Quantum algorithm for matrix functions by Cauchy's integral formula

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.

Quantum Algorithm for Matrix Logarithm by Integral Formula

“…In this method, we choose the Gauss-Legendre quadrature rule, while the trapezoidal rule was applied in [10]. We use the fact that the M -point Gauss Legendre quadrature is just the Padé approximation to the matrix logarithm [4].…”

“…And this is used for error estimation. The complex function must be analytical on the disk with center 0 in [10], while we do not have this restriction. When we prepare the manuscript, we notice the work [11].…”

“…Furthermore, LCU method can be also used for implementing matrix functions f (A). Recently, Takahira et al [10] proposed a quantum algorithm, which makes use of the contour integral of the matrix function f (A), to compute the state |f = f (A)|b / f (A)|b using HHL algorithm or LCU method as a subroutine. However, this quantum algorithm can be only applied when the contour is a circle centered at the origin, and cannot be applied to a function that has a singular at the origin, such as matrix logarithm [11].…”

“…For simplicity, we assume that the oracle O A returns the element with s bits of accuracy and τ k (k ∈ {1, 2, • • • , M }) has s bits. In step II-(2), it uses O(s 2 ) gates to obtain the value of τ k A ij with s bits of accuracy[10]. In step II-(3), the gate complexity for |i, j |0 a 2 → |i, j |δ ij a 2 is O(log N ).…”

Quantum algorithms for matrix operations and linear systems of equations

“…However, the classical computation is unsuitable for large matrices. After advening the HHL algorithm, more efficient quantum algorithms for matrix operations are given great expectation and researchers have obtained many substantial results [18][19][20][21][22]. Particularly, in [19] Zhao et al presented a new method for performing the elementary linear algebraic operations, for example matrix addition, multiplication, Kronecker sum, tensor product, Hadamard product and arbitrary real single-matrix functions, on a quantum computer for complex matrices.…”