State

Linear system Ax = b

One matrix and one right-hand side, with the whole time history folded into them. It has a condition number, and that number is what the cost of solving it will be measured against. Every solver in the literature is handed it as an access model for the matrix plus a routine preparing the right-hand side, which is why it counts as a matrix you can query.

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

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

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.

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

Discretize a PDE into one linear system
Replace every continuous variable at once — space together with time, or space together with velocity — so that the whole problem becomes a single matrix equation. Nothing is left to march: the grid values at every recorded point are unknowns of one system, solved in one go.
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.

Work that starts here

Quantum linear solve
Given access to a matrix $A$ and a unitary that prepares $|b>$ , produce a flagged quantum state that is $\varepsilon$ -close in $l2$ to the normalised $A^{-1}b$ . The deliverable is a state, not a classical vector.

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.

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.