Childs–Kothari–Somma sparse QLSP oracle model

Fix $N\ge 1$, a sparsity bound $d\ge 1$, and an $N\times N$ Hermitian matrix $A$ with $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1$. The sparse QLSP promises that the normalized input state $\mathrm{\mid }b⟩$ is supported on eigenvectors with $\mathrm{\mid }\lambda \mathrm{\mid }\ge 1\mathrm{/}\kappa$ for some declared $\kappa \ge 1$; equivalently these are singular values at least $1\mathrm{/}\kappa$. The QLSP target is the normalized state proportional to $A^{+}b$ from Quantum linear-systems state-output task. The zero matrix is excluded by the nonzero supported-state promise.

The access input supplies the following reversible operations and records every forward, inverse, and controlled call separately.

1. Row location. For each row $i$ and slot $\mathrm{\ell }\in [d]$, a total finite function $f(i,\mathrm{\ell })$ returns the $\mathrm{\ell }$th nonzero column in that row, or a designated invalid sentinel for a padded slot. The input supplies a reversible in-place row permutation implementing $(i,\mathrm{\ell })\mapsto (i,f(i,\mathrm{\ell }))$ on valid slots, together with a completion on every padded basis input and its inverse.
2. Reverse location. For each valid nonzero entry $(i,j)$, a reversible map returns its unique slot $\mathrm{\ell }$. Its completion on invalid and padded inputs is specified and charged. This reverse map is an explicit capability; it is not inferred from forward XOR access.
3. Values. A finite-precision reversible oracle returns the encoded complex entry $A_{ij}$ at valid pairs and zero at invalid pairs. Its value precision and the induced matrix approximation error are stated separately.
4. Input preparation. A supplied unitary $B$ prepares $\mathrm{\mid }b⟩$ from a designated zero state. $B$, $B^{†}$, controlled calls, and their circuit/query/gate costs are explicit. No state loading, QRAM, or coordinate list is free.

The paper's row-location and value XOR oracles may be included as component access, but they do not by themselves certify the in-place reversible permutations, reverse map, or their implementation costs. In particular Sparse Hamiltonian row and value oracles defines total XOR row-location and value maps; this item requires the stronger reversible-location capabilities above whenever the sparse block-encoding construction is used. The parameter $d$, support condition number $\kappa$, value precision, oracle query count, gate cost, and classical data cost remain separate.