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

Journal Article · Mon Dec 29 00:00:00 EST 2025 · PRX Quantum

DOI:https://doi.org/10.1103/1xkb-22cc· OSTI ID:3016106

Jennings, David ^[1]; Lostaglio, Matteo ^[1]; Pallister, Sam ^[1]; Sornborger, Andrew Tyler ^[2]; Subaşı, Yiğit ^[2]

---

1. PsiQuantum, Palo Alto, CA (United States)
2. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

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⁡(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 837⁢𝜅 at 𝜖 = 10⁻¹⁰ for Hermitian matrices.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 3016106
• Report Number(s): LA-UR--23-23886; 10.1103/1xkb-22cc
• Journal Information: PRX Quantum, Journal Name: PRX Quantum Journal Issue: 4 Vol. 6; ISSN 2691-3399
• Publisher: American Physical Society (APS)
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Computational complexity
Quantum Computing
Quantum algorithms
Quantum algorithms & computation
Quantum circuits
Quantum computation
Quantum information processing
Quantum protocols
Quantum simulation
System of Linear Equations