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.
Where the routes meet
Every circle is drawn once. Several ways of getting somewhere end on the same circle, and every way onward leaves from it — so a route you can take is any line in, followed by any line out, whether or not a paper has put those two together.
The circle between the ends is an object every way across passes through.
12 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
- 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) — open · opened: what was inside is drawn in its place
- Linear multistep method, all-at-once encoding — opens into 2 · a way across — click it to open it here
- Taylor propagator, all-at-once encoding — opens into 2 · a way across — click it to open it here
- Chebyshev spectral method, global collocation — opens into 2 · a way across — click it to open it here
- Dyson propagator, all-at-once encoding — opens into 2 · a way across — click it to open it here
- Time-marching with uniform singular value amplification — opens into 2 · a way across — click it to open it here
- LCHS — linear combination of Hamiltonian simulation — opens into 3 · a way across — click it to open it here
- Schrödingerisation (linear PDEs as Schrödinger equations) — opens into 3 · 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
- 1 way arrive at Linear ODE system and 3 lead on, so 3 routes cross it.
- Every line on this figure is one a recorded source takes.
Ways through Linear ODE system
66 combinations cross this circle. A source records 1 of them end to end. 18 cross slots a source does record, without naming which method fills them. 47 are compositions no recorded source takes.
- Carleman linearization → Linear multistep method, all-at-once encoding
- Carleman linearization → Taylor propagator, all-at-once encoding
- Carleman linearization → Chebyshev spectral method, global collocation
- Carleman linearization → Krovi's reanalysis of the all-at-once encoding
- Carleman linearization → Dyson propagator, all-at-once encoding
- Carleman linearization → Time-marching with uniform singular value amplification
- Carleman linearization → LCHS — linear combination of Hamiltonian simulation
- Carleman linearization → LCHS with the improved kernel
- Carleman linearization → Schrödingerisation (linear PDEs as Schrödinger equations)
- Carleman linearization → Simulate Hamiltonian evolution → Estimate an observable
- Koopman linearization → Linear multistep method, all-at-once encoding
- Koopman linearization → Taylor propagator, all-at-once encoding
- Koopman linearization → Chebyshev spectral method, global collocation
- Koopman linearization → Krovi's reanalysis of the all-at-once encoding
- Koopman linearization → Dyson propagator, all-at-once encoding
- Koopman linearization → Time-marching with uniform singular value amplification
- Koopman linearization → LCHS — linear combination of Hamiltonian simulation
- Koopman linearization → LCHS with the improved kernel
- Koopman linearization → Schrödingerisation (linear PDEs as Schrödinger equations)
- Koopman linearization → Choose a time discretization or propagator approximation → Quantum linear solve
- Koopman linearization → Simulate Hamiltonian evolution → Estimate an observable
- Carleman-Fourier linearization → Linear multistep method, all-at-once encoding
- Carleman-Fourier linearization → Taylor propagator, all-at-once encoding
- Carleman-Fourier linearization → Chebyshev spectral method, global collocation
- Carleman-Fourier linearization → Krovi's reanalysis of the all-at-once encoding
- Carleman-Fourier linearization → Dyson propagator, all-at-once encoding
- Carleman-Fourier linearization → Time-marching with uniform singular value amplification
- Carleman-Fourier linearization → LCHS — linear combination of Hamiltonian simulation
- Carleman-Fourier linearization → LCHS with the improved kernel
- Carleman-Fourier linearization → Schrödingerisation (linear PDEs as Schrödinger equations)
Showing 30 of 47.
These are derived from the two contracts each line carries, not proposed. A line here says the object one process hands back is the object the next one takes — nothing about whether it is a good idea, and nothing about whether the literature has missed it.
Open the cardRead the full write-up
Where you are
Path
- Solve a nonlinear ODE dy/dt = F(y)
Ways through: 4
Routes that skip it
No recorded route avoids this step.
Narrower kinds
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