the variational search to a selected sector.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Symmetry-verified VQE
Samples outside conserved symmetry sectors are rejected or reweighted as an error-mitigation step.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Gates
T phase gate
A non-Clifford phase gate that marks the boundary between a compact Clifford set and universal synthesis.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
T† gate
The Hermitian conjugate (and inverse) of the non-Clifford T gate, applying a -π/4 phase to |1⟩ and appearing throughout fault-tolerant circuits as the T-count partner of T.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
TETRIS-ADAPT-VQE
Operators with disjoint support are packed into the same adaptive layer to reduce circuit depth.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Given an invertible ideal I in a real quadratic field, determine whether there exists an α with I = αZ[√d] — that is, whether the ideal is principal — and if there is, find α.
- Cost, as recorded
- Polynomial in log Δ when the regulator is larger than some absolute constant, with success probability Ω(1/log Δ) per run; polynomially many repetitions in log Δ raise that to probability exponentially close to one.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
A quantum algorithm that produces approximate solutions for combinatorial optimization problems, tunable by a positive integer p.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum literature curation pass
- Gates
Toffoli (CCX) gate
The three-qubit doubly-controlled NOT gate that implements reversible classical AND/NAND logic and is universal for classical reversible computation.
- Cost, as recorded
- 3 qubits, 1 native / 6 CNOT decomposition
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Transverse-field Ising simulation
A compact Hamiltonian-simulation pattern for spin chains: alternate local field rotations with entangling ZZ evolution.
- Cost, as recorded
- 2 shown · chain generalizes to n qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Trotter–Suzuki Hamiltonian simulation
Approximates the time evolution of a Hamiltonian with non-commuting terms by alternating their individual exponentials.
- Cost, as recorded
- 1 (toy instance) qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
UCCSD VQE ansatz
A chemistry-inspired unitary coupled-cluster ansatz truncated to single and double excitations.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Evaluate risk measures of a financial position — Value at Risk and Conditional Value at Risk among them — where the classical route is a Monte Carlo simulation over sampled realisations of the uncertainty.
- Cost, as recorded
- A convergence rate of O(M⁻²ᐟ³) at the shortest possible circuit depth, which the abstract describes as growing polynomially in the number of qubits representing the uncertainty, against O(M⁻¹ᐟ²) for classical Monte Carlo; allowing the depth to grow faster but still polynomially brings the rate toward the optimum of O(M⁻¹), which the abstract characterises as a near quadratic speed-up compared to Monte Carlo methods for slowly increasing circuit depths. The abstract states the result only in those terms: it gives no gate count, no qubit count, no dependence on the number of risk factors, and no constant factor.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Variance-minimizing VQE
Hamiltonian variance supplements or replaces energy to target eigenstates and diagnose convergence.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Variational imaginary-time evolution
McLachlan-style projected imaginary-time dynamics update parameters toward low-energy states.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Algorithms
Variational quantum deflation
Overlap penalties against previously found states turn excited-state search into a sequence of VQE objectives.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
The Viterbi algorithm finds the most likely sequence of hidden states a hidden Markov model passed through, given a sequence of emissions. Applied to decoding, the hidden states are the encoder states of a classical convolutional code and the emissions are the received symbols, so the most likely path is the decoded message.
- Cost, as recorded
- A single iteration has gate complexity O(N |Q| F (log F)²) and time complexity O(N log F) if |Q| quantum systems can be manipulated simultaneously, where N is the frame length, Q the state set and F the fanout. The number of iterations can be up to O(√L) as in Grover's algorithm, with L = N F log F, giving O(√(F^N) N log F) overall when the maximal amount of parallelism is used.
- Speedup, as the sources state it
- Varies (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
VQE objective and optimization loop
The canonical hybrid loop: prepare an ansatz, estimate a Hamiltonian expectation, and update parameters classically.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
- Basic circuits
W state (three-qubit)
The three-qubit W state, an equal superposition of every single-excitation basis string that survives the loss of one qubit better than GHZ.
- Cost, as recorded
- 3 qubits, 2 controlled rotations + 2 CNOT
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Warm-start VQE
Classical approximations, smaller active spaces, or nearby geometries initialize the variational parameters.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Given oracle access to a graph built by joining two depth-n binary trees with a random weld, so that every node except the two roots has degree three, start from the label of one root and find the label of the other root.
- Cost, as recorded
- poly(n) oracle queries for a pair of depth-n trees, against a proved classical lower bound: no classical algorithm solves the problem with high probability in subexponential time.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Zero-noise-extrapolated VQE
Energy estimates at amplified noise levels are extrapolated toward an inferred zero-noise limit.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE literature expansion
Given a braid on n strands with m crossings and an integer k, compute a certain additive approximation to the Jones polynomial of the link obtained by closing the braid, evaluated at the primitive root of unity e^(2πi/k).
- Cost, as recorded
- Running time polynomial in m, n and k for an n-strand braid with m crossings, at any primitive root of unity e^(2πi/k); the guarantee is an additive approximation.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given a compact, orientable three-manifold presented by a Heegaard splitting, compute a certain additive approximation to its Turaev-Viro invariant, the scalar topological invariant that takes the same value on homeomorphic manifolds.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
Anti-Hermitian excitation generator
Generator used in unitary coupled-cluster circuits. Representative form: τ - τ†.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Given oracle access to the m × n payoff matrix of a zero-sum game with bounded entries, compute a classical representation of an ε-approximate Nash equilibrium of that game.
- Cost, as recorded
- Õ(√(m+n)·ε^(-2.5) + ε^(-3)) time, given a standard quantum oracle for accessing the payoff matrix, to output a classical representation of an ε-approximate Nash equilibrium; the abstract sets this against the best prior quantum runtime Õ(√(m+n)·ε^(-3)) and the classic Õ((m+n)·ε^(-2)), and states the improvement whenever ε = Ω((m+n)^(-1)). The Zoo quotes the same quantum runtime and the same classical Õ((m+n)·ε^(-2)), and states no runtime for the prior result it says this algorithm improves on.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Give polynomial-time quantum algorithms for three average-case lattice problems in parameter regimes where none was known: the short integer solution problem under the infinity norm, the learning-with-errors problem when the input is supplied as LWE-like quantum states rather than classical samples, and the extrapolated dihedral coset problem.
- Cost, as recorded
- Polynomial time in the lattice dimension n, for each of the three problems, in the stated parameter regimes. The paper gives the regimes rather than an exponent: the short integer solution result holds for a number of samples m in Ω((q−c)³ · n^(c+1) · q · log q), and the two quantum-state problems for m in Ω(n · q/η²).
- Speedup, as the sources state it
- Exponential (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
Bose–Hubbard Hamiltonian
Interacting lattice-boson model. Representative form: H = -tΣ⟨ij⟩(b†ᵢbⱼ+h.c.) + U/2Σᵢnᵢ(nᵢ-1).
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Bravyi–Kitaev mapping
Balances locality of parity and occupation updates. Representative form: occupation and parity stored in logarithmic update sets.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
The phase-only sibling of the Toffoli gate that applies a -1 phase exactly when all three qubits are |1⟩, symmetric in all three qubits and the natural three-qubit generalization of CZ.
- Cost, as recorded
- 3 qubits, 1 native / 3-gate decomposition around Toffoli (H, CCX, H)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Classical Ising Hamiltonian operator
The classical (longitudinal-field) Ising Hamiltonian expressed as a diagonal SparsePauliOp: a foundational Z-only spin model with no quantum superposition dynamics of its own.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Basic circuits
Clifford brickwork benchmark · 16 qubits
A deterministic H/S/CNOT brickwork circuit suited to stabilizer and compiler regression checks.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Clifford brickwork benchmark · 3 qubits
A deterministic H/S/CNOT brickwork circuit suited to stabilizer and compiler regression checks.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
Determine the ground state of weakly-interacting, non-covalently bonded molecules — the weakly-bound intermolecular regime that variational quantum algorithms applied to strongly-bound, covalently-bonded systems with full molecular-orbital bases had left largely unexplored — using a coarse-grained representation of the electronic response suited to a VQA.
- Cost, as recorded
- Qubit numbers that grow linearly with the number of molecules. Separately, the abstract states that scaling behaviour for the number of circuits and the number of measurements required is derived and compares favourably to traditional variational quantum eigensolver methods, but it gives no explicit function of either quantity and no exponent or constant for the qubit scaling beyond linearly.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Operators
Commuting observable group
Measurement partition for shared basis estimation. Representative form: G_k={P_j : [P_i,P_j]_qw=0}.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Constraint-penalty operator
Raises energy outside a feasible subspace. Representative form: H_penalty = λ(Ax-b)².
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
A controlled version of the Hadamard gate that puts the target into superposition only when the control qubit is |1⟩, a canonical example of promoting a single-qubit gate to a controlled operation.
- Cost, as recorded
- 2 qubits, 1 native / 2 CNOT decomposition
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
A parametrized controlled gate that applies an RZ(θ) rotation to the target exactly when the control is |1⟩, the entangling generalization of RZ used when a Z-rotation itself needs to be conditioned on another qubit.
- Cost, as recorded
- 2 qubits, 1 native / 4-gate decomposition (RZ, CX, RZ, CX)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Gates
Controlled-Y gate
A controlled version of the Pauli-Y gate that flips the target bit and attaches a ±i phase exactly when the control qubit is |1⟩, completing the CX/CY/CZ family of controlled Paulis.
- Cost, as recorded
- 2 qubits, 1 native / 3-gate decomposition (S†, CX, S)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Coulomb interaction operator
Represents two-electron repulsion in an orbital basis. Representative form: V = 1/2 Σ hₚqrs a†ₚa†_q a_r a_s.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Given N coins of which exactly k are counterfeit, where the real coins all share one weight and the counterfeit coins all share another, and given a pan balance that can compare the weight of any pair of subsets of the coins but reports only whether they balance or tilt, identify all of the counterfeit coins in as few weighings as possible.
- Cost, as recorded
- Q(k,N) = O(k^(1/4)) quantum queries to the balance oracle for any k and N with k < N/2, where Q(k,N) is the quantum query complexity of finding all k false coins among the N given coins; the classical query complexity is Ω(k log(N/k)), a bound that depends on N, so the paper reports a quartic speed-up. No matching lower bound is given: the paper shows only that any algorithm that satisfies certain properties, its own algorithm included, needs Ω(k^(1/4)) queries.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
A two-qubit permutation gate defined as two CNOTs with reversed control and target applied back to back — a genuinely different two-CNOT gate from SWAP (which uses three), and notable for not being self-inverse.
- Cost, as recorded
- 2 qubits, 2 CNOTs (definitional)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Deflation projector
Penalizes an already found eigenstate. Representative form: H' = H + β|ψ⟩⟨ψ|.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Given a real symmetric N × N matrix A whose rows are sparse — the non-zero positions and values of any row returned by an efficiently computable function rather than stored — together with an index j, an integer power m, an a priori bound b on the norm of A, a threshold g and a precision ε, decide whether the diagonal entry (A^m)_jj is at least g + εb^m or at most g − εb^m.
- Cost, as recorded
- Polylogarithmic in the matrix dimension. The abstract places the problem in BQP when m and ε are respectively polylogarithmic and inverse polylogarithmic in N, and the proof states that the estimate can be produced with time and space polynomial in n, m and 1/ε, where N = 2^n. Phase estimation is run with p := 2⌈log((48m)/ε)⌉ control qubits.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Basic circuits
Dicke state |D²₄⟩
The symmetric Dicke state |D²₄⟩: a uniform superposition over every 4-qubit basis string with exactly 2 excitations.
- Cost, as recorded
- 4 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Find quantum circuits that diagonalize a given input Hamiltonian, that is, approximate its eigenstates, without resorting to brute-force optimization of an unstructured variational circuit, which runs into barren plateaus.
- Cost, as recorded
- Exponential in the number of recursion steps, and stated as such by the paper about its own method. The query count to the evolution oracle satisfies N(k+1) = 3N(k) + 1, whose solution with N(1) = 1 gives at most (3 + o(1))^K queries for K group-commutator recursion steps.
- Speedup, as the sources state it
- Unknown (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
The maximally entangling two-qubit gate native to IBM's newer superconducting processors, built from an 'echoed' pair of opposite-sign cross-resonance pulses around a control-qubit X flip, and locally equivalent to CX.
- Cost, as recorded
- 2 qubits, 1 native pulse (2× RZX + 1× X in the echo decomposition)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Given oracle access to a weighted graph on N vertices of maximum degree d whose edge weights are conductances, so that an edge of weight w carries resistance 1/w, estimate the effective resistance between a chosen pair of vertices s and t to within a factor of 1 + ε. Wang poses it as ENA-ER: for a network with |V| = N, deg(G) ≤ d, edge conductances normalized to 1 ≤ w_e ≤ c for every edge — equivalently edge resistances in [1/c, 1] — and spectral gap λ2(L_G) ≥ λ > 0, estimate R_eff(s, t) up to multiplicative error ε, succeeding with probability at least 2/3, given a procedure Pv that on input a vertex index and a number k returns the k-th edge incident to that vertex and a procedure Pe that on input an edge index returns that edge's two endpoints and its weight, both of which Wang assumes can be implemented in time poly(log(N)).
- Cost, as recorded
- Wang's abstract states a running time of poly(d, c, log(N), 1/λ, 1/ε) for computing voltages, currents, dissipated powers and effective resistances, where N is the number of vertices, d the maximum unweighted degree, c the ratio of largest to smallest edge resistance, λ the spectral gap of the normalized Laplacian and ε the accuracy; that this is a time bound and not only a query count rests on his assumption that the graph-access procedures can themselves be implemented in time poly(log(N)). For the resistance problem alone the paper gives two counts, both in uses of the two graph-access procedures Pv and Pe and both gate-efficient in its own sense: the linear-systems route is O((c d^2/(λ ε)) · poly log(c d/(λ ε))), and the quantum-walk route is O(min{c^0.5 d^1.5/(ε λ), c d^0.5/(ε λ^1.5)} · poly log(c d/(ε λ))). Wang also proves a lower bound of Ω(1/√λ) queries for any of his four electrical-network problems, and reads it as saying his algorithms are optimal up to polynomial factors. Ito and Jeffery report bounds of a different kind in a different query model: Õ(n √(R_st(G)) / ε^(3/2)) for estimating R_st(G) to relative accuracy ε, and Õ(n √(R_st(G)/µ) / ε) when µ is a lower bound on λ2(G), both in O(log n) space, and both linear rather than logarithmic in n.
- Speedup, as the sources state it
- Exponential (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
Electronic density operator
Spatial electron-density observable. Representative form: ρ(r) = Σₚq φ*ₚ(r)φ_q(r)a†ₚa_q.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Electronic-structure Hamiltonian
Second-quantized molecular energy operator. Representative form: H = Σ hₚq a†ₚa_q + 1/2 Σ hₚqrs a†ₚa†_q a_r a_s.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Encode a linear kinetic plasma problem — modeling electrostatic linear waves, driven by a spatially localized external current, in a one-dimensional Maxwellian electron plasma — into a quantum circuit that solves the resulting linear system.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Given the specification of a nontrivial multiplicative character χ and an additive character indexed by β over a finite field F_{p^r}, estimate the angle γ modulo 2π in the Gauss sum G(F_{p^r}, χ, β) = √(p^r) · e^(iγ). The same question is then asked for Dirichlet characters over Z/nZ.
- Cost, as recorded
- O((1/ε) · polylog(p^r)) time to estimate the angle to within ε over a finite field F_{p^r}, and O((1/ε) · polylog(n)) over Z/nZ, with the norm |G(Z/nZ, χ, β)| determinable in polylog(n) time. The Jacobi-sum corollary carries the same O((1/ε) · polylog(p^r)) bound.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given a function f and a positive semi-definite matrix A whose eigenvalues are λⱼ, estimate the spectral sum Tr[f(A)] = Σⱼ f(λⱼ), a family whose typical examples the paper gives as the von Neumann entropy, the trace of A⁻¹, the log-determinant and the Schatten p-norm, the last of which it says does not require the matrix to be positive semi-definite.
- Cost, as recorded
- Sub-linear in the matrix size, assuming access to a block-encoding of the matrix, and at most quadratic in other parameters such as the condition number and the approximation error, against what the abstract calls the current best classical randomized algorithms for these quantities, whose runtime it states is at least linear in the number of nonzero entries of the matrix and quadratic in the estimation error; that abstract states no closed-form runtime expression, exponent or constant for the quantum algorithms. The Zoo supplies the surrounding costs for its own route: the unitary e^(-iAt) is approximable by a circuit with only poly(n,t) gates for a 2ⁿ × 2ⁿ Hermitian matrix with poly(n) nonzero entries per row, while computing such a sum classically has worst case computational cost that is exponential in n.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given elements a, b, c, f, g of the multiplicative group of a finite field F_q, decide whether the exponential congruence af^x + bg^y = c has a solution in nonnegative integers x and y, and find one if it does. The problem generalizes the discrete logarithm and is connected to the hidden subgroup problem over semidirect product groups.
- Cost, as recorded
- q^(3/8)(log q)^(O(1)) on a quantum computer in the worst case, against q^(9/8)(log q)^(O(1)) for the best known classical algorithm. In the typical case the figures are q^(1/3)(log q)^(O(1)) quantum and q(log q)^(O(1)) classical. A sharper quantum variant runs in q^(1/2)(st)^(−1/4)(log q)^(O(1)) in terms of the multiplicative orders s and t, which the paper bounds above by O(q^(1/8)(log q)^(O(1))).
- Speedup, as the sources state it
- Polynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Solve the transport equation — for variable grid sizes and discrete particle velocities, in two and three spatial dimensions — on a fault-tolerant universal quantum computer, including the reflection of particles at the walls, edges and corners of obstacles.
- Cost, as recorded
- Quadratic scaling, per the paper's own comparison, in the number of qubits needed to encode the grid and the number needed to encode the discrete velocities, stated for a single spatial dimension; the abstract calls this scaling superior to state-of-the-art approaches in the literature. Three further findings in the same abstract are each attached to a different subroutine and are qualitative rather than numeric: the streaming approach is said to reduce the number of CNOT gates against state-of-the-art quantum streaming methods; the object-encoding method for walls is said to make the CNOT-gate cost of encoding a wall independent of the wall's size; and the velocity encoding is said to give a linear speed-up in the cost of reflecting a particle's velocity and to make that cost independent of the number of velocities encoded. None of the four is accompanied by an explicit formula, a constant, or an error dependence in the abstract.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
The two-site Fermi-Hubbard dimer Hamiltonian, mapped to qubits via the Jordan-Wigner transformation: nearest-neighbor hopping competing with on-site Coulomb repulsion.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Fermi–Hubbard Hamiltonian
Correlated lattice-fermion benchmark Hamiltonian. Representative form: H = -tΣ⟨ij⟩σ(c†ᵢσcⱼσ+h.c.) + UΣᵢnᵢ↑nᵢ↓.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Fermion-parity operator
Z2 symmetry used for sectors and tapering. Representative form: Π = (-1)^N.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Fermionic annihilation operator
Removes a fermion from spin orbital p. Representative form: aₚ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Fermionic creation operator
Adds a fermion in spin orbital p subject to antisymmetry. Representative form: a†ₚ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Fermionic hopping operator
Moves amplitude between orbitals or lattice sites. Representative form: Tₚq = a†ₚa_q + a†_q aₚ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Fermionic number operator
The Jordan-Wigner-encoded fermionic number operator n = c†c = (I-Z)/2, whose eigenvalues count mode occupation exactly.
- Cost, as recorded
- 1 per mode qubits, Diagonal (no gates needed to define)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Fermionic pairing operator
Creates and annihilates correlated fermion pairs. Representative form: Δₚq = a†ₚa†_q + a_q aₚ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Given oracle access to a spherically symmetric function f from R^d to an arbitrary set S, locate its center of symmetry to a fixed precision using as few queries as possible.
- Cost, as recorded
- A constant number of quantum queries independent of the dimension, written O(1) oracle queries, against a classical lower bound of Ω(d) queries, the Zoo naming that dimension d where the paper names it n. The Zoo states the constant-query result flatly, but the paper puts the count forward as a conjecture rather than a theorem: it conjectures that the algorithms succeed with constant probability using one quantum-sample and O(1) oracle queries respectively, independent of the dimension n. What the paper reports as proved are rigorous bounds on the distribution of probability mass for the continuous curvelet transform, which show that the algorithms work in an idealized continuous model. The precision to which the center is located is fixed for simplicity in the Zoo statement, and neither source states a gate count.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Gates
Fredkin (CSWAP) gate
A three-qubit controlled-swap gate that conditionally exchanges two target qubits, useful for reversible routing logic and as a component of the SWAP test for state overlap.
- Cost, as recorded
- 3 qubits, 1 native / 5 two-qubit gates (Smolin–DiVincenzo)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Graph collision on a known graph
We are given an undirected graph G on n vertices, known explicitly in advance, together with oracle access to a labeling of the vertices by 1 and 0; the graph collision problem is to decide, by querying that labeling, whether there exist a pair of vertices, connected by an edge, both of which are labeled 1. Only the labeling is queried and never the graph, which is why Magniez, Santha and Szegedy can restate the task as deciding whether the set of vertices of value 1 forms an independent set in G. It is not either of the two problems it is easily confused with: collision finding asks for two arguments of a two-to-one function that share a value, and element distinctness drops that promise and asks whether any two of N items are equal. Those two share a single other Zoo entry, Collision Finding and Element Distinctness, and a single record here, element-distinctness; neither of them has a graph or an edge relation in it.
- Cost, as recorded
- Õ(n^(2/3)) quantum queries on an arbitrary n-vertex graph, which is Theorem 3 of Magniez, Santha and Szegedy and which the Zoo reports without the tilde as O(N^(2/3)) and calls the best upper bound on quantum query complexity known for this problem on general graphs; against that, the Zoo gives quantum query complexity Ω(√n) and classical query complexity Θ(n), both from embedding Grover's unstructured search in a star graph. The sharper bounds are parameterized rather than restricted to a class of graphs: each holds on every graph and degrades to the general bound as its parameter grows. They are O(√n α^(1/6)) in the independence number α (Belovs, Theorem 7); O(√n + √(α*(G))) where α*(G) is the maximum total degree of the vertices in an independent set of G (Gavinsky and Ito, Theorem 1), with O(√(n log n)) queries on most graphs when every edge is present independently with a fixed probability; and O(√n t^(1/6)) in the treewidth t together with O(√n + √(α**(G))) where α**(G) is the smallest, over the vertex covers of G, of the largest total degree of an independent subset of that cover (Ambainis, Balodis, Iraids, Ozols and Smotrovs, Theorems 1 and 4). Only two of the bounds are genuinely restricted to a class: Õ(√n + √m) for a balanced bipartite graph on 2n vertices, where m counts the pairs of the two sides that are not edges, with Õ(√(nλ) + √m) for finding all λ collisions rather than deciding that one exists (Jeffery, Kothari and Magniez, Theorem 3.1), and O(√n) for a subclass of circulant graphs (Ambainis, Balodis, Iraids, Ozols and Smotrovs, Theorem 5).
- Speedup, as the sources state it
- Polynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Basic circuits
Graph state on a four-node ring
A graph state built on a 4-cycle rather than an open chain, illustrating how stabilizer generators follow directly from graph adjacency.
- Cost, as recorded
- 4 qubits, 1 layer H + 4 CZ
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Given a finite group G presented only through an oracle that takes an ordered pair of element labels and returns the label of their product, compute the order of G from the labels of a set of generators, and decide whether a given bitstring is the label of a group element — in the constructive form of the membership question, exhibiting that element as a product of the generators.
- Cost, as recorded
- polylog(|G|) queries to compute the order and to decide membership when G is solvable, and likewise when G is Abelian, against a classical query cost that the Zoo says cannot be brought down to polylog(|G|) even for Abelian G. Watrous states the solvable case as a polynomial-time quantum algorithm for computing orders of solvable groups, and the abstract names no query count and no parameter that its polynomial time is measured in.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
The minimal-basis (STO-3G) H₂ electronic Hamiltonian after Jordan-Wigner/parity mapping and two-qubit tapering: the canonical small-molecule target for variational quantum eigensolver (VQE) demonstrations.
- Cost, as recorded
- 2 (tapered from 4) qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Basic circuits
Hadamard-basis states |+⟩ and |−⟩
The Hadamard-basis (X-basis) states |+⟩ and |−⟩: the eigenstates of the Pauli-X operator, produced from |0⟩/|1⟩ by a single Hadamard gate.
- Cost, as recorded
- 1 qubits, 1 gate (H, or X+H)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Hamiltonian variance operator
Eigenstate diagnostic and alternative objective. Representative form: H² - ⟨H⟩².
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Heisenberg Hamiltonian
Isotropic interacting-spin model. Representative form: H = ΣJ(XᵢXⱼ + YᵢYⱼ + ZᵢZⱼ).
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Heisenberg XXZ spin-chain operator
The anisotropic Heisenberg (XXZ) spin-chain Hamiltonian: exchange-coupled spins with tunable easy-axis/easy-plane anisotropy Δ, U(1)-symmetric under total-Sz rotation.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
HHL linear-systems reference
A linear-algebra reference that forces the catalog to show input loading, conditioning, and output observability.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
Given oracle access to a hidden subset over a finite field that is not a lattice, that is, a hidden nonlinear structure, identify that subset.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Hybrid HHL++ for portfolio optimization
Adapt the Harrow-Hassidim-Lloyd (HHL) quantum linear-systems algorithm, most of whose components current noisy quantum hardware cannot reach, into a form that near-term devices can actually execute, and demonstrate it on an application.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Ideals in a finite black-box ring
Given black-box access to a finite ring R, not necessarily commutative, together with a list of generators for an ideal I in R, find an additive basis representation for I.
- Cost, as recorded
- poly(log |R|) time to find an additive basis representation for an ideal I in a finite ring R, the same order the Zoo states for finding additive generators of (R,+) itself, against known classical algorithms that the Zoo says scale as poly(|R|); neither source quotes an exponent, a constant factor or a gate count, and neither quotes a separate cost for any of the ideal problems derived from the basis.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Gates
Identity (I) gate
The single-qubit no-op gate that leaves every state exactly unchanged, used as a placeholder in circuit diagrams, a timing/idle slot on real hardware, and the base case for gate-composition identities.
- Cost, as recorded
- 1 qubits, 1 gate (0-duration no-op semantically)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Ising cost Hamiltonian
Diagonal optimization and spin-model objective. Representative form: H_C = Σᵢ hᵢZᵢ + ΣᵢⱼJᵢⱼZᵢZⱼ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Given a hard combinatorial problem, rewrite it as an Ising spin model whose lowest-energy spin configurations are exactly that problem's solutions, so that a machine which minimizes energy can be pointed at the problem at all.
- Cost, as recorded
- At most cubic in the size of the problem, counted in spins — an encoding size, not a running time. No speedup is claimed: the abstract states no time to solution, no query or gate count and no comparison with classical solvers, and its only forward-looking statement is that the work may be useful in designing adiabatic quantum optimization algorithms.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Gates
iSWAP gate
A two-qubit gate that swaps |01⟩ and |10⟩ while attaching a factor of i, arising naturally from XY-type physical couplings and native to several superconducting and photonic platforms.
- Cost, as recorded
- 2 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Joint parity operator measurement
The joint parity operator Z^{⊗n} and its standard non-destructive ancilla-based measurement circuit, the core primitive behind stabilizer syndrome extraction.
- Cost, as recorded
- 4 CNOT + 1 measurement
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Jordan–Wigner hopping mapping
Qubit representation of fermion transport. Representative form: a†ₚa_q+h.c. ↦ Pauli strings with a Z parity chain.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Maps fermionic antisymmetry into a parity string. Representative form: a†ₚ ↦ (Xₚ-iYₚ)/2 ⊗ Z₀···Zₚ₋₁.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Jordan–Wigner number mapping
Local qubit representation of orbital occupation. Representative form: nₚ ↦ (I-Zₚ)/2.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Kitaev-chain Hamiltonian
Topological superconducting-chain model. Representative form: H = -μΣnᵢ - tΣ(c†ᵢcᵢ₊₁+h.c.) + ΔΣ(cᵢcᵢ₊₁+h.c.).
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Basic circuits
Linear cluster-state benchmark · 16 qubits
A linear graph-state preparation circuit using one Hadamard per qubit and nearest-neighbor CZ edges.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Linear cluster-state benchmark · 3 qubits
A linear graph-state preparation circuit using one Hadamard per qubit and nearest-neighbor CZ edges.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Algorithms
Linear combination of unitaries
A block-encoding primitive that turns a weighted sum of unitary operations into one larger unitary circuit.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Basic circuits
Linear SWAP-network benchmark · 16 qubits
A forward-and-reverse nearest-neighbor SWAP network for testing routing and qubit-order preservation.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Linear SWAP-network benchmark · 3 qubits
A forward-and-reverse nearest-neighbor SWAP network for testing routing and qubit-order preservation.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Operators
Magic T state
The single-qubit magic state |T⟩ = T|+⟩, the standard resource state consumed by gate teleportation to implement a fault-tolerant non-Clifford T gate.
- Cost, as recorded
- 1 qubits, 2 gates (H, T)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Matrix products over semirings
Compute the product of two n × n matrices over a semiring rather than a ring — the (max, min) product, the distance product, and the Boolean product — where the absence of subtraction rules out the fast algebraic algorithms that make ring matrix multiplication cost n^ω. The question the paper poses is whether anything beats the Õ(n^(5/2)) that follows from applying Grover search or quantum minimum-finding entry by entry.
- Cost, as recorded
- O(n^2.473) for the (max, min) product of two n × n matrices, against O(n^2.687) for the best known classical algorithm. For the distance product, the ℓ most significant bits of each entry are computed in O(2^(0.64ℓ) n^2.46). The sparse Boolean product is given as a four-case piecewise bound in the number of non-zero entries, and the paper says its algorithm performs better when n^1.151 < m < n^(ω−1/2).
- Speedup, as the sources state it
- Polynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Matrix rank by a span program
Given oracle access to the integer entries of an n×m matrix A, determine the rank of A.
- Cost, as recorded
- O(√(r(n-r+1)) L T) queries to the entries of an n×m integer matrix A under a promise that the rank of A is at least r, where L is the root-mean-square of the reciprocals of the r largest singular values of A and T is a factor set by the sparsity of A: T = O(√(nm)) for general A, and T = O(k log(n+m)) when A has at most k nonzero entries in any row or column. The Zoo gives the classical cost as order nm queries. Every figure here is the Zoo's; the paper's own abstract states no query count.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
MaxCut cost operator
QAOA objective for graph cuts. Representative form: C = 1/2 Σ(i,j)∈E (I - ZᵢZⱼ).
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Maximally mixed state I/2ⁿ
The maximally mixed state ρ = I/2ⁿ: the unique n-qubit state invariant under every unitary, carrying zero information and maximal (n-bit) von Neumann entropy.
- Cost, as recorded
- n (n=1,2 shown) qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
Molecular dipole operator
Observable for molecular polarity and response. Representative form: μ = -Σᵢrᵢ + Σ_AR_AZ_A.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Determine the value at the root of a read-once Boolean formula, canonically a NAND tree on N variables, given only oracle access to the variables and using as few queries as possible.
- Cost, as recorded
- O(N^(1/2 + ε)) queries in the conventional quantum query model for any fixed ε > 0, converted from a continuous-time algorithm running in time O(√(N log N)) in the Hamiltonian query model; the Zoo states the walk's cost separately, in exponential form, as NAND trees on 2^n variables evaluable in time O(2^(0.5n)), against Ω(2^(0.753n)) queries required classically.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
NOON state (two-mode, N=2)
The NOON state (|N,0⟩+|0,N⟩)/√2: a two-mode entangled state with all N particles in one mode or the other, the standard resource for Heisenberg-limited interferometric phase estimation.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Basic circuits
Occupation-seeded VQE ansatz · 16 qubits
A Hartree–Fock-like computational-basis seed followed by tunable rotations and a CNOT chain.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Occupation-seeded VQE ansatz · 3 qubits
A Hartree–Fock-like computational-basis seed followed by tunable rotations and a CNOT chain.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Operators
One-body fermionic operator
Kinetic, external-potential, or orbital-rotation term. Representative form: O₁ = Σ hₚq a†ₚa_q.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Basic circuits
One-dimensional cluster state
The canonical resource state for one-way (measurement-based) quantum computing: qubits in |+⟩ entangled by CZ along a line.
- Cost, as recorded
- 4 qubits, 1 layer H + 3 CZ
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
One-particle reduced density matrix
Compact one-body state descriptor and orbital gradient input. Representative form: γₚq = ⟨a†ₚa_q⟩.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Given a set of constraints over a finite field, such as a max-XORSAT instance or the task of finding a degree-n polynomial over 𝔽ₚ that approximates a given data set as well as possible, produce an assignment that satisfies as many constraints as possible.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
Orbital number operator
Measures occupation of one spin orbital. Representative form: nₚ = a†ₚaₚ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Algorithms
Ordered search
Given oracle access to a list of N numbers held in order from least to greatest, together with a number x, determine where in the list x would fit.
- Cost, as recorded
- 4 log₆₀₅ N ≈ 0.433 log₂ N oracle queries for the exact quantum algorithm, against log₂ N queries for classical binary search; a lower bound of (ln 2)/π · log₂ N quantum queries is known for the problem.
- Speedup, as the sources state it
- Constant factor (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
Parity fermion-to-qubit mapping
Alternative encoding that can expose removable symmetries. Representative form: occupation ↦ cumulative parity bits.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Pauli time-evolution operator
Primitive for Trotter simulation and problem-inspired ansätze. Representative form: U_P(t) = exp(-itP).
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Pauli-string observable
Basic measured term in a qubit Hamiltonian. Representative form: P = P₀ ⊗ ··· ⊗ Pₙ₋₁.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Pauli-X operator
A bit-flip operator that makes the relationship between a circuit and its truth table explicit.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Basic circuits
Phase feature-map benchmark · 16 qubits
A fixed-data feature-map scaffold with Hadamards, local phase rotations, and pairwise ZZ encodings.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Phase feature-map benchmark · 3 qubits
A fixed-data feature-map scaffold with Hadamards, local phase rotations, and pairwise ZZ encodings.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
Given an oracle that returns the value p(x) of an unknown degree-d polynomial p over the finite field GF(q) at any queried point x, determine the coefficients of p using as few queries as possible.
- Cost, as recorded
- Univariate over GF(q): d/2 + 1/2 quantum queries suffice to determine a degree-d polynomial with bounded error, which matches the lower bound, and d/2 + 1 queries suffice for success probability approaching 1 at large q, against d + 1 queries necessary and sufficient classically; the algorithm can be implemented with gate complexity poly(log q) with negligible decrease in the success probability. Multivariate, for degree d in n variables: ⌈(1/(n+1))·(n+d choose d)⌉ queries suffice for probability 1 over ℂ and twice that over ℝ, except for d = 2 and four other special cases, and ⌈(d/(n+d))·(n+d choose d)⌉ queries suffice for probability approaching 1 over GF(q) at large field order, against a classical query complexity of (n+d choose d).
- Speedup, as the sources state it
- Varies (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Simulate strongly correlated chemical systems on near-term quantum hardware, whose noise and limited size otherwise confine such simulations to small chemical systems, by embedding a quantum treatment of a strongly correlated fragment within a larger classical calculation.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Given only an oracle that, for a vertex of an N-vertex graph of degree d together with an index j in {1, 2, ..., d}, returns that vertex's j-th neighbor or null when the vertex has degree less than d, decide in as few queries as possible whether the graph is bipartite or far from bipartite — far meaning that a constant fraction of the edges would have to be removed to achieve bipartiteness — and, in the same model, whether the graph is an expander or far from being one.
- Cost, as recorded
- Õ(N^(1/3)) quantum complexity, in the Zoo's words, for deciding bipartiteness on an N-vertex bounded-degree graph promised to be either bipartite or far from bipartite, and Õ(N^(1/3)) together with an Ω̃(N^(1/4)) quantum lower bound, the two not meeting, for distinguishing expanders from graphs far from being expanders, against a classical Θ̃(√N); the primary paper states the same upper bound as time O(N^(1/3)) for both problems, its quantum lower bound as Ω(N^(1/4)) and the classical side as an Ω(√N) lower bound holding for both problems, all without tildes. The Zoo adds, from papers outside this record, Θ(√(NM)) quantum query complexity for finding a minimal spanning tree, Θ(N) for deciding connectivity in the undirected case and Θ̃(√(NM)) in the directed case, Θ̃(√(NM)) for computing the lowest weight path from a given source to all other vertices on a weighted graph, and Õ(N√d) running time for st-connectivity, bipartiteness and forest testing using only logarithmically many qubits, where N is the number of vertices, M the number of edges and d the degree.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Predict the three-dimensional structure a protein takes from its primary sequence of amino acids, posed here on the model Hamiltonian the paper defines for a chain of N monomers placed on a tetrahedral lattice.
- Cost, as recorded
- O(N⁴) scaling for the model Hamiltonian of a polymer chain with N monomers on a tetrahedral lattice. The abstract states no running time for the variational algorithm, no query or gate count, and no speed-up factor against classical folding methods; its only concrete figures are qubit counts for two named instances, 22 qubits for the 10 amino acid Angiotensin peptide and 9 qubits for a 7 amino acid neuropeptide on an IBM Q 20-qubit quantum computer. On the classical side it states that the problem is intrinsically NP-hard even reduced to its simplest Hydrophobic-Polar model, while classical algorithms provide practical solutions by sampling the conformation space of small proteins.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Quadratically signed weight enumerators
A quadratically signed weight enumerator is the sum S(A, B, x, y) = Σ (−1)^(bᵀBb) x^|b| y^(n−|b|) over the 0/1 vectors b with Ab = 0. Given that the diagonal of A is the identity, that k and l are positive integers, and a promise that |S(A, lwtr(A), k, l)| is at least (k² + l²)^(n/2)/2, determine the sign of S.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Quantum adiabatic evolution
An optimization pattern that slowly deforms an easy ground state into the ground state of a problem Hamiltonian.
- Cost, as recorded
- 2 in the toy interpolation qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Given quantum access to a historical record of asset returns, determine the optimal risk-return tradeoff curve of a portfolio and provide a way to sample from the optimal portfolio.
- Cost, as recorded
- In principle poly(log N), where N is the size of the historical return dataset — rendered from the abstract's inline TeX poly(log(N)) — against poly(N) for the direct classical algorithms the abstract compares it to. The abstract states this only as an order in N: it gives no constant, no dependence on any error or precision parameter, and no qubit or gate count, and it qualifies the comparison itself by pointing to recent work on efficient classical sampling approaches rather than treating the gap as settled.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Simulate the wave equation under Dirichlet and Neumann boundary conditions on a quantum computer, using Hamiltonian simulation and quantum linear system algorithms as subroutines.
- Cost, as recorded
- Improved scaling in truncation error and improved scaling for the cost of state preparation, relative to general-purpose linear differential-equation algorithms, for the wave-equation algorithm this abstract presents. The abstract states this comparison only in those qualitative terms — no complexity expression, exponent, or constant is given for the wave-equation algorithm or for the general-purpose algorithms it is compared against, and no bound at all is given for the separately mentioned, merely considered use of the same approach for Klein-Gordon or Maxwell's equations.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Given a first-order linear differential equation d/dt x = A(t)x + b(t) with N-dimensional vectors x and b and an N×N matrix A, and given an initial condition x(0), produce the solution x(t) at a later time t to precision ε, in the sense that the normalized vector x(t)/‖x(t)‖ returned is at distance at most ε from the exact solution.
- Cost, as recorded
- Time O(t² poly(1/ε) poly log N) for the Berry algorithm, against O(t poly N) for the fastest classical algorithms; for systems with constant coefficients the Berry, Childs, Ostrander and Wang variant is polynomial in the logarithm of the inverse error, an exponential improvement in the precision dependence.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given a system of nonlinear ordinary differential equations, in the primary algorithm's case a dissipative quadratic n-dimensional system, produce the solution at a chosen evolution time T to error ε, encoded in the amplitudes of a quantum state.
- Cost, as recorded
- For dissipative quadratic n-dimensional ODEs with R < 1, complexity T² q poly(log T, log n, log 1/ε)/ε, where T is the evolution time, ε the allowed error and q measures decay of the solution; a worst-case lower bound in the same paper shows the general quadratic problem is intractable for R ≥ √2. The earlier Leyton and Osborne algorithm has expected resource requirements polylogarithmic in the number of variables but exponential in the integration time, against a best classical algorithm scaling linearly in the number of variables.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Simulate strongly correlated fermionic systems — notoriously hard for classical computers — on a quantum computer with 2D or linear (1D) nearest-neighbor qubit-qubit couplings, of the kind typical of superconducting transmon qubit arrays, including preparing the relevant quantum states and evolving the system in time, with the Fermi-Hubbard model as a worked example.
- Cost, as recorded
- Three separate bounds, each attached to the subroutine the abstract attaches it to, not merged into one figure for the algorithm as a whole: preparing an arbitrary fermionic Gaussian state costs O(N²) gates and O(N) circuit depth; the 2D fermionic Fourier transform on a 2D qubit array costs O(N^1.5) gates and O(√N) circuit depth, described as the minimum depth needed for quantum information to cross the array; and simulating each time step of the 2D Fermi-Hubbard model, again on a 2D qubit array, costs O(N) gates and O(√N) circuit depth. None of the three is a total cost for a full simulation, and none is stated by the abstract specifically for the one-dimensional (linear) geometry this record's reference implementation demonstrates: the abstract names linear geometry as within its scope but attaches no gate or depth bound to it.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Solve the heat equation in a rectangular region of spatial dimension d, in the sense of approximately computing the amount of heat in a given region.
- Cost, as recorded
- An at most quadratic quantum speedup for spatial dimension d ≥ 2, from an approach that applies amplitude estimation to an accelerated classical random walk; the alternative approach, based on a quantum algorithm for linear equations, is never faster than the best classical algorithms. The abstract states the comparison only in those terms: it quotes no complexity expression, no dependence on the error or the discretization, and no qubit or gate count, and the complexities of the ten algorithms it compares are given inside the paper.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Given a target position for a robot manipulator's end effector, find joint angles that reach it — the inverse kinematics problem, which has no analytical solution for a general 6-degree-of-freedom arm and admits many joint configurations at once for a redundant one.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Quantum counting
Estimates how many items in an unstructured search space satisfy an oracle, without checking them one by one.
- Cost, as recorded
- 5 (3 counting + 2 system, N=4) qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Given a subgraph of the Boolean hypercube on bit strings of length n, whose edges all run from smaller to larger Hamming weight, decide whether it contains a path from the all-zeros vertex 0^n to the all-ones vertex 1^n.
- Cost, as recorded
- O*(1.817^n) for path in the hypercube on bit strings of length n, where O* omits polynomial factors, against O*(2^n) for the fastest known classical algorithm; the abstract reports the same approach for vertex ordering problems in O*(1.817^n), which the Zoo sets against O*(2^n) classically, and for graph bandwidth in O*(2.946^n), set against O*(4.383^n) classically, and reports similar ideas for the travelling salesman problem and minimum set cover in O*(1.728^n), with minimum set cover stated by the Zoo as O(poly(m,n) 1.728^n) against O(nm2^n) classically. The two sources differ on one figure: where the abstract writes 1.728^n for the travelling salesman problem and minimum set cover together, the Zoo writes 1.729^n for traveling salesman and feedback arc set against O*(2^n) classically and gives minimum set cover a figure of its own.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Detect the edges of a digital image: the pixel positions at which the image values change sharply. Processing digital images keeps growing in volume, with matching demands on data storage, transmission and processing power.
- Cost, as recorded
- One single-qubit operation for the edge-detection step, independent of the size of the image; for the image processing algorithms the paper presents, an exponential speed-up over their classical counterparts. The qubit claim is comparative rather than absolute — the representation reduces the required number of qubits compared to existing implementations — and the abstract states no gate count for preparing the encoded image or for reading the result back out.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Quantum imaginary-time evolution
A ground-state preparation method that projects non-unitary imaginary-time evolution onto a parameterized quantum circuit.
- Cost, as recorded
- 2 in the ansatz sketch qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Detect anomalous, potentially fraudulent, transactions in a credit-card dataset, framed as an anomaly-detection task and compared against classical kernel-based benchmarks such as one-class support vector machines.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Quantum kernel support-vector classifier
A quantum feature-map record paired with a classical SVM so model quality and data costs stay visible.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
Given access only to an oracle that, for a pair of integers in {1, 2, ..., n}, says whether the corresponding vertices of an n-vertex graph are joined by an edge, decide a property of that graph or find a structure in it, such as connectivity, a minimum spanning tree, a lowest weight path, or a triangle, using as few queries as possible.
- Cost, as recorded
- Θ(n^(3/2)) quantum queries in the adjacency matrix model for Minimum Spanning Tree and for Connectivity, against Θ(√(nm)) and Θ(n) respectively in the array model; the Zoo adds O(n^(3/2) log² n) for finding lowest weight paths, Õ(n^(3/2)) queries and quantum gates for bipartiteness, cycle detection and st-connectivity, Θ(n^(2/3)) for sparse graph properties that cannot be characterized by a list of forbidden subgraphs, and O(n^(5/4)) queries for finding a triangle. On the classical side the Zoo puts every problem it lists ahead of triangle finding at Ω(n²) queries only under the widely-believed Aanderaa-Karp-Rosenberg conjecture, whereas for triangle finding it states Ω(n²) classical queries outright. Durr and Hoyer's minimum-finding algorithm runs in time O(c√N) on a table of size N and returns the index of the minimum with probability at least 1 - 1/2^c.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Quantum signal processing
A single-qubit rotation sequence that transforms an encoded signal into a polynomial response.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Quantum simulated annealing
Sample from the final limiting distribution πₙ of a slowly varying sequence of Markov chains, where πₙ is chosen to concentrate on good solutions of a combinatorial optimization problem.
- Cost, as recorded
- Order 1/√δ steps to find an optimal solution with bounded error probability, where δ is the minimum spectral gap of the stochastic matrices used in the classical annealing process; this is a quadratic improvement over the order 1/δ steps required by the classical procedure.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Simulate the quantum kicked rotator model — used to study quantum chaos, localization and the Anderson transition — with a quantum algorithm that scales better than classical simulation of the same model.
- Cost, as recorded
- Exponentially faster than classical algorithms, for simulating the quantum kicked rotator model — a statement about the cost of simulating that one dynamical system, not a bound on solving a decision, search, or optimization problem. The abstract gives no big-O expression, no base for the exponential, no qubit or gate count, and no error dependence; a second, separate claim — that a similar algorithm simulates classical chaos in certain area-preserving maps efficiently — is likewise unquantified.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Algorithms
Quantum singular value transformation
A framework for applying bounded polynomial transformations to singular values of a block-encoded matrix.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Measure, as a single number, how large a random circuit of equal width and depth a given quantum computer successfully implements, so that progress toward improved system-wide gate error rates can be measured and compared across near-term devices.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
Given a network's vulnerabilities and how their connectivity creates kill-chains — paths to security compromise — decide which vulnerabilities to prioritize for patching, so that those paths are removed.
- Cost, as recorded
- No proven asymptotic bound. The paper reports, as an experimental finding for vulnerability graphs of the density expected in the real world, that the quantum computer's solve time is almost constant while the classical solve time increases exponentially; the abstract gives no formula, no growth rate, and no qubit, gate, or annealing-time count for either quantity.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Operators
Qubit-ADAPT Pauli pool
Qubit-space anti-Hermitian generator candidates. Representative form: {iP_k}.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
QUBO operator mapping
Maps binary quadratic objectives into diagonal Pauli form. Representative form: xᵀQx, xᵢ ↦ (I-Zᵢ)/2.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Reference-state projector
Overlap, fidelity, and penalty observable. Representative form: Π_ref = |φ⟩⟨φ|.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Basic circuits
Ring graph-state benchmark · 16 qubits
A cyclic graph-state preparation circuit with Hadamards followed by CZ interactions around a ring.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Ring graph-state benchmark · 3 qubits
A cyclic graph-state preparation circuit with Hadamards followed by CZ interactions around a ring.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
Estimate an average error rate for a set of operations (gates) on a quantum information processor, under a noise model general enough to allow errors that depend on both the time and the gate at which they occur.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
A parametrized two-qubit gate implementing evolution under the Ising XX coupling, the RZZ gate's basis-rotated sibling used in Heisenberg-model simulation and trapped-ion Mølmer–Sørensen-style entangling operations.
- Cost, as recorded
- 2 qubits, 1 native / 5-gate decomposition (H, H, CX, RZ, CX, H, H)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Basic circuits
RY-CX hardware-efficient ansatz · 16 qubits
A reproducible hardware-efficient VQE layer with parameterized RY rotations and a linear CNOT entangler.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
RY-CX hardware-efficient ansatz · 3 qubits
A reproducible hardware-efficient VQE layer with parameterized RY rotations and a linear CNOT entangler.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
RZ-RY-CZ hardware-efficient ansatz · 16 qubits
A two-axis variational layer followed by a nearest-neighbor CZ entangling pattern.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
RZ-RY-CZ hardware-efficient ansatz · 3 qubits
A two-axis variational layer followed by a nearest-neighbor CZ entangling pattern.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
Sample from the output distribution of a rudimentary optical device: identical photons are generated, sent through a network of beamsplitters and phase shifters, and then non-adaptively measured to count the photons in each mode. The question is whether a classical computer can do the same sampling efficiently.
- Cost, as recorded
- Stated as complexity-theoretic consequences rather than as a running time. Exact sampling is not efficiently solvable classically unless P^#P = BPP^NP and the polynomial hierarchy collapses to the third level. For approximate sampling, a classical sampler running in time polynomial in the input size and 1/ε would put the Gaussian permanent estimation problem in BPP^NP.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Search with wildcards
Identify a hidden n-bit string x using an oracle that, for a chosen subset S of the n positions and a string y of length |S|, returns one when the substring of x specified by S equals y and zero otherwise.
- Cost, as recorded
- O(√n log n) quantum queries to recover the hidden n-bit string, against the classical lower bound of Ω(n) queries. The Zoo states the problem's quantum query complexity as Θ(√n) and its classical query complexity as Θ(n); the paper's abstract claims only the upper bound and calls its algorithm nearly optimal, so the Zoo's Θ(√n) is the stronger of the two statements and both are recorded rather than reconciled. For combinatorial group testing, the second problem in the same paper, O(k log k) queries against the classical lower bound of Ω(k log(n/k)) queries, where n counts the items in the set and k bounds the special items among them rather than the bits of the hidden string above.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given a list of m + 1 Hermitian n × n matrices C, A_1, A_2, …, A_m and m numbers b_1, …, b_m, find the positive semidefinite n × n matrix X that maximizes tr(CX) subject to the constraints tr(A_j X) ≤ b_j for j = 1, 2, …, m, approximately, to within ± ε of the optimum. The Zoo states that semidefinite programming has many applications in operations research, combinatorial optimization, and quantum information, and that it includes linear programming as a special case. All three papers reduce that optimization to a feasibility question by binary search on a guess for the optimal value, but not to the same feasibility question, and the difference carries a parameter that the runtimes below then depend on. Brandão and Svore add the constraint tr(X) ≤ R, with R a measure of the size of the optimal solution, and test the dual: they sample from the distribution y divided by the 1-norm of y for a dual feasible vector y. Van Apeldoorn, Gilyén, Gribling and de Wolf likewise test membership in a dual polytope induced by the current primal candidate, with R bounding the trace of an optimal primal X. Brandão, Kalev, Li, Lin, Svore and Wu instead normalise the primal to tr(X) = 1 and test it directly, deciding whether the constraints tr(A_i X) ≤ a_i + ε admit a positive semidefinite X of unit trace; Chia, Gilyén, Li, Lin, Tang and Wang adopt that same trace-one form for the dequantized version.
- Cost, as recorded
- Brandão and Svore's abstract gives a worst-case running time n^(1/2) m^(1/2) s^2 poly(log(n), log(m), R, r, 1/δ), with n the dimension and s the row-sparsity of the input matrices, m the number of constraints, δ the accuracy of the solution, and R and r upper bounds on the size of the optimal primal and dual solutions — in that paper r is a solution-size bound, not a rank — together with an Ω(n^(1/2) + m^(1/2)) quantum lower bound for constant s, R, r and δ. Their own Corollary 5 instantiates that poly as Õ(n^(1/2) m^(1/2) s^2 R^32/δ^18), and van Apeldoorn, Gilyén, Gribling and de Wolf record that this figure is for multiplicative error 1 ± δ in the special case b_j ≥ 1 and OPT ≥ 1, and that the reduction from a general SDP to that special case significantly worsens the dependence on R, r and δ; the paper's own reduction sends δ to δ/r and R to 2R + 1. Van Apeldoorn et al. keep the same dependence on m, n and s and improve the rest, reporting Õ(√(mn) s^2 (Rr/ε)^8), and Õ(√(mn) (Rr/ε)^5) for the special case of linear programs; in their paper R bounds the trace of an optimal primal X and r the sum of entries of an optimal dual y, and they argue that R, r and 1/ε trade against one another and should be read as the single parameter Rr/ε. Brandão, Kalev, Li, Lin, Svore and Wu report Õ(s^2 (√m ε^(-10) + √n ε^(-12))) in the entry-oracle model, but for their own normalised feasibility problem, which carries tr(X) = 1: they state that converting a general primal with width bound tr(X) ≤ R into that form changes ε to ε/R, and to ε divided by R times their dual bound if a strictly feasible solution is wanted, so the ε in those exponents is not the ε of a general SDP and the bound is not a version in which the size parameters have vanished. The same paper performs exactly that substitution when it restates Brandão and Svore's cost as Õ(√(mn) s^2 (R R-tilde/ε)^32), R-tilde being its name for the bound on the optimal dual solution. Its second bound, Õ(√m + poly(r)) · poly(log m, log n, B, ε^(-1)) with r the rank of the constraint matrices and B an upper bound on their trace norm, belongs to a fully quantum input model in which the matrices are supplied as quantum states. The Zoo's composite figure O(√m log m · poly(log n, r, ε^(-1))), with r read as the rank, is that second bound: it is the paper's Remark D.7, which takes the decomposition of each A_j to be its eigen-decomposition so that under the low-rank assumption B collapses to r, and s is absent because that model does not use sparsity at all. It is therefore not a bound in the entry-oracle model, where the same authors' Ω(√n) lower bound applies.
- Speedup, as the sources state it
- Polynomial (with some exceptions) (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
Shifted Hamiltonian square
Folded-spectrum objective around target energy ω. Representative form: (H-ωI)².
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
A foundational code record that compares quantum protection with the narrower classical repetition-code idea.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
- Operators
Single-qubit thermal (Gibbs) state
The thermal (Gibbs) state of a single qubit in a longitudinal field, prepared exactly via a one-ancilla purification circuit.
- Cost, as recorded
- 1 system + 1 ancilla qubits, 2 gates (RY, CNOT)
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Given a text T of length n and a pattern P of length m < n, both over a finite alphabet, find an occurrence of P as a substring of T or report that P is not a substring of T. The Zoo also states the problem for d-dimensional arrays rather than strings, where the task is to return the location of P as an m × m × ... × m block within the n × n × ... × n array T or report that no such location exists.
- Cost, as recorded
- Õ(√n + √m) quantum time to determine whether a pattern of length m occurs in a text of length n, with inverse polynomial failure probability, where Õ allows for logarithmic factors in m and n/m; the Zoo gives Ω(√n + √m) as the worst-case quantum query complexity of the problem, implied by the Ω(√N) lower bound for unstructured search. For the average case in d dimensions, Montanaro's algorithm solves the pattern matching problem for random patterns and texts in time Õ((n/m)^(d/2) 2^(O(d^(3/2) √(log m)))), against Ω̃((n/m)^d + n^(d/2)) for the best possible classical algorithm, a separation Montanaro's abstract calls super-polynomial for large m.
- Speedup, as the sources state it
- Superpolynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
String rewriting derivation counts
Fix a relation that permits replacing one substring by another. Given strings s, t and t′ of length L, let Δ(n) be the number of ways of reaching t from s in exactly n replacements minus the number of ways of reaching t′. Given a promise that |Δ(m)| ≥ εc^m for the input m, and a growth promise that Δ(n) ≤ c^n for every n, determine the sign of Δ(m).
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
A fault-tolerance record for comparing physical error, syndrome extraction, decoder choice, and logical failure.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
- Gates
SX (√X) gate
A Clifford square root of the Pauli-X gate that is the native physical single-qubit gate (an X_{π/2} pulse) on most superconducting hardware.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
Represent the spectrum of a quantum chemistry Hamiltonian, given in an arbitrary (for example molecular) orbital basis, as a block-encoded quantum circuit cheap enough to support phase estimation of a molecular eigenvalue.
- Cost, as recorded
- Õ(N) Toffoli complexity for the block-encoding circuits, in a basis of N orbitals, with O(λ/ε) repetitions of those circuits needed for phase estimation to sample in the molecular eigenbasis, where λ is the 1-norm of the Hamiltonian coefficients and ε is the target precision. The abstract states this is the lowest complexity shown for quantum computations of chemistry within an arbitrary basis, and that up to logarithmic factors it matches the scaling of the most efficient prior block encodings restricted to an orthogonal basis.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
In the spiked tensor model, an unknown signal vector v_sig in R^N of magnitude √N is hidden inside a p-th order tensor T0 = λ v_sig^⊗p + G, where G is noise whose entries are chosen independently from a Gaussian distribution of zero mean and unit variance and λ is a scalar representing a signal-to-noise ratio; the task called recovery is to infer v_sig to some accuracy given T0, and the simpler task called detection is to distinguish the case λ = 0 from λ = λ̄ for some λ̄ > 0, again just given T0. The paper treats the symmetrized and non-symmetrized cases as reducible to each other, and says that for odd p it is convenient in the analysis not to symmetrize T0 and to take complex G. Recovery is information-theoretically possible for λ much larger than N^((1-p)/2), but no polynomial-time algorithm is known that achieves that performance; the two best known algorithms are spectral and sum-of-squares, and for even p the spectral method works for λ much larger than N^(-p/4), with a variant conjectured to perform similarly for odd p. The regime this record is about is the hard one at and below that spectral threshold: write λ = α N^(-p/4), and the question is what recovery costs as α shrinks.
- Cost, as recorded
- Both sides of the comparison are indexed by nbos, and the object whose size sets the cost is D(N, nbos), the dimension of the symmetric subspace of nbos qudits of dimension N, which the paper approximates by N^nbos / nbos! for N much greater than nbos; Emax is its high-probability bound on the largest eigenvalue of the noise Hamiltonian H(G) and E0 the signal energy scale, with the algorithms analysed for E0 at least (1 + c) Emax. Classically, one iteration of the power method on H(T0) takes time Õ(D(N, nbos)) in space Õ(D(N, nbos)), and O(log(D(N, nbos)) / log(E0/Emax)) iterations suffice for detection and recovery, so the paper's total is Õ(D(N, nbos))O(1 / log(E0/Emax)) with space exponential in nbos. Theorem 6 bounds the quantum algorithm's expected runtime by poly(N, nbos, 1/(E0 − Emax), log(D(N, nbos)/ε)) exp(O(nbos)) log(N)^(4 nbos) (N^(-p/4)/λ)^(nbos/p) D(N, nbos)^(1/4); the intermediate versions of the same algorithm cost D(N, nbos) and D(N, nbos)^(1/2) times that same poly factor, with plain phase estimation and with amplitude amplification respectively. The polynomial-space claim is not part of theorem 6: it is section 5.2's statement that, in contrast to the classical algorithms above, all these quantum algorithms take only polynomial space. The quartic is a statement about exponents in a stated limit — the log of the quantum runtime divided by the log of the classical runtime approaches 1/4 as N → ∞ at fixed N^(-p/4)/λ, that is, at fixed α. The regime is Assumption 1 together with the paper's standing assumption on λ: nbos = O(N^θ) for a p-dependent constant θ > 0 chosen sufficiently small, λ = Ω(N^(-θ′)) for a p-dependent constant θ′ > p/4, and λ = O(N^(-p/4)) — above that last bound the paper says simple spectral methods already succeed. Below the threshold the price is paid in nbos, which the paper states increases polynomially in λ^(-1) as polylog(N)(N^(-p/4)/λ)^(4/(p-2)) while the runtime increases exponentially. The Zoo's own form for the classical cost, exponential in α^(-1), is the one the paper gives for the sum-of-squares sequence and for the Kikuchi-hierarchy spectral algorithms of Wein, El Alaoui and Moore, not for Hastings's own nbos ladder, whose exponent 4/(p-2) equals 1 only at p = 6.
- Speedup, as the sources state it
- Polynomial (quartic) (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given a list of k generators for a group G together with black-box access to group multiplication, decide whether G is commutative using as few queries to that black box as possible.
- Cost, as recorded
- O(k^(2/3)) for a black-box group given by k generators, with a lower bound of Ω(k^(2/3)) obtained by reduction from a special case of Element Distinctness; the Zoo records the resulting quantum query complexity as Θ̃(k^(2/3)). The classical baseline is Pak's randomized algorithm involving O(k) group operations, which the same paper states it proves optimal for the randomized model.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
Given oracle access to two unknown probability distributions P and Q on an N-element set — in the Zoo's setup, black boxes A and B whose domain is the integers 1 through T and whose range is the integers 1 through N, with the distribution over outputs induced by choosing uniformly at random among allowed inputs — approximate the L1 distance between the two distributions to constant precision.
- Cost, as recorded
- About N^(1/2) queries to estimate the L1 distance between P and Q with constant precision, where classical computers need Ω(N) queries; O(N^(1/3)) queries for testing Uniformity and Orthogonality, whose classical query complexity the abstract says is known to be Ω(N^(1/2)). The Zoo states the first figure as O(√N) and describes the classical requirement as scaling essentially linearly with N.
- Speedup, as the sources state it
- Polynomial (from a secondary index; not yet checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
The Abelian hidden subgroup problem
Let G be a finitely generated Abelian group and let H be a subgroup of G such that G/H is finite, and let f be a function on G with the promise that f(g1) = f(g2) if and only if g1 and g2 lie in the same coset of H. The task is to find H, that is, a set of generators for H, by making queries to f. Mosca and Ekert work order finding and period finding as G = Z with hidden subgroup rZ, Simon's problem as G = Z_2^l, and the discrete logarithm as G = Z_r × Z_r.
- Cost, as recorded
- O(log |G|) quantum queries to f against Ω(|G|) classically, both the Zoo's own, and black-box counts, not gate counts. Boneh and Lipton state their results as random quantum polynomial time in n = log q, not a query count, under conditions on the multiplicity m: Theorem 1 needs m and the variable count k at most n^O(1) and m below the smallest prime divisor of q, Theorem 2 the same on m alone. Beaudrap, Cleve and Watrous give a 2n-bit query problem over GF(2^n) solved exactly by one quantum query plus polynomially many auxiliary operations, where standard technique spends Θ(n) queries and bounded-error classical Ω(2^(n/2)).
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
The class group of a number field is the finite abelian group of ideals modulo principal ideals. Computing it means computing the structure of that abelian group, not merely its order.
- Cost, as recorded
- Quantum polynomial time for a number field of constant degree, assuming the generalized Riemann hypothesis. Both conditions are stated in the theorem itself and neither is dropped elsewhere in the paper.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
The unit group of a number field is the set of invertible algebraic integers inside it. By Dirichlet's unit theorem it is, up to a root of unity, free abelian of a rank determined by the field's real and complex embeddings, so computing it means finding a fundamental system of units.
- Cost, as recorded
- Quantum polynomial time, for a number field of constant degree. The paper states the bound as a theorem in exactly those terms and gives no exponent, and the degree appears as a restriction rather than as a parameter of the bound.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Algorithms
Thermal Gibbs state sampling
Prepare the thermal Gibbs state of an interacting quantum system on a quantum computer, and use that preparation to evaluate the system's partition function to a target accuracy.
- Cost, as recorded
- The preparation sets a universal upper bound D^α on the thermalization time, where D is the Hilbert space dimension and α < 1/2 is proportional to the Helmholtz free energy density; the derived partition-function algorithm runs in time proportional to that thermalization time and inversely proportional to the targeted accuracy squared.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake
- Operators
Total particle-number operator
Defines fixed-number sectors and symmetry checks. Representative form: N = Σₚ a†ₚaₚ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Total spin-X operator
Collective transverse spin observable. Representative form: Sₓ = 1/2 Σᵢ Xᵢ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Total spin-Y operator
Collective quadrature spin observable. Representative form: Sᵧ = 1/2 Σᵢ Yᵢ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Total spin-Z operator
Collective magnetization observable. Representative form: S_z = 1/2 Σᵢ Zᵢ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Total-spin-squared operator
Labels total-spin sectors and spin contamination. Representative form: S² = Sₓ² + Sᵧ² + S_z².
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Transverse-field Ising Hamiltonian
Non-commuting spin-chain ground-state benchmark. Representative form: H = -JΣZᵢZᵢ₊₁ - hΣXᵢ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
The transverse-field Ising model (TFIM): the standard minimal Hamiltonian exhibiting a quantum (zero-temperature) phase transition driven by competing Z-Z order and X-field disorder.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Basic circuits
Transverse-field Ising VQE layer · 16 qubits
A problem-inspired layer combining ZZ interactions and transverse X rotations for a ring Ising model.
- Cost, as recorded
- 16 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Basic circuits
Transverse-field Ising VQE layer · 3 qubits
A problem-inspired layer combining ZZ interactions and transverse X rotations for a ring Ising model.
- Cost, as recorded
- 3 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum portable benchmark expansion · MQT Bench methodology reference
- Operators
Two-body fermionic operator
Electron-electron interaction term. Representative form: O₂ = 1/2 Σ hₚqrs a†ₚa†_q a_r a_s.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Two-particle reduced density matrix
Two-body correlation descriptor used in energy and response. Representative form: Γₚqrs = ⟨a†ₚa†_q a_s a_r⟩.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
The three-parameter gate that can reach any single-qubit unitary up to global phase, used as the universal target when a compiler synthesizes one-qubit operations from Euler angles.
- Cost, as recorded
- 1 qubits, 1 gate
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Operators
UCC doubles operator pool
Correlated double-excitation generator set. Representative form: {a†_a a†_b a_j a_i - h.c.}.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
UCC singles operator pool
Adaptive or fixed single-excitation generator set. Representative form: {a†_a a_i - a†_i a_a}.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
A hybrid chemistry workflow that compares a quantum expectation loop with classical eigensolvers.
- Cost, as recorded
- Problem mapped qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Literature reference
- Operators
VQE gradient commutator
Ranks adaptive ansatz generators. Representative form: ∂E/∂θ|₀ = ⟨ψ|[H,A]|ψ⟩.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Weighted Pauli-sum Hamiltonian
Canonical qubit-space objective used by VQE. Representative form: H = Σⱼ cⱼPⱼ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Werner state (two-qubit)
The Werner state ρ_W(p) = p|Ψ⁻⟩⟨Ψ⁻| + (1−p)I/4: a one-parameter family interpolating between the maximally entangled singlet and the maximally mixed state, historically significant as the first known example separating entanglement from Bell-inequality violation.
- Cost, as recorded
- 2 qubits
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Curated reference
- Algorithms
Workflow scheduling by QUBO modeling
Schedule a workflow of tasks — an instance of the workflow scheduling problem, a known NP-hard class of scheduling problems — for an industrial use case, in a way that can be represented and solved by quantum, classical, and hybrid quantum-classical algorithms.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum Classiq-parity intake
- Operators
XY spin Hamiltonian
Excitation-preserving spin-exchange model. Representative form: H = ΣJₓXᵢXⱼ + JᵧYᵢYⱼ.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
XYZ spin Hamiltonian
Anisotropic extension of the Heisenberg model. Representative form: H = Σ(JₓXX + JᵧYY + J_zZZ).
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
- Operators
Z2 symmetry generator
Defines conserved sectors and qubit tapering constraints. Representative form: S = ⊗ᵢ Pᵢ, S²=I, [S,H]=0.
- Cost, as recorded
- Not stated in the source.
- Speedup, as the sources state it
- Not stated in the source.
- Source
- Leona Quantum VQE operator expansion
Given a curve C of genus g over a finite field F_q, compute the numerator P(t) of its zeta function Z(C,t) = P(t)/((1−t)(1−qt)), a polynomial of degree 2g with integer coefficients, in time polynomial in g and log q jointly rather than in one at the expense of the other.
- Cost, as recorded
- Polynomial in g and log q jointly. The paper states this as a theorem for computing the numerator P(t), and states the subroutine bounds separately: Watrous's black-box group order routine runs in time polynomial in mn, and the class number #Cl(C_e) is computable in time polynomial in g, log q and e. No exponent or constant factor is given.
- Speedup, as the sources state it
- Superpolynomial (checked against the record's own primary paper)
- Source
- Leona Quantum Zoo-parity intake