Takes

An access model for $A$ — sparse row/column entry oracles, or a block-encoding; a unitary preparing $\lvert b\rangle$; a known upper bound $\kappa$ on the condition number; the normalisation $\lVert A\rVert \le 1$; and a target state error $\varepsilon$.

Returns

A flagged state $\varepsilon$-close in $\ell_2$ to $A^{-1}\lvert b\rangle/\lVert A^{-1}\lvert b\rangle\rVert$. It does not return $\lVert x\rVert$, any entry of $x$, or any classical functional of $x$ — those cost extra and are decided a layer above.

Callers above — differential equations, regression, finite-element methods — need exactly this contract and are indifferent to how it is met. The routes below are not interchangeable in their assumptions: some require sparse entry oracles, some need only $e^{-iAt}$, some use no amplitude amplification at all, and their costs differ by up to a full factor of $\kappa$. $\kappa$ itself is an input here; nothing in this layer estimates it for you.

• HHL: eigenvalue inversion by phase estimation

  Prepare $|b>$, run phase estimation against $e^{-iAt}$ to write eigenvalue estimates into an ancilla register, apply a controlled rotation with amplitude proportional to $1/λ̃$, uncompute the estimation and post-select on the rotation ancilla. The success amplitude is about $1/κ$, so the procedure is amplified $O(κ)$ times.

• QSVT matrix inversion

  Block-encode $A$, apply the quantum singular value transformation with an odd polynomial approximating a scaled $1/x$ away from the origin, then amplify. Because it acts on singular values, $A$ need not be Hermitian or sparse — only block-encodable.

• Chebyshev LCU inversion

  Expand an approximation of $1/x$ in Chebyshev polynomials and implement that expansion directly through the quantum walk associated with a sparse $A$, rather than through Hamiltonian simulation. Phase estimation is never used, which is what removes the $\mathrm{poly}(1/\varepsilon)$ bottleneck.

• Discrete adiabatic inversion

  Encode the solution as the null eigenstate of a Hamiltonian path built from $A$ and $|b>$, follow that path with a sequence of qubitization walk operators to fixed precision, then finish with an eigenstate filter implemented as a linear combination of walk operators rather than by quantum signal processing. Costa and co-authors prove an adiabatic theorem for intrinsically discrete-time evolutions, which removes the residual $\log(κ)$ that continuous adiabatic treatments carried.

• Eigenstate filtering inversion

  Construct the minimax-optimal polynomial that is 1 at a target eigenvalue and uniformly small outside a spectral gap, and apply it through quantum signal processing. For a linear system $|x>$ is the null eigenstate of a Hamiltonian built from $A$ and $|b>$, so one application of the filter solves it once a starting state with non-trivial overlap is supplied.

• 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.

• Time-marching with uniform singular value amplification

  Propagate the solution one step at a time and defeat the exponentially vanishing success probability by repeatedly invoking uniform singular value amplification, improved further by a compression gadget lemma. Fang, Lin and Tong present it explicitly as a design path alternative to solvers based on quantum linear systems algorithms.

• LCHS — linear combination of Hamiltonian simulation

  Express a general non-unitary evolution operator as a linear combination of unitary evolution operators, each of which solves a Hamiltonian simulation problem, rather than converting the problem into a dilated linear system. An, Liu and Lin state that the method can achieve optimal cost in terms of state preparation.

• LCHS with the improved kernel

  A family of identities expressing non-unitary evolution as a linear combination of unitary evolutions, built on the kernel $f(z) = 1/(C_β e^{(1+iz)^β})$ with $β ∈ (0,1)$ and $C_β = 2π e^{-2^β}$. The kernel decays at a near-exponential rate $e^{-c|k|^β}$, replacing the original Cauchy kernel's quadratic decay and exponentially enhancing accuracy.

• Schrödingerisation (linear PDEs as Schrödinger equations)

  A simple change of variable — the warped phase transformation, which introduces one extra variable — recasts any linear PDE or ODE system into a system of Schrödinger equations in real time, which ordinary Hamiltonian simulation then runs. The original solution is recovered from the auxiliary dimension.