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
Measure what the machine can actually do
Run a protocol whose answer is already known, on the hardware, and read the machine's own performance off how far the result falls short. Nothing here computes anything a user wanted — the point is that the answer is known in advance, because that is what makes the shortfall a measurement.
Open the full recordExpand it here — a map of just this
- Takes
- A programmable device — its qubits, its native gate set, its connectivity and its measurement — plus how many circuits and how many shots you are willing to spend, and the confidence level the answer has to be established at.
- Returns
- A number characterising the hardware, the protocol that produced it, and the statistical confidence it holds at — never an answer to a computational problem, because no computational problem was posed.
What a reader is choosing between is **what the number is about**: one method isolates a gate set from the rest of the machine, the other refuses to. Randomized benchmarking reports the average error rate of a gate set, from the decay of fidelity over random Clifford sequences, deliberately independent of everything around it. Quantum volume reports the largest random square circuit the whole machine can actually execute — gates, connectivity, crosstalk and compiler together — and Cross et al. introduce it precisely because they hold that the first kind of number does not predict the second: "performance of isolated gates may not predict the behavior of the system. Methods such as randomized benchmarking, state and process tomography, and gateset tomography are valued for measuring the performance of operations on a few qubits, yet they fail to account for errors arising from interactions with spectator qubits." So the two do not merely differ in what they measure — one of them is an argument about the other's category, which is a stronger condition than a layer needs and the reason this one is not a topic tag.
- Quantum volume from random square circuits
Run random circuits that are as deep as they are wide, and ask how often the machine returns one of the outputs that should be more likely than the median. Widen and deepen together until it can no longer beat that bar; the last size it managed is the number.
- Randomized benchmarking over Clifford sequences
Apply a random sequence of Clifford gates, then the one gate that undoes all of them, and see how often the machine comes back to where it started. Lengthen the sequence and the return probability decays; the decay rate is the average error a single gate costs, and preparation and measurement errors fall out of the fit rather than contaminating it.
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.
2 recorded ways of doing Measure what the machine can actually do. 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
- Measure what the machine can actually do
Ways through: 2
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