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Randomized Adiabatic Quantum Linear Solver Algorithm with Optimal Complexity Scaling and Detailed Running Costs

David Jennings1, Matteo Lostaglio1,*, Sam Pallister1, Andrew T. Sornborger2, and Yiğit Subaşı2

  • 1PsiQuantum, 700 Hansen Way, Palo Alto, California 94304, USA
  • 2Computer, Computational, and Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

  • *Contact author: mlostaglio@psiquantum.com
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PRX Quantum 6, 040373 – Published 29 December, 2025

DOI: https://doi.org/10.1103/1xkb-22cc

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. Subaşı et al. [Phys. Rev. Lett. 122, 060504 (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 and Tong, Quantum 4, 361 (2020)] and Poissonization techniques [Cunningham and Roland, ArXiv:2406.03972 (2024)], the algorithm complexity is improved to the optimal scaling O(κlog(1/ϵ))—an exponential improvement in ϵ, and a shaving of a logκ 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 837κ at ϵ=10−10 for Hermitian matrices.

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The task of solving high-dimensional linear equations is a ubiquitous problem that arises across numerous areas of science: from solving differential equations, to machine-learning, optimization, and statistics to name a few. We know rigorously that fault-tolerant quantum computers can provide exponential advantage over classical computers in the context of solving such linear equations. Moreover, in the coming decade it is widely expected that fault-tolerant devices will come online and are expected to find transformative impact.

In this context, our randomized algorithm provides a novel method to realize such quantum advantage for solving linear systems in a way that is both (a) optimally efficient in key properties (condition number and error tolerance) and (b) is built out of elementary methods and techniques suited to earlier implementations. In particular, we introduce a cheaper randomized walk operator method replacing Hamiltonian simulation, which also removes the need for potentially challenging classical precomputations.

We also provide a detailed analysis giving rigorous guarantees for the non asymptotic performance, making this algorithm readily deployable for resource estimates of high-impact use cases. Significantly, this places our work in a very small pool of optimal algorithms known for the fundamental task of solving linear equations, and showcases the competitiveness of randomized methods.

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Outline

Information

PRX Quantum 6, 040373– Published 29 December, 2025
  • Received 27 June 2023
  • Revised 1 September 2025
  • Accepted 7 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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