Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra

arXiv (Cornell University)

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 $N\times N$ 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 sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

Authors 3

1. Samson Wang corresponding

   Imperial College London

   Affiliation as printed

   Department of Physics , Imperial College London , London , UK

   Department of Physics, Imperial College London, London, UK

2. Sam McArdle

   California Institute of Technology · Amazon (United States)

   Affiliation as printed

   AWS Center for Quantum Computing , Pasadena , USA

   California Institute of Technology , Pasadena , USA

   AWS Center for Quantum Computing, Pasadena, USA

   California Institute of Technology, Pasadena, USA

3. Mario Berta Aachen Institute for Quantum Information

   RWTH Aachen University · Imperial College London

   Affiliation as printed

   Department of Computing , Imperial College London , London , UK

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

   Department of Computing, Imperial College London, London, UK

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