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
Recover a noiseless expectation value by post-processing
Estimate what an observable would have measured on a noiseless device by running modified or repeated circuits on the noisy one and combining the results classically. No qubits are spent on redundancy; the whole price is paid in shots.
- Takes
- A circuit, a target observable, a noisy device, a shot budget, and — for the model-based methods — a learned characterization of the device noise.
- Returns
- A bias-reduced expectation-value estimate, with a variance — equivalently a sampling overhead — that grows with circuit volume.
Mitigation and correction are different contracts, not different degrees of the same one. Correction returns logical qubits a later subroutine can consume coherently; mitigation returns a statistic, and nothing downstream can take its output as a quantum input, so an algorithm whose result is a sampled bitstring cannot use it at all. It is also the one layer here with a proven ceiling: the sampling overhead of any protocol in a broad class — including protocols with nonlinear post-processing and protocols not yet invented — grows exponentially with circuit depth for layered circuits under local depolarizing and, more broadly, Markovian noise, and for random circuits under local noise with qubit count as well. Those are worst-case, class-wide bounds, so particular structured instances can still be mitigated cheaply — which is why mitigation remains a useful small-scale instrument and not a scaling path.
- Zero-noise extrapolation (ZNE)
Deliberately amplify the device noise by a set of known factors, measure the observable at each, and extrapolate the resulting curve back to zero noise. The fit is Richardson's deferred approach to the limit, or another model.
- Probabilistic error cancellation (PEC)
Write the inverse of the characterized noise channel as a quasi-probability distribution over implementable operations, sample circuits from it, and combine the results with signed weights. Unlike extrapolation it inverts the noise rather than fitting through it, so it is unbiased in principle.
- Readout (measurement) error mitigation
Correct the classical readout channel by deconvolving the assignment matrix that maps true bitstrings to observed ones. The naive form calibrates and inverts the full matrix; the scalable form never forms it.
- Check the symmetries the answer must have
The physics fixes quantities the true state cannot change — particle number, spin. Measure them alongside the run and discard, or post-process away, the outcomes that violate them: an error that moves the state out of the sector announces itself, and one that keeps it inside does not.
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
Every circle is drawn once. This step has no smaller object recorded inside it, so the strands between its two circles are the recorded ways of taking it — one strand per method.
4 recorded ways of doing Recover a noiseless expectation value by post-processing. Nothing smaller is recorded inside it, so there is no object in the middle to draw.
Everything on this figure that opens is open.
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
- Every line on this figure is one a recorded source takes.
Open the cardRead the full write-up
Where you are
Path
- Recover a noiseless expectation value by post-processing
Ways through: 4
Routes that skip it
No recorded route avoids this step.
Narrower kinds
Nothing recorded is a narrower kind of this.
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