Hermitian generator

This is a kind of

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

• Hamiltonian you can query

  A Hermitian operator reachable as a sum of terms you can exponentiate one at a time, as sparse-access oracles, or as a block-encoding — plus the norm parameter the simulation cost will be quoted against.

• Matrix you can query

  Some way of asking about a matrix without writing it down — sparse row and column oracles, a Pauli or LCU decomposition, a purification, or an explicit arithmetic rule. Which one you have decides which routes are open.

• Linear ODE system

  A generator A(t), any inhomogeneity, and an initial vector — du/dt = A(t)u + b(t) on a space that may be much larger than the one you started in, together with the maps that lifted you into it and will read you back out.

Narrower kinds of this

No state in the vocabulary is recorded as a narrower kind of this one.

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

• Graph-Laplacian finite differences

  Discretize the domain onto a lattice and read the discrete Laplacian off the resulting graph: off-diagonal entries minus one between neighbours, each diagonal entry the degree of its vertex. Higher-order stencils are obtained by factorizing the operator through hypergraph incidence matrices, which is what lets the error fall faster than the second power of the spacing while keeping a form a simulator can consume.

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.

• Solve a linear ODE du/dt = A(t)u + b(t)

  Given block-encoding access to $A(t)$ and $b(t)$ and a preparation unitary for $u_0$, output a normalized state $\varepsilon$-close to $u(T)/||u(T)||$. Matrix-query and state-preparation-query counts are stated separately, because methods here differ in them independently.

• Recast a non-Hermitian generator as Hamiltonian evolution

  Given a generator $A(t)$ whose evolution is not unitary, produce a Hermitian generator — or a quadrature-indexed family of them — on a space at least as large, whose unitary evolution reproduces the original dynamics, together with the map that recovers the original solution. Both halves are required: a construction that reaches a Hamiltonian and cannot get back is not a route.

• Choose a time discretization or propagator approximation

  Reduce continuous evolution over $[0,T]$ to a finite algebraic object — a banded linear system, a product of step propagators, or a spectral coefficient system — with a stated truncation error. When a linear system is formed, a conditioning bound is stated with it.

• Block-encode a matrix

  Wrap an operator $A$ inside a larger unitary $U$ so that $A/α$ sits in $U$'s top-left block, giving every routine above it one uniform way to touch the matrix. The subnormalization $α$ and the ancilla count are outputs of this layer, not free parameters.

• Simulate Hamiltonian evolution

  Implement $e^{-iHt}$ to error $\varepsilon$ given some access model for $H$. It is an application in its own right and also the engine inside phase estimation and several linear-system solvers.