Randomized adiabatic quantum linear solver algorithm with optimal complexity scaling and detailed running costs

TL;DR

The authors introduce a randomized adiabatic quantum linear solver that attains optimal κ log(1/ε) scaling by integrating Poissonization, eigenstate filtering, and a novel randomized walk operator to replace Hamiltonian simulation. They provide a detailed, non-asymptotic cost analysis with explicit constants, including a Hermitian-specialized bound, and show how to manage rescaling, block-encoding construction, and error fusion between adiabatic and filtering stages. The approach relies on a Hermitian extension of A, a carefully engineered H(s) path with a known spectral gap, and a walk-based dephasing mechanism that achieves dephasing without Hamiltonian simulation. The work demonstrates competitive, rigorous resource guarantees for QLSA and identifies practical directions for hardware-friendly optimizations and problem-structure exploiting refinements.

Abstract

Solving linear systems of equations is a fundamental problem with a wide variety of applications across many fields of science, and there is increasing effort to develop quantum linear solver algorithms. [Subaşi et al., Phys. Rev. Lett. (2019)] proposed a randomized algorithm inspired by adiabatic quantum computing, based on a sequence of random Hamiltonian simulation steps, with suboptimal scaling in the condition number $κ$ of the linear system and the target error $ε$. Here we go beyond these results in several ways. Firstly, using filtering [Lin et al., Quantum (2019)] and Poissonization techniques [Cunningham et al., arXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling $O(κ\log(1/ε))$ - an exponential improvement in $ε$, and a shaving of a $\log κ$ scaling factor in $κ$. Secondly, the algorithm is further modified to achieve constant factor improvements, which are vital as we progress towards hardware implementations on fault-tolerant devices. We introduce a cheaper randomized walk operator method replacing Hamiltonian simulation - which also removes the need for potentially challenging classical precomputations; randomized routines are sampled over optimized random variables; circuit constructions are improved. We obtain a closed formula rigorously upper bounding the expected number of times one needs to apply a block-encoding of the linear system matrix to output a quantum state encoding the solution to the linear system. The upper bound is $867 κ$ at $ε=10^{-10}$ for Hermitian matrices.

Theorems & Definitions (17)

• Theorem 1: Optimal QLSA with explicit counts
• Lemma 2: Constructing a block-encoding of $H(s)$
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5: Filtering cost
• proof
• Lemma 6