About this map
Sections
What this is
Quantum algorithms are not written from scratch. They are assembled from a small number of reusable steps, and almost every published method is a different route through the same handful of them.
This is a map of those routes. Circles are the things an algorithm can be holding. Lines are the steps that carry you from one to the next. A method is a path across.
Nothing here is generated. Every line was read out of a paper and checked against it.
How to read it
- Something you can hold — a state, a matrix, a circuit, an answer.
- The same, in the middle of a step you have opened.
- A step. Someone has published a way through it.
- A step whose way through has not been pinned to one method.
- A step nothing published fills yet.
- A step you have opened. What is drawn inside it is how it was done.
- There is a record in the repository for this one.
How to move around
- Two fingers move the map. Pinch to zoom, or hold ctrl and scroll.
- Click a step to open it in place — everything else stays where it is.
- Click a name to read the full record without leaving the map.
- Arrow keys move, plus and minus zoom, zero puts it back.
What a line is claiming
A solid line means a paper puts those two steps together and we have the citation. A long-dashed line means the route is recorded but no single method has been named for that step. A short-dashed line means nothing published fills it — the step is real, the way through is not written yet.
A count after a step's name — ×T/h, ×O(κ) — means the route walks that step that many times rather than once. It is the source's own symbol, and the card says what it stands for and what one turn costs. A step with no count is a step no source we read said is repeated, which is not the same as one taken once.
A line drawn nested under another, on the soft shaded band behind it, is a narrower version of the line above it: the same construction, re-analysed or re-tuned, filling the same step. It is why two lines can draw the identical interior and still be two entries. Lines outside the band are alternatives to their neighbours, not versions of them.
The map does not hide the gaps. An empty step is drawn as an empty step.
What is not here yet
The map covers the algorithm literature. The repository covers circuits and primitives. They overlap less than you would expect, and where a method has no record we say so on its page rather than leaving the space blank.
Where something named here does have a record, its name links straight to it.
Process
Simulate Hamiltonian evolution
Implement to error given some access model for . It is an application in its own right and also the engine inside phase estimation and several linear-system solvers.
- Takes
- An access model for — a sum of efficiently exponentiable terms, sparse-access oracles, or a block-encoding — plus an evolution time and a target error .
- Returns
- A circuit approximating to within , with a stated query or gate count, an ancilla count, and the norm parameter — sparsity times , or the LCU 1-norm — that the cost is measured against.
The families here consume genuinely different inputs and pay in different currencies. Product formulas need only a term decomposition and build no block-encoding, but carry a polynomial dependence on . LCU and qubitization need a block-encoding and its ancillas and reach a logarithmic dependence on . Which is cheaper depends on the precision regime and on the structure of , so a cost model that names "Hamiltonian simulation" without naming the family has not costed anything. The norm parameter is inherited from the block-encoding layer below, which is where the constant that dominates real resource estimates is actually fixed.
- Product-formula (Trotter-Suzuki) simulation
Split into terms that can each be exponentiated directly and alternate short evolutions of them — the Lie-Trotter formula and its higher-order generalizations. No block-encoding is built and there is no all-zeros flag to amplify.
- Truncated-Taylor LCU simulation
Truncate the Taylor series of over short segments and implement the truncated sum as a linear combination of unitaries — PREPARE loads the coefficients, SELECT applies the terms, PREPARE unprepares — with the all-zeros ancilla flag amplified.
- Qubitization walk simulation
From a block-encoding pair with , build a walk operator that splits the Hilbert space into invariant two-dimensional subspaces, one per eigenvalue of , with eigenvalues . Quantum signal processing phases applied to then produce .
- Chebyshev LCU inversion
Expand an approximation of in Chebyshev polynomials and implement that expansion directly through the quantum walk associated with a sparse , rather than through Hamiltonian simulation. Phase estimation is never used, which is what removes the bottleneck.
No field holds this yet — the model is still being designed.
- Trotter–Suzuki Hamiltonian simulation
Approximates the time evolution of a Hamiltonian with non-commuting terms by alternating their individual exponentials.
- Transverse-field Ising simulation
A compact Hamiltonian-simulation pattern for spin chains: alternate local field rotations with entangling ZZ evolution.
- Quantum algorithms to simulate many-body physics of correlated fermions
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.
- Quantum simulation of the kicked rotator model
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.
Covered in the repository.
Where the routes meet
11 problems nothing else needs — the places a reader arrives. Open a line to see what is recorded inside it, or click its name to go there.
13 lines have something recorded inside that you have not opened.
Of the routes that have been taken apart, 15 are built entirely from named slots, 15 hand off part of the work and finish the rest themselves, and 20 are one undivided act. None of the three is a defect; they are different things to reuse.
Every line on this figure, in words
The lines on this figure
Solve a nonlinear ODE dy/dt = F(y)
- Embed a nonlinear system into a linear one — opens into 6 · a way across — click it to open it here
- Solve a linear ODE du/dt = A(t)u + b(t) — opens into 9 · a way across — click it to open it here
- Choose a time discretization or propagator approximation → Quantum linear solve — open
- Choose a time discretization or propagator approximation — opens into 6 · a way across — click it to open it here
- Quantum linear solve — opens into 5 · a way across — click it to open it here
- Simulate Hamiltonian evolution → Estimate an observable — open
- Simulate Hamiltonian evolution — opens into 3 · a way across — click it to open it here
- Estimate an observable — opens into 4 · a way across — click it to open it here
Compile a circuit to a specific device
- NISQ transpilation (retargetable pass pipeline) — open · opened: what was inside is drawn in its place
- Satisfy the hardware connectivity constraint — opens into 3 · a way across — click it to open it here
- Fault-tolerant compilation (Clifford+T pipeline) — opens into 2 · a way across — click it to open it here
Estimate an excited-state energy
- Variational quantum deflation — opens into 3 · a way across — click it to open it here
- Subspace-search variational eigensolver — opens into 3 · a way across — click it to open it here
- Quantum subspace expansion
- Quantum equation of motion
- Folded-spectrum variational eigensolver — opens into 3 · a way across — click it to open it here
- Penalty-constrained variational eigensolver — opens into 3 · a way across — click it to open it here
- Multistate contracted variational eigensolver — opens into 3 · a way across — click it to open it here
Every step you can open
1 of these have an object recorded in the middle; the rest open into the methods that fill them.
- Solve a nonlinear ODE dy/dt = F(y)
- Replace a spatial domain with a finite grid
- Discretize a PDE into one linear system
- Embed a nonlinear system into a linear one
- Solve a linear ODE du/dt = A(t)u + b(t)
- Recast a non-Hermitian generator as Hamiltonian evolution
- Choose a time discretization or propagator approximation
- Quantum linear solve
- Matrix function
- QSP phase factors
- Polynomial approximation
- Block-encode a matrix
- Prepare an input state
- Amplify a success branch
- Simulate Hamiltonian evolution
- Estimate an observable
- Compile a circuit to a specific device
- Satisfy the hardware connectivity constraint
- Approximate a continuous rotation in a discrete gate set
- Recover a noiseless expectation value by post-processing
- Build logical qubits at a target logical error rate
- Estimate a Hamiltonian's ground-state energy
- Choose a parameterised trial state
- Minimise the objective over the parameters
- Estimate an excited-state energy
- Measure what the machine can actually do
- Recover the period of a periodic function
- Estimate the eigenphase of a unitary
- Find the item a check accepts
- Walk a graph to the vertex you want
- Search a cost Hamiltonian for the assignment it minimises
What is on this map, counted
What is here, counted
147 nodes — 31 slots and 116 methods.
76 of the 147 link to a record in the Atlas, between them naming 89 records. The rest name papers and nothing else: this graph describes work the catalogue has not got yet, and the nodes with no record are the list of what a corpus pass has to go and read.
0 slots have no method recorded, and 32 methods have not been taken apart. Both are shown as what they are rather than left blank.
Every claim here rests on a source. This graph cites 140 papers; they and the 172 the Atlas cites alone are registered in one place, with what each reports and everywhere it is cited from. Papers