State

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.

A state is an object you can be holding, named once so that two routes reaching the same thing are drawn as reaching the same thing. It says nothing about how you got here or where you can go next — that is entirely in the processes below.

This is a kind of

This state is not recorded as a kind of anything else. It stands on its own in the vocabulary.

Narrower kinds of this

Hermitian generator
A linear generator that happens to be Hermitian, so the evolution it drives is unitary and a simulator can run it as it stands. This is why some lifts reach an answer without ever assembling a linear system.

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

Replace a spatial domain with a finite grid
Approximate the spatial derivatives of a PDE on finitely many points, leaving time continuous, so that what remains is a system of ordinary differential equations in the grid values. The method of lines: the continuum is gone, the clock is not.
Embed a nonlinear system into a linear one
Given a nonlinear vector field $F$ , produce a (truncated) linear generator on a lifted space, a lift of the initial condition into that space, and a decoding of the target quantity, such that linear evolution reproduces the nonlinear dynamics to accuracy $\varepsilon$ . The truncation or lift parameter fixes both the accuracy and the dimension.

Work that starts here

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.