Evolution circuit, input in hand

This is a kind of

Anything that asks for one of these will accept this, because it is narrower. The reverse does not hold.

• State you can prepare

  Not the state itself but the routine that makes it — which is the useful form, because a routine can be run again, controlled, and inverted, and a state that has already collapsed can do none of those.

• Circuit for e^{-iHt}

  Time evolution under a Hamiltonian, approximated to a stated error, with the query or gate count and the norm parameter the count is measured against. It is a circuit, not an answer — something still has to run it on a state and read the result.

• Abstract circuit

  Arbitrary rotation angles, arbitrary two-qubit gates, and any qubit able to talk to any other. No machine runs this. Everything between here and hardware is the business of closing that gap and counting what it costs.

Narrower kinds of this

• Unitary whose eigenphase is wanted

  A circuit you can apply controlled powers of, together with the routine preparing the state it acts on, plus the declaration that what is being asked for is the phase that state picks up — not the state, and not an expectation value read off it. The second half is not decoration: the same pair handed to a readout returns an average over a distribution, and an eigenphase is a single number sitting in the operator's spectrum.

Records that are this object

Nothing in the catalogue has been joined to this state. That is a gap in the join rather than a claim that no such object exists; the shelf on /repository lists what is joined and what is not, with the reason.

Work that arrives here

No contract in this graph returns this. It is reached only by narrowing, below — which is a real arrival, and the reason the state is named at all.

Routes that reach it by narrowing

These do not declare it in a contract. They record that one of their steps lands on something narrower than the slot promises, and this is that narrower thing.

• Quantum simulation of the KvN representation

  Because the Koopman-von Neumann generator is Hermitian and its propagator unitary, the lifted evolution can be run by Hamiltonian simulation directly. No linear system is assembled and no linear solver is called.

Work that starts here

No process asks for this by name. The ones below ask for something broader, and this is a kind of it — so they take it as it stands.

Also accepted where something broader is wanted

These ask for an object this one is a kind of. Narrowing composes in that direction and only that direction: handing on something broader than a process asks for would be a skipped conversion.

• Estimate an observable

  Given the ability to prepare $|ψ⟩$ and a description of an observable $O$, return a classical scalar within $\varepsilon$ of $⟨O⟩$ at confidence $1−δ$. The state is never returned; only the number is.

• Compile a circuit to a specific device

  Turn a circuit written as arbitrary unitaries over abstract qubits into an executable instruction sequence for one machine's own gate set and connectivity graph. The result is functionally equivalent, or equivalent to within a stated approximation error.

• Satisfy the hardware connectivity constraint

  Place logical qubits on physical ones and schedule connectivity-repair operations — usually SWAPs — so that every two-qubit gate acts on a coupled pair. The problem combines subgraph isomorphism with token swapping.

• Approximate a continuous rotation in a discrete gate set

  Given a target single-qubit unitary — typically a z-rotation by an arbitrary angle — and a precision $\varepsilon$, produce a finite word over a fixed discrete gate set such as Clifford+T whose product is within $\varepsilon$ of the target in a stated metric. The cost is charged in non-Clifford gates.