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
Walk a graph to the vertex you want
When candidates sit on a graph and the work done at one vertex is still worth something at its neighbour, walk the graph instead of sampling it — and pay per move rather than per candidate.
Open the full recordExpand it here — a map of just this
- Takes
- A rule naming the neighbours of any vertex, a rule saying whether a vertex is marked, somewhere to start — one named vertex, or a distribution over them — and the size bounds the schedule is computed from: the number of vertices, the depth, the maximum degree, or the size of the subsets a vertex stands for. Which of those a method needs is a property of the walk, not of the slot.
- Returns
- A marked vertex, with the evidence it is marked already in hand from the work stored on it — or the report that the graph holds none, at a stated failure probability.
Both routes here inherit a detector — a bit saying whether a marked vertex exists — and have to turn it into a finder, and they do it differently — which is the choice a reader actually makes. Szegedy's quantization of a Markov chain, and Belovs's later reading of it through effective resistance, certify a bit: whether a marked vertex exists. Ambainis converts that into a witness by construction, because the vertex his walk lands on carries the queried values that prove it marked, so measuring it hands over the colliding pair. Montanaro converts it by search, running the detector on subtree after subtree down the tree, which costs him a factor of for the descent and another for tightening each detection's confidence, and buys the right to a walk on a graph nobody laid out in advance — he says so against the earlier work directly: "in prior work it is usually assumed that the input graph is known in advance, and moreover that the initial state of the quantum walk is the stationary distribution of the corresponding random walk". That the two families are genuinely different, rather than one framework at two settings, is the literature's own verdict rather than this map's: Magniez and Nayak, using Szegedy's route where Ambainis's had not worked, conclude that group commutativity and triangle finding together "give strong evidence that the walks due to Ambainis are not comparable with the ones due to Szegedy".
- Walk over subsets that remember their queries
Make each vertex a subset of the input together with the values already queried for it, so that stepping to a neighbouring subset costs one query rather than a fresh batch. Grover diffusion over which element to add or remove supplies the moves, and a phase flip on subsets that already contain the answer supplies the direction.
- Walk the backtracking tree itself
Take the tree a classical backtracking algorithm would have explored — never built, never known in advance — and walk it from the root. Phase estimation on the walk operator says whether a solution is down there at all; running that test on subtree after subtree turns the answer into the solution.
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 Walk a graph to the vertex you want. 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
- Walk a graph to the vertex you want
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