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
Minimise the objective over the parameters
Search the parameters of a fixed circuit family for the ones that minimise a chosen objective, given that every evaluation of that objective is a noisy estimate someone paid shots for. The output is a routine that prepares one particular state — the family collapsed to a member.
Open the full recordExpand it here — a map of just this
- Takes
- A parameterised circuit family; an objective function of its parameters, evaluated only through estimates bought with a finite shot budget; a starting point; and a stopping rule — a tolerance, an iteration cap, or an exhausted budget.
- Returns
- A preparation routine for the state at the parameters the search stopped at, and the total number of objective evaluations it consumed. The routine is returned whether or not the search found a minimum; that it stopped is not evidence that it converged.
Once the family is fixed, what is left is a non-convex minimisation in which every function evaluation is a noisy estimate bought with shots — so the classical optimiser is not an implementation detail, it is the thing that decides the total run count, and the run count is what the whole method is charged. Methods here differ in the currency they spend: a gradient-free search buys robustness to shot noise with many cheap evaluations, an analytic-gradient method buys direction with extra circuits per step, and a metric-aware method buys better steps with the cost of estimating the metric. The objective is a choice too, and a separate one: the plain energy expectation is not the only function whose minimiser is the answer, and changing it changes the landscape the optimiser is walking rather than the way it walks. Nothing about that is decided by the ansatz above or the readout below, which is what makes it a slot instead of a step in someone's write-up.
- Conditional-value-at-risk objective
Change what the classical loop is minimising rather than how it minimises. Instead of averaging every measurement outcome into an expectation value, keep only the best tail of them and average that — which is defensible precisely when the answer is a single good bitstring rather than a physical average.
- Analytic-gradient parameter search
Get the gradient of the objective exactly, rather than by finite differences, by running the same circuit again at shifted parameter values. The direction is then not an estimate of a slope taken from two noisy numbers; it is the slope, estimated to whatever precision the shots allow.
- Minimise the energy variance
Minimise how much the energy fluctuates rather than the energy itself. Any eigenstate has zero variance, so the objective's own value tells you whether you have arrived — which the energy never does, since a low number is only low relative to a minimum nobody knows.
- Follow the steepest descent in the state's own geometry
Take the step that moves the state fastest, not the one that moves the parameters fastest. The two differ because equal changes in parameters do not make equal changes in the state, and the metric measuring that difference has to be estimated before every step.
- Grow the circuit a layer at a time while training it
Do not settle the circuit before optimising it. Start shallow, train what is there, then hold most of it fixed and add the next layer on top — so every step of the search runs on a shallow circuit with few free parameters, which is where a gradient is still large enough to follow.
- Simultaneous-perturbation optimization
Perturb every parameter at once, in one random direction, and take the difference of two objective evaluations as the gradient estimate. The estimate is bad in any single round and unbiased across rounds, so the cost of a step stops growing with the number of parameters.
None found yet.
No field holds this yet — the model is still being designed.
None found yet.
No record covers this yet. Whether that is a gap or deliberate is not something anything on this record can say.
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.
15 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
Estimate an excited-state energy
- Variational quantum deflation — open · opened: what was inside is drawn in its place
- Choose a parameterised trial state — opens into 13 · a way across — click it to open it here
- Minimise the objective over the parameters — opens into 6 · a way across — click it to open it here
- Estimate an observable — opens into 4 · 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