A Catalyst Framework for the Quantum Linear System Problem via the Proximal Point Algorithm

TL;DR

This work addresses solving the quantum linear system problem (QLSP) by embedding it in a proximal point algorithm (PPA) framework. By applying a QLSP solver to the modified matrix $\frac{I + \eta A}{\|I + \eta A\|}$ and using an adaptive initialization $x_0$, the method preconditions the system and reduces the effective condition number from $\kappa$ to $\hat{\kappa} = \frac{\kappa(1+\eta)}{\kappa + \eta}$, enabling improved quantum query complexity. The analysis decomposes the total error into solver error and PPA error, showing that the final accuracy scales as $\mathcal{O}(\hat{\kappa} \log(1/\varepsilon))$ with controllable overhead, and warm-starting can further reduce practical cost. The framework is solver-agnostic and can yield constant-factor improvements over state-of-the-art baselines (e.g., Costa et al. 2022) when ill-conditioning is present, with demonstrated benefits from using a gradient-descent warm start. Overall, the paper provides a flexible, tunable approach to accelerate QLSP solvers by manipulating conditioning through $\eta$ and $x_0$.

Abstract

Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system problem (QLSP) in terms of the problem dimension, but the advantage is bottlenecked by condition number of the coefficient matrix. In this work, we propose a new quantum algorithm for QLSP inspired by the classical proximal point algorithm (PPA). Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing \texttt{QLSP\_solver}, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size $η$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches. Importantly, this is the first iterative framework for QLSP where a tunable parameter $η$ and initialization $x_0$ allows controlling the trade-off between the runtime and approximation error.

Figures (2)

• Figure 1: (Left) Query complexity scaling with respect to the condition number $\kappa$. Here, the SOTA quantum algorithm costa2022optimal, which enjoys (asymptotically) optimal query complexity $\Omega(\kappa \log \frac{1}{\varepsilon})$, is used as QLSP_solver for the subroutine in Algorithm \ref{['alg:ppa-qlsp']}. Simply by "wrapping" the QLSP_solver, one can achieve much better scaling with respect to the condition number $\kappa$. (Right) Query complexity improvement with warm start. Baseline (blue) is again the (asymptotically) optimal query complexity by costa2022optimal. The other three lines are the improved query complexity in \ref{['eq:decomp-thm6']} using Algorithm \ref{['alg:ppa-qlsp']} where $x_0$ is initialized with the output of $\{200, 500, 1000\}$ steps of GD.
• Figure 2: How the "improvement" term, the "overhead" term, and their sum "total" allows improvement compared to the baselines from childs2017quantum and costa2022optimal (c.f., Theorem \ref{['thm:improve-costa']}). (Right): using CKS childs2017quantum as baseline (c.f., Theorem \ref{['thm:cks-complexity']}); (Left): using costa2022optimal as baseline (c.f., Theorem \ref{['thm:costa']}). The key insight is that the rate of "improvement" is faster than the rate of "overhead," which plateaus quickly thanks to the logarithm as can be seen in Theorem \ref{['thm:improve-costa']}.

Theorems & Definitions (23)

• Definition 2.1: Quantum Linear System Problem
• Definition 2.2: State preparation oracle harrow2009quantum
• Definition 2.3: Sparse matrix access childs2017quantum
• Definition 2.4: Matrix block-encoding gilyen2019quantum
• Remark 3.1: Access to $(I + \eta A) / \| I + \eta A \|$
• Remark 3.2: Access to $|x_0 + \eta b \rangle$
• Lemma 3.3
• Remark 3.4: $\eta$ trade-off
• Proposition 4.1: QLSP_solver error
• Example 4.2: CKS polynomial childs2017quantum