Beyond asymptotic scaling: Comparing functional quantum linear solvers

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 without access to quantum hardware. 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 QSVT-based method shows the best performance.

💡 Research Summary

The paper conducts a detailed, instance‑level comparison of four prominent functional quantum linear solvers (QLS): the original Harrow‑Hassidim‑Lloyd (HHL) algorithm, two linear‑combination‑of‑unitaries (LCU) based methods (denoted QLS‑Fourier and QLS‑Chebyshev), and a quantum singular‑value‑transformation (QSVT) based method (QLS‑QSVT). All algorithms are assumed to operate under the same sparse‑access input model (SAIM), meaning they have identical access to two oracles: O_F for locating non‑zero entries in a d‑sparse Hermitian matrix A, and O_A for reading the corresponding values. The goal is to prepare a quantum state proportional to the solution x = A⁻¹b with ℓ₂‑error ≤ ε, given a state preparation oracle for |b⟩.

To move beyond asymptotic big‑O analysis, the authors introduce a “hybrid benchmarking” methodology that computes the exact number of oracle queries each algorithm requires for a concrete problem instance, characterized by its condition number κ, sparsity d, target accuracy ε, and the norm of the exact solution ‖x‖. The dominant cost is the number of calls to O_F and O_A; preparation of |b⟩ and error‑correction overhead are assumed equal across algorithms and thus omitted.

The technical core consists of a series of lemmas that translate each algorithm’s logical steps into query counts:

• Lemma 1 gives the query complexity of Hamiltonian simulation via qubitization: Q