Quantum Kaczmarz Algorithm for Solving Linear Algebraic Equations

TL;DR

This paper presents a quantum algorithm based on the Kaczmarz method for solving linear systems efficiently, especially when the system has low rank or structured rows, avoiding oracle queries and improving circuit complexity.

Contribution

The authors develop a quantum Kaczmarz algorithm that reduces circuit complexity and depth, relaxing the need for oracle access and outperforming previous quantum solvers in certain regimes.

Findings

Achieves circuit complexity $rac{1}{ ext{epsilon}} ext{log} m$ for low-rank systems.

Provides circuit depth $rac{1}{ ext{epsilon}} ext{log} s$ for structured rows with $s$ nonzero entries.

Offers exponential depth improvement over existing quantum algorithms when sparsity $s$ is $O( ext{log} m)$.

Abstract

We introduce a quantum linear system solving algorithm based on the Kaczmarz method, a widely used workhorse for large linear systems and least-squares problems that updates the solution by enforcing one equation at a time. Its simplicity and low memory cost make it a practical choice across data regression, tomographic reconstruction, and optimization. In contrast to many existing quantum linear solvers, our method does not rely on oracle access to query entries, relaxing a key practicality bottleneck. In particular, when the rank of the system of interest is sufficiently small and the rows of the matrix of interest admit an appropriate structure, we achieve circuit complexity $\mathcal{O}\left(\frac{1}{\varepsilon}\log m\right)$, where $m$ is the number of variables and $\varepsilon$ is the target precision, without dependence on the sparsity $s$, and could possibly be without explicit dependence on condition number $\kappa$. This shows a significant improvement over previous quantum linear solvers where the dependence on $\kappa,s$ is at least linear. At the same time, when the rows have an arbitrary structure and have at most $s$ nonzero entries, we obtain the circuit depth $\mathcal{O}\left(\frac{1}{\varepsilon}\log s\right)$ using extra $\mathcal{O}(s)$ ancilla qubits, so the depth grows only logarithmically with sparsity $s$. When the sparsity $s$ grows as $\mathcal{O}(\log m)$, then our method can achieve an exponential improvement with respect to circuit depth compared to existing quantum algorithms, while using (asymptotically) the same amount of qubits.