Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra

PRX Quantum, vol. 5

Abstract

We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic and no additional qubits are required for quantum data structures. Our algorithms start from a classical data structure in which the matrix of interest is specified in the Pauli basis. For N × N Hermitian matrices, the space cost is log  ( N ) + 1 qubits and, depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size O ( N 2 ) , when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as algorithms for sampling properties of ground states and Gibbs states of Hamiltonians. As a concrete application, we combine these subroutines to present a scheme for calculating Green’s functions of quantum many-body systems. Published by the American Physical Society 2024

Authors 3

1. Samson Wang

   Imperial College London

   Affiliation as printed

   Department of Physics, Imperial College London

   Department of Physics, Imperial College London, London SW7 2BW, United Kingdom

2. Sam McArdle

   California Institute of Technology · Amazon (United States)

   Affiliation as printed

   AWS Center for Quantum Computing

   Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125

   AWS Center for Quantum Computing, Pasadena, CA 91106, USA

   Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA

3. Mario Berta Aachen Institute for Quantum Information

   RWTH Aachen University · Imperial College London

   Affiliation as printed

   Department of Computing, Imperial College London

   Institute for Quantum Information, RWTH Aachen University

   Department of Computing, Imperial College London, London SW7 2BW, United Kingdom

   Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany