Strong empiricalAlgorithmsAlgorithmAmplitude amplification2 toy q
Grover unstructured search
The canonical search primitive, paired with the linear classical baseline it is meant to improve.
Every quantum algorithm worth knowing about, written down the same way: what it takes, what it returns, what it costs, and who proved it.
The Map draws our corpus as one connected structure: Open the Map
Every source behind both surfaces: See the papers
A speedup class on a record is quoted: See whose claim it is
Every record is classified by how it was verified. The badge shows the strongest tier of evidence; the chips list each method that applies.
The defining behavior was checked exactly: a mathematical identity, a full statevector or stabilizer simulation, or an exhaustive basis-state truth table.
The design was verified by construction plus measured evidence: statistical re-execution, small-instance analytic agreement, sub-block, echo, or invariant checks. Scale-specific bugs can still survive.
The record rests on external authority: peer-reviewed papers, standard textbooks, expert review, or evidence carried over from related verified entries. Nothing here was re-executed by this catalog.
Only automated (LLM-assisted) review or an unreviewed community submission backs this record so far. Treat it as a starting point, not evidence.
153 public entries
Atlas stars stay in this public list. Saving an entry to your workspace starts an unstarred private copy.
The canonical search primitive, paired with the linear classical baseline it is meant to improve.
A finance- and Monte-Carlo-facing primitive that makes the oracle and error model visible.
Price discretely monitored Asian options over T monitoring points, where the underlying asset is modeled by a geometric Brownian motion.
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.
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.
Price rainbow options — a type of path-independent multi-asset derivative — on a quantum computer.
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.
A block-encoding primitive that turns a weighted sum of unitary operations into one larger unitary circuit.
A framework for applying bounded polynomial transformations to singular values of a block-encoded matrix.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
The eigenphase-estimation primitive behind chemistry, simulation, and period-finding workflows.
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.
A communication primitive that makes entanglement, measurement, and classical feed-forward explicit.
An entanglement-assisted communication protocol that encodes two classical bits into one transmitted qubit.
The unitary that maps computational basis states to Fourier-basis phase patterns, and the phase-estimation primitive behind Shor's and HHL's speedups.
The general framework behind Grover's algorithm: boosting the success probability of any subroutine with a known reflection structure.
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.
A ground-state preparation method that projects non-unitary imaginary-time evolution onto a parameterized quantum circuit.
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.
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.
A compact Hamiltonian-simulation pattern for spin chains: alternate local field rotations with entangling ZZ evolution.
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.
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.
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.
The cryptography-facing quantum algorithm record, with the assumptions that make its security relevance precise.
Estimates the eigenphase of a unitary using a single reused ancilla qubit instead of a full phase-estimation register.
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.
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.
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.
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.
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.
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.
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.
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.
An optimization pattern that slowly deforms an easy ground state into the ground state of a problem Hamiltonian.
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.
Approximates the time evolution of a Hamiltonian with non-commuting terms by alternating their individual exponentials.
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.
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.
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).
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.
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.
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.
A quantum algorithm that produces approximate solutions for combinatorial optimization problems, tunable by a positive integer p.
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.
A p=1 QAOA circuit with a classical MaxCut comparison and reproducible simulation evidence.
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.
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.
Estimates how many items in an unstructured search space satisfy an oracle, without checking them one by one.
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.
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.
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.
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.
Simulate the wave equation under Dirichlet and Neumann boundary conditions on a quantum computer, using Hamiltonian simulation and quantum linear system algorithms as subroutines.
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.
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.
A resource-focused QFT construction with a readable compile comparison.
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.
A linear-algebra reference that forces the catalog to show input loading, conditioning, and output observability.
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.
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.
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.
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.
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.
A quantum feature-map record paired with a classical SVM so model quality and data costs stay visible.
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.
A query circuit that recovers the hidden bit string in f(x) = s · x.
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.
A one-query promise-problem circuit that separates constant from balanced Boolean oracles.
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.
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)).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Given oracle access to the integer entries of an n×m matrix A, determine the rank of A.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
A coin-and-shift circuit that spreads a walker's position amplitude ballistically, in contrast to the diffusive spread of a classical random 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.
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.
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.
A single-qubit rotation sequence that transforms an encoded signal into a polynomial response.
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.
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).
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.
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.
A hybrid chemistry workflow that compares a quantum expectation loop with classical eigensolvers.
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.
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.
Frozen-core and active-orbital choices define the Hamiltonian size before variational optimization.
An adaptive ansatz grows one operator at a time using measured energy gradients from a predefined pool.
Several high-gradient operators are appended per adaptive iteration to reduce optimization and measurement rounds.
Randomized measurements are reused to estimate many observables from a shared data set.
Conditional value-at-risk averages only a selected low-energy tail of samples for combinatorial objectives.
Minimizing the squared shifted Hamiltonian targets eigenstates near a chosen energy shift.
Generalized singles and doubles relax occupied-to-virtual restrictions to enlarge the variational manifold.
Parameter-shift or analytic derivative measurements supply gradients to a classical optimizer.
Alternating native one-qubit rotations and entanglers reduce compilation overhead but can change trainability.
McLachlan-style projected imaginary-time dynamics update parameters toward low-energy states.
Iterative QCC repeatedly dresses the Hamiltonian and selects new entanglers instead of fixing one deep circuit.
Repeated paired generalized doubles with generalized singles trade expressivity against shallower chemistry circuits.
Circuit depth grows in stages so each newly introduced layer can be initialized and optimized locally.
A contracted reference subspace is jointly entangled before a small effective Hamiltonian is diagonalized.
Commuting Pauli terms are partitioned into compatible bases to reduce distinct measurement circuits.
The Fubini–Study metric preconditions parameter updates according to circuit-state geometry.
The canonical hybrid loop: prepare an ansatz, estimate a Hamiltonian expectation, and update parameters classically.
Orbital rotations are optimized alongside circuit parameters to improve compact active-space descriptions.
Givens-style or excitation-preserving blocks keep evolution inside a fixed-particle-number sector.
Orthogonality or symmetry penalties augment the energy objective to exclude previously identified sectors.
Qubit coupled-cluster uses Pauli-word entanglers and a product-state reference directly in qubit space.
Commutator matrix elements over a VQE reference produce excitation energies through an equation-of-motion problem.
Measured response operators around a VQE state define a generalized eigenproblem for excitations and mitigation.
Qubit-space Pauli generators replace fermionic excitation operators to seek shorter adaptive circuits.
Known Z2 symmetries remove qubits and constrain the variational search to a selected sector.
Calibrated assignment errors are inverted or regularized before Pauli expectations are assembled.
Spin-complemented generators reduce leakage from a target total-spin sector.
Simultaneous perturbation estimates a stochastic gradient with two objective evaluations per iteration.
One shared unitary transforms several orthogonal inputs while a weighted objective orders multiple eigenstates.
The ansatz is constrained to preserve selected particle-number, parity, or spin symmetries.
Samples outside conserved symmetry sectors are rejected or reweighted as an error-mitigation step.
Operators with disjoint support are packed into the same adaptive layer to reduce circuit depth.
A chemistry-inspired unitary coupled-cluster ansatz truncated to single and double excitations.
Hamiltonian variance supplements or replaces energy to target eigenstates and diagnose convergence.
Overlap penalties against previously found states turn excited-state search into a sequence of VQE objectives.
Classical approximations, smaller active spaces, or nearby geometries initialize the variational parameters.
Energy estimates at amplified noise levels are extrapolated toward an inferred zero-noise limit.