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
Frame an operator as a ground-state question
Given a Hamiltonian reachable as hamiltonian-access, plus the caller's own declaration that the quantity wanted is its lowest eigenvalue rather than its time evolution, return the same operator typed as a ground-state-problem -- the entry `ground-state-energy`, and every method realising it, is written against.
Open the full recordExpand it here — a map of just this
- Takes
- A Hermitian operator reachable as a sum of terms, as sparse-access oracles or as a block-encoding -- exactly what hamiltonian-access already promises -- plus the one thing no prior process can hand over: the caller's own declaration that the quantity being asked for is that operator's lowest eigenvalue rather than, say, its state at a later time.
- Returns
- The identical operator, now typed as a ground-state-problem. Nothing about the Hamiltonian is transformed; the whole content of this step is the declaration attached to it.
This is the thinnest layer the map draws -- it changes no bit of the operator, only what is declared about it -- and it is still drawn rather than folded into `ground-state-energy`'s own contract, for the reason `ground-state-problem` is a narrower state than `hamiltonian-access` at all: the declaration is not decoration (state-vocabulary.ts says so directly), and a route that has not made it should not be able to walk into this region by accident. Jiang, Kalev, Mruczkiewicz and Neven name exactly this framing, in their own first section, while introducing an unrelated ternary-tree fermion-to-qubit encoding: "fermion-to-qubit mapping is a key ingredient in any quantum simulation protocol, e.g., the variational quantum eigensolver (VQE)" -- one sentence about VQE's own inputs, not about their encoding, which is why it sources a framing step rather than a region. W32 read that same paper's encoding for a competing-methods slot at `hamiltonian-access` and refused it (R2, plans/leona-map-scaling-rules.md); this slot is not that proposal -- it takes a bare Hamiltonian and the caller's own declaration and returns the pair, and the declaration is precisely what an automatic encoding cannot supply. It stands on one method rather than two contested ones, by the owner's own ruling on ai-ops 195 (which closes ai-ops 64's "some problems can be solved using VQE by different framing and preparation of the problem itself"): a slot that only names what a caller already knows they want has nothing to contest the way `ansatz-construction` or `ground-state-energy` itself do, and inventing a second method here would manufacture a competition the paper making the claim does not contain.
- The mapping paper that names VQE as the reason to frame a ground-state question
Realises `ground-state-framing` on one citation: a ternary-tree fermion-to-qubit mapping paper whose own first section states, in passing, that an efficient encoding matters because a caller downstream -- naming VQE by name -- is going to ask for a ground state. The mapping itself is not drawn here; only the sentence that motivates it is.
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.
1 recorded way of doing Frame an operator as a ground-state question. 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
- Estimate an excited-state energy
- Estimate a Hamiltonian's ground-state energy
- Frame an operator as a ground-state question
Ways through: 1
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
- Frame an operator as a ground-state question
- 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
149 nodes — 32 slots and 117 methods.
76 of the 149 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 33 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 141 papers; they and the 193 the Atlas cites alone are registered in one place, with what each reports and everywhere it is cited from. Papers