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arXiv:1301.2340 · Physical Review Letters

Preconditioned Quantum Linear System Algorithm

B. D. Clader, B. C. Jacobs, C. Sprouse

Published Jun 18, 2013• 288 citations
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Abstract

We describe a quantum algorithm that generalizes the quantum linear system algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] to arbitrary problem specifications. We develop a state preparation routine that can initialize generic states, show how simple ancilla measurements can be used to calculate many quantities of interest, and integrate a quantum-compatible preconditioner that greatly expands the number of problems that can achieve exponential speedup over classical linear systems solvers. To demonstrate the algorithm's applicability, we show how it can be used to compute the electromagnetic scattering cross section of an arbitrary target exponentially faster than the best classical algorithm.

DOI
10.1103/physrevlett.110.250504

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We describe a quantum algorithm that generalizes the quantum linear system algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] to arbitrary problem specifications. We develop a state preparation routine that can initialize generic states, show how simple ancilla measurements can be used to calculate many quantities of interest, and integrate a quantum-compatible preconditioner that greatly expands the number of problems that can achieve exponential speedup over classical linear systems solvers. To demonstrate the algorithm's applicability, we show how it can be used to compute the electromagnetic scattering cross section of an arbitrary target exponentially faster than the best classical algorithm.
The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.
Abstract In a sense, gate-based quantum computers are just machines that apply (extremely) large matrices to input states (vectors), and so it is natural to ask whether linear algebra applications can benefit from quantum computing. In the case of solving linear systems, this question has been answered with an emphatic ‘yes’. In 2009, Aram Harrow, Avinatan Hassidim, and Seth Lloyd proposed a quantum algorithm to solve linear systems exponentially faster than the best classical counterpart. The quantum linear systems solver is ubiquitously known by the initials of its inventors, ‘HHL’. As well as detailing HHL, this chapter also explores some of the quantum machine learning applications that have been proposed based on HHL along with results showing that these algorithms are often ‘dequantised’ in the problem space of interest – i.e., there exists an efficient classical ‘quantum-inspired’ algorithm.
We consider the quantum linear solver for Ax=bAx=bAx=b with the circulant preconditioner CCC. The main technique is the singular value estimation (SVE) introduced in [Kerenidis and Prakash, Quantum recommendation system, in ITCS (2017)]. However, the SVE should be modified to solve the preconditioned linear system ${C}^{\ensuremath{-}1}Ax={C}^{\ensuremath{-}1}b$. Moreover, different from the preconditioned linear system considered in [Phys. Rev. Lett. 110, 250504 (2013)], the circulant preconditioner is easy to construct and can be directly applied to general dense non-Hermitian cases. The time complexity depends on the condition numbers of CCC and ${C}^{\ensuremath{-}1}A$, as well as the Frobenius norm ${\ensuremath{\parallel}A\ensuremath{\parallel}}_{F}$.
Even in cases where quantum linear solvers provide significant speedup compared to their classical counterparts, their performance depends on some of the same parameters. In particular, the condition number of the matrix which is to be inverted is a decisive parameter. A well known classical, and now quantum, remedy is to precondition the linear system Ax=bA x = bAx=b by premultiplying it by a matrix HHH in such a way that the condition number of HAHAHA is significantly smaller than the condition number of AAA. In this work, we focus on a family of preconditioners called domain decomposition. First, we prove that it is feasible to apply quantum domain decomposition. We provide upper bounds for the block-encoding parameters of the Poisson problem discretized by the finite element method and preconditioned by the two-level Additive Schwarz preconditioner (one of the most fundamental domain decomposition techniques). From these bounds, we deduce the complexity of the quantum linear system solver. Second, we focus on a particular choice of local solver within the domain decomposition preconditioner by applying recent work by [Deiml and Peterseim, \textit{Math. Comput.}, 2025] on the Bramble--Pasciak--Xu (BPX) preconditioner. Finally, we provide details on how the operators are implemented.
Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system problem (QLSP) in terms of the problem dimension, but the advantage is bottlenecked by condition number of the coefficient matrix. In this work, we propose a new quantum algorithm for QLSP inspired by the classical proximal point algorithm (PPA). Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing QLSP solver, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size eta, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches. Importantly, this is the first iterative framework for QLSP where a tunable parameter eta and initialization x_0 allows controlling the trade-off between the runtime and approximation error.