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arXiv:2106.08075 · Quantum Information and Computation

Quantum algorithm for matrix functions by Cauchy's integral formula

Souichi Takahira, Asuka Ohashi, Tomohiro Sogabe, Tsuyoshi S. Usuda

Published Feb 1, 2020• 10 citations
arXivHTMLPDF

Abstract

For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{\im A t}. However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime, where \epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, \e^{\im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.

DOI
10.26421/qic20.1-2-2

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For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{\im A t}. However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime, where \epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, \e^{\im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.
The matrix logarithm is one of the important matrix functions. Recently, a quantum algorithm that computes the state ∣f⟩|f\rangle∣f⟩ corresponding to matrix-vector product f(A)bf(A)bf(A)b is proposed in [Takahira, et al. Quantum algorithm for matrix functions by Cauchy's integral formula, QIC, Vol.20, No.1\&2, pp.14-36, 2020]. However, it can not be applied to matrix logarithm. In this paper, we propose a quantum algorithm, which uses LCU method and block-encoding technique as subroutines, to compute the state ∣f⟩=log⁡(A)∣b⟩/∥log⁡(A)∣b⟩∥|f\rangle = \log(A)|b\rangle / \|\log(A)|b\rangle\|∣f⟩=log(A)∣b⟩/∥log(A)∣b⟩∥ corresponding to log⁡(A)b\log(A)blog(A)b via the integral representation of log⁡(A)\log(A)log(A) and the Gauss-Legendre quadrature rule.
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues and normal matrices. The output of the quantum algorithms is a superposition of the eigenvalues and the corresponding eigenvectors. The complexities are dominated by solving a linear system of ODEs and performing quantum singular value estimation, which usually can be solved efficiently in a quantum computer. In the special case when the matrix MMM is sss-sparse, the complexity is O~(sρ2κ2/ε2)\widetilde{O}(sρ^2 κ^2/ε^2)O(sρ2κ2/ε2) for diagonalizable matrices that only have real eigenvalues, and O~(sρ∥M∥max⁡/ε2)\widetilde{O}(sρ\|M\|_{\max} /ε^2)O(sρ∥M∥max​/ε2) for normal matrices. Here ρρρ is an upper bound of the eigenvalues, κκκ is the conditioning of the eigenvalue problem, and εεε is the precision to approximate the eigenvalues. We also extend the quantum algorithm to diagonalizable matrices with complex eigenvalues under an extra assumption.
Eigenvalue transformations appear ubiquitously in scientific computation, ranging from matrix polynomials to differential equations, and are beyond the reach of the quantum singular value transformation framework. In this work, we study the efficiency of quantum algorithms based on contour integral representation for eigenvalue transformations from both theoretical and practical aspects. Theoretically, we establish a complete complexity analysis of the contour integral approach proposed in [Takahira, Ohashi, Sogabe, and Usuda. Quant. Inf. Comput., 22, 11\&12, 965--979 (2021)]. Moreover, we combine the contour integral approach and the sampling-based linear combination of unitaries to propose a quantum algorithm for estimating observables of eigenvalue transformations using only 333 additional qubits. Practically, we design contour integral based quantum algorithms for linear ordinary differential equations, φ\varphiφ-functions, Fréchet derivatives of matrix functions, and matrix functions. In particular, we show that the contour integral algorithm can outperform all the existing quantum algorithms in the case of solving asymptotically stable differential equations.
he matrix functions can be defined by Cauchy's integral formula and can be approximated by the linear combination of inverses of shifted matrices using a quadrature formula. In this paper, we propose a quantum algorithm for matrix functions based on a procedure to implement the linear combination of the inverses on quantum computers. Compared with the previous study [S. Takahira, A. Ohashi, T. Sogabe, and T.S. Usuda, Quant. Inf. Comput., \textbf{20}, 1\&2, 14--36, (Feb. 2020)] that proposed a quantum algorithm to compute a quantum state for the matrix function based on the circular contour centered at the origin, the quantum algorithm in the present paper can be applied to a more general contour. Moreover, the algorithm is described by the block-encoding framework. Similarly to the previous study, the algorithm can be applied even if the input matrix is not a Hermitian or normal matrix. This is an advantage compared with quantum singular value transformation.
Abstract We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines quantum phase estimation and quantum amplitude estimation to achieve a quadratic speedup with respect to the best classical algorithm in terms of matrix dimensionality, i.e. O ~ ( N / ε ) 9 9 In this work O ~ ignores terms that are polylogarithmic in N or 1 / ε . black-box queries to an oracle encoding the matrix, where N is the matrix dimension and ɛ is the desired precision. In contrast, the best classical algorithm for the same task requires Ω ( N ) polylog ( 1 / ε ) queries. In addition, this algorithm allows the user to select any constant success probability. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix’s low-energy subspace. We implement simulations of both algorithms and demonstrate their application to problems in quantum chemistry and materials science.
The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given interval. Our quantum algorithm is based on a method that approximates the eigenvalue counts by applying the numerical contour integral and the stochastic trace estimator applied to a matrix involving resolvent matrices. As components of our algorithm, the HHL solver is applied to an augmented linear system of the resolvent matrices, and the quantum Fourier transform (QFT) is adopted to represent the operation of the numerical contour integral. To reduce the size of the augmented system, we exploit a certain symmetry of the numerical integration. We also introduce a permutation formed by CNOT gates to make the augmented system solution consistent with the QFT input. The eigenvalue count in a given interval is derived as the probability of observing a bit pattern in a fraction of the qubits of the output state.