Quantum Kaczmarz Algorithm for Solving Linear Algebraic Equations
Nhat A. Nghiem, Tuan K. Do, Trung V. Phan
TL;DR
This work introduces a quantum analogue of the Kaczmarz row-action method for solving linear systems $A x = b$ that avoids relying on oracle access to matrix entries. It leverages block-encoding and state-preparation techniques to implement the Kaczmarz update as a quantum operation, producing a quantum state $|x^{(T)}\rangle$ after $T$ iterations that approximates the solution. The algorithm achieves favorable circuit depths when the system has small rank or structured rows, with complexities scaling as $\mathcal{O}\left( \frac{1}{\varepsilon} \log m\right)$ or $\mathcal{O}\left( \frac{1}{\varepsilon} \log s\right)$ in ideal cases, though in general the dependence on the rank $r_A$ introduces a factor of $2^{r_A}$ and the iteration count $T$ induces a $1/\varepsilon$ scaling. Overall, the quantum Kaczmarz method provides a practical, QRAM-free alternative that can outperform existing quantum linear solvers in rectangular-system regimes, while highlighting a key open challenge of reducing the iteration-depth bottleneck to achieve broader speedups.
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 $κ$. This shows a significant improvement over previous quantum linear solvers where the dependence on $κ,s$ 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.