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arXiv:2510.13766

The Practicality of Randomized Quantum Linear Systems Solvers

Siddharth Hariprakash, Roel Van Beeumen, Katherine Klymko, Daan Camps

Published Oct 15, 2025• 0 citations
arXivHTMLPDF

Abstract

Randomized quantum algorithms have been proposed for quantum linear algebra with the goal of constructing shallower circuits than methods based on block encodings, and have been speculated to offer benefits in the early fault-tolerant era. In this work, we derive explicit, non-asymptotic error bounds on every algorithmic parameter of a randomized quantum linear systems solver that combines sampling from a Fourier series with Hamiltonian simulation, and confirm these bounds numerically. We show that even a $4 \times 4$ instance with condition number $κ= 100$ requires on the order of $10^{15}$ non-Clifford gates to converge, calling into question the practicality of such randomized schemes. Comparing the two Hamiltonian-simulation kernels, product formulas (PFs) and the random Taylor expansion (RTE), both our analytical bounds and experiments show RTE reaches a given target error with roughly an order of magnitude smaller total gate budget, although neither is practical. Our analysis bridges asymptotic proposals and hardware implementation.

DOI
10.48550/arxiv.2510.13766

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Randomized quantum algorithms have been proposed for quantum linear algebra with the goal of constructing shallower circuits than methods based on block encodings, and have been speculated to offer benefits in the early fault-tolerant era. In this work, we derive explicit, non-asymptotic error bounds on every algorithmic parameter of a randomized quantum linear systems solver that combines sampling from a Fourier series with Hamiltonian simulation, and confirm these bounds numerically. We show that even a 4×44 \times 44×4 instance with condition number κ=100κ= 100κ=100 requires on the order of 101510^{15}1015 non-Clifford gates to converge, calling into question the practicality of such randomized schemes. Comparing the two Hamiltonian-simulation kernels, product formulas (PFs) and the random Taylor expansion (RTE), both our analytical bounds and experiments show RTE reaches a given target error with roughly an order of magnitude smaller total gate budget, although neither is practical. Our analysis bridges asymptotic proposals and hardware implementation.
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Quantum simulation, the simulation of quantum processes on quantum computers, suggests a path forward for the efficient simulation of problems in condensed-matter physics, quantum chemistry, and materials science. While the majority of quantum simulation algorithms are deterministic, a recent surge of ideas has shown that randomization can greatly benefit algorithmic performance. In this work, we introduce a scheme for quantum simulation that unites the advantages of randomized compiling on the one hand and higher-order multi-product formulas, as they are used for example in linear-combination-of-unitaries (LCU) algorithms or quantum error mitigation, on the other hand. In doing so, we propose a framework of randomized sampling that is expected to be useful for programmable quantum simulators and present two new multi-product formula algorithms tailored to it. Our framework reduces the circuit depth by circumventing the need for oblivious amplitude amplification required by the implementation of multi-product formulas using standard LCU methods, rendering it especially useful for early quantum computers used to estimate the dynamics of quantum systems instead of performing full-fledged quantum phase estimation. Our algorithms achieve a simulation error that shrinks exponentially with the circuit depth. To corroborate their functioning, we prove rigorous performance bounds as well as the concentration of the randomized sampling procedure. We demonstrate the functioning of the approach for several physically meaningful examples of Hamiltonians, including fermionic systems and the Sachdev–Ye–Kitaev model, for which the method provides a favorable scaling in the effort.
Solving systems of linear equations is a key subroutine in many quantum algorithms. In the last 15 years, many quantum linear solvers (QLS) have been developed, competing to achieve the best asymptotic worst-case complexity. Most QLS assume fault-tolerant quantum computers, so they cannot yet be benchmarked on real hardware. Because an algorithm with better asymptotic scaling can underperform on instances of practical interest, the question of which of these algorithms is the most promising remains open. In this work, we implement a method to partially address this question. We consider four well-known QLS algorithms which directly implement an approximate matrix inversion function: the Harrow-Hassidim-Lloyd algorithm, two algorithms utilizing a linear combination of unitaries, and one utilizing the quantum singular value transformation (QSVT). These methods, known as functional QLS, share nearly identical assumptions about the problem setup and oracle access. Their computational cost is dominated by query calls to a matrix oracle encoding the problem one wants to solve. We provide formulas to count the number of queries needed to solve specific problem instances; these can be used to benchmark the algorithms on real-life instances, under benevolent, idealized assumptions, without requiring access to quantum hardware to implement the procedure. We select three data sets: random generated instances that obey the assumptions of functional QLS, linear systems from simplex iterations on MIPLIB, and Poisson equations. Our methods can be easily extended to other data sets and provide a high-level guide to evaluate the performance of a QLS algorithm. In particular, our work shows that HHL underperforms in comparison to the other methods across all data sets, often by orders of magnitude, while the Chebyshev-approximation based methods typically have the best performance.
Optimal quantum linear equation solvers provide complexity O(κlog⁡(1/ε))O(κ\log(1/ε))O(κlog(1/ε)), where κκκ is the condition number and εεε is the allowable error. The optimal solver using a discrete adiabatic approach [PRX Quantum 3, 040303 (2022)] has large analytically proven constant factors for the upper bound on the complexity. The constant factors were later found to be about 1,200 times smaller in numerical testing [Quantum 9, 1887 (2025)]. This meant it is about an order of magnitude more efficient than using a randomised approach from [PRX Quantum 6, 040373 (2025)], which has far smaller analytically proven constant factors. Recently, a ``Shortcut'' method has been found to provide an optimal solver which also has small proven constant factors. In the present work, we conduct a comprehensive numerical analysis comparing this method with the adiabatic solver for two families of random linear systems. We find that, in the case where the solution norm is unknown, the adiabatic solver provides slightly better performance. If the solution norm is known, then the shortcut method provides significantly better performance for non-Hermitian matrices.
Randomized quantum algorithms for the estimation of expectation values of molecular observables offer a promising route to early fault-tolerant quantum advantage by trading circuit depth for an increased number of samples. However, this begs the question whether the resulting measurement overhead remains feasible for industrially relevant molecular systems. In this work, we benchmark the randomized sampling algorithm of Wang et al. [1] through numerical simulations on a hydrogen molecule and resource estimates for various active spaces of the cytochrome P450, an industry-relevant system in the pharmaceutical industry. Our numerical simulations reveal that the analytical error bounds significantly overestimate the actual sampling error, whereas the analytical bound on the required imaginary evolution time is comparatively tight, exceeding a direct DMRG estimate by roughly 44% for the active space where the comparison is feasible. The measurement overhead nevertheless remains prohibitively high: even for the smallest active spaces, the expectation-value estimation alone requires around 1010 samples, scaling to beyond 1015 for the largest active spaces considered. These results suggest that substantial improvements in measurement strategies or algorithmic scaling are necessary before randomized early-fault-tolerant algorithms become competitive for industrial quantum chemistry applications.