Quantum algorithm for matrix functions by Cauchy's integral formula

• URL: http://arxiv.org/abs/2106.08075v1
• Date: Tue, 15 Jun 2021 12:10:16 GMT
• Title: Quantum algorithm for matrix functions by Cauchy's integral formula
• Authors: Souichi Takahira, Asuka Ohashi, Tomohiro Sogabe, and Tsuyoshi Sasaki Usuda
• Abstract summary: We consider the problem of obtaining quantum state $lvert f rangle$ corresponding to vector $f(A)boldsymbolb$. We propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation.
• Score: 1.399948157377307
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For matrix $A$, vector $\boldsymbol{b}$ and function $f$, the computation of vector $f(A)\boldsymbol{b}$ arises in many scientific computing applications. We consider the problem of obtaining quantum state $\lvert f \rangle$ corresponding to vector $f(A)\boldsymbol{b}$. There is a quantum algorithm to compute state $\lvert f \rangle$ using eigenvalue estimation that uses phase estimation and Hamiltonian simulation $\mathrm{e}^{\mathrm{{\bf i}} A t}$. However, the algorithm based on eigenvalue estimation needs $\textrm{poly}(1/\epsilon)$ runtime, where $\epsilon$ is the desired accuracy of the output state. Moreover, if matrix $A$ is not Hermitian, $\mathrm{e}^{\mathrm{{\bf i}} A t}$ 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 $\mathrm{poly}(\log(1/\epsilon))$ and the algorithm outputs state $\lvert f \rangle$ even if $A$ is not Hermitian.

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