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arXiv:1807.04563 · Physical Review A

Quantum circulant preconditioner for a linear system of equations

Changpeng Shao, Hua Xiang

Published Dec 18, 2018• 30 citations
arXivHTMLPDF

Abstract

We consider the quantum linear solver for $Ax=b$ with the circulant preconditioner $C$. 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 $C$ and ${C}^{\ensuremath{-}1}A$, as well as the Frobenius norm ${\ensuremath{\parallel}A\ensuremath{\parallel}}_{F}$.

DOI
10.1103/physreva.98.062321

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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.
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.
Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that Ax=b. Based on the technique of the singular value estimation, the paper proposes a modified quantum scheme to obtain the quantum state |x⟩ corresponding to the solution of the linear system of equations in O(κ2rpolylog(mn)/ϵ) time for a general m×n dimensional A, which is superior to existing quantum algorithms, where κ is the condition number, r is the rank of matrix A and ϵ is the precision parameter. Meanwhile, we also design a quantum circuit for the homogeneous linear equations and achieve an exponential improvement. The coefficient matrix A in our scheme is a sparsity-independent and non-square matrix, which can be applied in more general situations. Our research provides a universal quantum linear system solver and can enrich the research scope of quantum computation.
A recent breakthrough by Tang (STOC 2019) showed how to "dequantize" the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but "quantum-inspired", efficiently computes a low-rank approximation of the users' preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gilyén and Jeffery (ICALP 2019) and Gilyén, Su, Low and Wiebe (STOC 2019).
We transform the problem of solving linear system of equations Ax=bA\mathbf{x}=\mathbf{b}Ax=b to a problem of finding the right singular vector with singular value zero of an augmented matrix CCC, and present two quantum algorithms for solving this problem. The first algorithm solves the problem directly by applying the quantum eigenstate filtering algorithm with query complexity of $O\left( sκ\log \left( 1/ε\right) \right) $ for a sss-sparse matrix CCC, where κκκ is the condition number of the matrix AAA, and εεε is the desired precision. The second algorithm uses the quantum resonant transition approach, the query complexity scales as $O\left[sκ+ \log\left( 1/ε\right)/\log \log \left( 1/ε\right) \right] $. Both algorithms meet the optimal query complexity in κκκ, and are simpler than previous algorithms.
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.