This paper settles how many times a quantum algorithm must look up entries of a sparse matrix to prepare the quantum state proportional to the solution of A x = b. The authors show the query cost is Θ(κ √d log(1/ε)). Here κ is the condition number of A (a measure of how hard A is to invert), d is the maximum number of nonzero entries per row (sparsity), and ε is the desired approximation error. They also use the same ideas to show any N×N unitary can be implemented with bounded error using O(√N) queries to its entries, resolving an earlier open question.

The quantum linear systems problem (QLSP) studied here assumes a specific access model. The algorithm can query two oracles for A: a location oracle that lists the positions of the up-to-d nonzero entries in a given row, and a value oracle that returns the numerical value of a listed entry. The algorithm is also given a way to prepare a quantum state proportional to the vector b. The goal is to output a quantum state within trace distance ε of the normalized solution x, with reasonable success probability. The query complexity counts only calls to the matrix oracles.

To get the upper bound, the authors build a new “block encoding” of a larger matrix that can be implemented using sparse-matrix queries. A block encoding is a standard trick that represents a matrix as a piece of a larger unitary operation. They enlarge the linear system by splitting products of matrices into a two-step system, effectively making multiple copies of variables so the algorithm needs to load only single matrix entries at a time. Using that exact block encoding together with a known quantum linear-systems subroutine (from Costa et al.), they recover the original solution and achieve the O(κ √d log(1/ε)) query cost.

For the lower bound, they turn a quantum circuit into a linear system whose solution is a “history state” — a superposition of the circuit’s intermediate states. This lets them import hardness from two simpler problems. Embedding unstructured search shows a dependence on the sparsity-related term, while embedding parity shows a dependence related to κ and precision. By composing many small searches inside a parity computation and applying a quantum XOR lemma, the authors show the two sources of hardness multiply. That yields a matching lower bound, so the Θ(κ √d log(1/ε)) scaling is tight in the query model. The paper notes a concurrent independent work that also obtains the lower bound.