🤖 AI Summary
Existing quantum algorithms for the Quantum Linear Systems Problem (QLSP) exhibit high sensitivity to the condition number κ of the coefficient matrix, resulting in prohibitive computational complexity and limited practical applicability. To address this, we propose the first quantum meta-framework grounded in the classical Proximal Point Algorithm (PPA). Our method avoids explicit matrix inversion and instead employs a tunable-step quantum proximal operator for implicit preconditioning, integrated with Hamiltonian simulation and a QLSP subroutine. Crucially, it preserves quantum exponential speedup while reducing the κ-dependence in query complexity from κ to √κ. The key contribution lies in the systematic incorporation of PPA into quantum algorithm design—marking the first such application—which significantly enhances feasibility and practicality for ill-conditioned systems.
📝 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 even such a theoretical advantage is bottlenecked by the 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 exttt{QLSP_solver}, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size $eta$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches.