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arXiv:2406.12086

A shortcut to an optimal quantum linear system solver

Alexander M. Dalzell

Published Jun 17, 2024• 1 citation
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Abstract

Given a linear system of equations $A\boldsymbol{x}=\boldsymbol{b}$, quantum linear system solvers (QLSSs) approximately prepare a quantum state $|\boldsymbol{x}\rangle$ for which the amplitudes are proportional to the solution vector $\boldsymbol{x}$. Asymptotically optimal QLSSs have query complexity $O(κ\log(1/\varepsilon))$, where $κ$ is the condition number of $A$, and $\varepsilon$ is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm $\lVert\boldsymbol{x}\rVert$ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is $(1+O(\varepsilon))κ\ln(2\sqrt{2}/\varepsilon)$. If the norm is unknown, our method allows it to be estimated up to a constant factor using $O(\log\log(κ))$ applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal $O(κ\log\log(κ)\log\log\log(κ)+κ\log(1/\varepsilon))$ total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that $O(κ)$ complexity can be achieved for norm estimation, yielding an optimal QLSS with $O(κ\log(1/\varepsilon))$ complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of $56κ+1.05κ\ln(1/\varepsilon)+o(κ)$ for the complexity of our optimal QLSS.

DOI
10.48550/arxiv.2406.12086

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Given a linear system of equations Ax=bA\boldsymbol{x}=\boldsymbol{b}Ax=b, quantum linear system solvers (QLSSs) approximately prepare a quantum state ∣x⟩|\boldsymbol{x}\rangle∣x⟩ for which the amplitudes are proportional to the solution vector x\boldsymbol{x}x. Asymptotically optimal QLSSs have query complexity O(κlog⁡(1/ε))O(κ\log(1/\varepsilon))O(κlog(1/ε)), where κκκ is the condition number of AAA, and ε\varepsilonε is the approximation error. However, runtime guarantees for existing optimal and near-optimal QLSSs do not have favorable constant prefactors, in part because they rely on complex or difficult-to-analyze techniques like variable-time amplitude amplification and adiabatic path-following. Here, we give a conceptually simple QLSS that does not use these techniques. If the solution norm ∥x∥\lVert\boldsymbol{x}\rVert∥x∥ is known exactly, our QLSS requires only a single application of kernel reflection (a straightforward extension of the eigenstate filtering (EF) technique of previous work) and the query complexity of the QLSS is (1+O(ε))κln⁡(22/ε)(1+O(\varepsilon))κ\ln(2\sqrt{2}/\varepsilon)(1+O(ε))κln(22​/ε). If the norm is unknown, our method allows it to be estimated up to a constant factor using O(log⁡log⁡(κ))O(\log\log(κ))O(loglog(κ)) applications of kernel projection (a direct generalization of EF) yielding a straightforward QLSS with near-optimal O(κlog⁡log⁡(κ)log⁡log⁡log⁡(κ)+κlog⁡(1/ε))O(κ\log\log(κ)\log\log\log(κ)+κ\log(1/\varepsilon))O(κloglog(κ)logloglog(κ)+κlog(1/ε)) total complexity. Alternatively, by reintroducing a concept from the adiabatic path-following technique, we show that O(κ)O(κ)O(κ) complexity can be achieved for norm estimation, yielding an optimal QLSS with O(κlog⁡(1/ε))O(κ\log(1/\varepsilon))O(κlog(1/ε)) complexity while still avoiding the need to invoke the adiabatic theorem. Finally, we compute an explicit upper bound of 56κ+1.05κln⁡(1/ε)+o(κ)56κ+1.05κ\ln(1/\varepsilon)+o(κ)56κ+1.05κln(1/ε)+o(κ) for the complexity of our optimal QLSS.
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.
The spectral condition number is a widely adopted measure of worst-case cost for quantum linear system solvers. Yet it can significantly overestimate the actual runtime for a typical problem instance. We present two quantum algorithms that produce the normalized solution ∣x⟩|x\rangle∣x⟩ of linear system Ax=∣b⟩Ax=| b \rangleAx=∣b⟩ to accuracy εεε with complexity independent of the condition number κ=∥A−1∥κ=\lVert A^{-1}\rVertκ=∥A−1∥. We focus on the standard input model where AAA is accessed through a block encoding and ∣b⟩| b \rangle∣b⟩ is prepared by a unitary. But we also introduce an affine dilation model that encodes AAA and ∣b⟩| b \rangle∣b⟩ jointly, allowing further refinements of the query complexity. Our truncation-based solver makes an optimal number of queries to ∣b⟩| b \rangle∣b⟩ and O⁡(κeffpolylog⁡(κeffε))\operatorname{\mathbf{O}}\left(κ_{\mathrm{eff}}\operatorname{polylog}\left(\frac{κ_{\mathrm{eff}}}ε\right)\right)O(κeff​polylog(εκeff​​)) queries to AAA. We prove a family of upper bounds on the effective condition number, including κeff≤∥(A†A)−t/2∣x⟩∥1/tε1/tκ_{\mathrm{eff}}\leq\frac{\lVert(A^\dagger A)^{-t/2}|x\rangle\rVert^{1/t}}{ε^{1/t}}κeff​≤ε1/t∥(A†A)−t/2∣x⟩∥1/t​ for positive even integer ttt and κeff≤∥A−1†(A†A)−(t−1)/2∣x⟩∥1/tε1/tκ_{\mathrm{eff}}\leq\frac{\lVert A^{-1\dagger}(A^\dagger A)^{-(t-1)/2}|x\rangle\rVert^{1/t}}{ε^{1/t}}κeff​≤ε1/t∥A−1†(A†A)−(t−1)/2∣x⟩∥1/t​ for positive odd ttt, overcoming the κκκ-barrier. Our filtering-based solver is extremely simple with a favorable runtime prefactor. In particular, the solver has query complexity 3∥A−1†∣x⟩∥ε3\frac{\lVert A^{-1\dagger}|x\rangle\rVert}ε3ε∥A−1†∣x⟩∥​ to leading order when the solution norm is known. We then present a similarly simple solution norm estimator with the same asymptotic cost up to logarithmic factors. Our quantum linear system solvers thus substantially improve a recent algorithm of Li, enabling faster quantum linear system solving beyond the condition number.
In the quantum linear systems problem (QLSP), we are given query access to a ddd-sparse N×NN\times NN×N matrix AAA with condition number κκκ, and the ability to prepare a quantum state proportional to a vector b⃗\vec bb. The goal is to output an εεε-approximation to the quantum state proportional to the solution x⃗\vec xx of Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities O(κdlog⁡(1/ε))O(κd\log(1/ε))O(κdlog(1/ε)) and κd(κd/ε)o(1)κ\sqrt d(κd/ε)^{o(1)}κd​(κd/ε)o(1), while the best known lower bounds were Ω(κlog⁡(1/ε))Ω(κ\log(1/ε))Ω(κlog(1/ε)) and Ω(κd)Ω(κ\sqrt d)Ω(κd​). We improve these bounds and show that the complexity of the QLSP is Θ(κdlog⁡(1/ε))Θ(κ\sqrt d\log(1/ε))Θ(κd​log(1/ε)). We also resolve an open problem of Berry and Childs by showing that any N×NN\times NN×N unitary can be implemented with bounded error using O(N)O(\sqrt N)O(N​) queries to its matrix entries.
Recently, several approaches to solving linear systems on a quantum computer have been formulated in terms of the quantum adiabatic theorem for a continuously varying Hamiltonian.Such approaches have enabled near-linear scaling in the condition number κ of the linear system, without requiring a complicated variable-time amplitude amplification procedure.However, the most efficient of those procedures is still asymptotically suboptimal by a factor of log(κ).Here, we prove a rigorous form of the adiabatic theorem that bounds the error in terms of the spectral gap for intrinsically discrete-time evolutions.In combination with the qubitized quantum walk, our discrete adiabatic theorem gives a speed-up for all adiabatic algorithms.Here, we use this combination to develop a quantum algorithm for solving linear systems that is asymptotically optimal, in the sense that the complexity is strictly linear in κ, matching a known lower bound on the complexity.Our O[κ log(1/ )] complexity is also optimal in terms of the combined scaling in κ and the precision .Compared to existing suboptimal methods, our algorithm is simpler and easier to implement.Moreover, we determine the constant factors in the algorithm, which would be suitable for determining the complexity in terms of gate counts for specific applications.
Quantum linear-system algorithms access the coefficient matrix through a block-encoding oracle UAU_AUA​ and the right-hand side through a state-preparation oracle UbU_bUb​. The optimal query complexities of these two oracles are known separately: O(κlog⁡(1/ε))O(κ\log(1/\varepsilon))O(κlog(1/ε)) queries to UAU_AUA​ and O(κ/s)O(κ/s)O(κ/s) queries to UbU_bUb​, where s=α∥A−1b∥s=α\|A^{-1}b\|s=α∥A−1b∥. Whether a single algorithm can achieve both bounds simultaneously has remained open. We resolve this question affirmatively. Given a constant-factor estimate of sss, we give a quantum linear-system algorithm that prepares the normalized solution to error ε\varepsilonε using simultaneously $ O\left(κ\log\frac1\varepsilon\right)$ queries to UAU_AUA​ and O(κ/s)O\left(κ/{s}\right)O(κ/s) queries to UbU_bUb​. The algorithm separates state preparation from precision refinement, so only the matrix-query count depends on the target accuracy. We also prove a matching same-instance worst-case lower bound: for every quantum linear-system algorithm and every solution-norm scale, there exists a single Hermitian instance on which that algorithm requires both Ω(κlog⁡(1/ε))Ω(κ\log(1/\varepsilon))Ω(κlog(1/ε)) matrix queries and Ω(κ/s)Ω(κ/s)Ω(κ/s) state-preparation queries. Thus the two oracle complexities can be optimized simultaneously, and the resulting bounds are jointly optimal.
Quantum linear systems algorithms (QLSAs) aim to solve linear systems $A\bb{x}=\bb{b}$ exponentially faster than classical methods under certain conditions. In this work, we develop quantum algorithms for solving linear algebraic equations from an ODE-based perspective. Inspired by the linear combination of Hamiltonian simulation (LCHS) representation in the Fourier approach \cite{Childs2017QLSA}, we express the solution $\bb{x}$ as a linear combination of solutions to a system of linear convection equations, which become Schrödinger-type equations with unitary evolutions in the Fourier domain. We refer to this representation as LC-Schrödingerization. Based on this result, we construct an LCHS-based quantum algorithm with two LCHS instances: one for time-marching and one for numerical integration. The key construction uses the derivative of a Gaussian-smoothed hat function and recovers the solution over a fixed auxiliary interval. This permits a truncation time independent of the target accuracy and avoids the loss in success probability from selecting a single grid point. Periodization and explicit Fourier coefficients provide the corresponding projection error bounds. Under the stated oracle assumptions and given a constant-factor estimate of the solution norm, direct simulation of the select operators and block preconditioning achieve the optimal matrix-query complexity O(κAlog⁡1ε)\mathcal{O}(κ_A\log\frac1\varepsilon)O(κA​logε1​) without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.