Low Rank Approximation in Simulations of Quantum Algorithms

TL;DR

This paper explores the use of low-rank tensor approximations to simulate certain quantum algorithms efficiently on classical computers, highlighting which algorithms retain low-rank structures and are thus more classically simulable.

Contribution

It introduces methods for low-rank tensor decomposition in quantum simulation and analyzes which quantum algorithms preserve low-rank structures for efficient classical simulation.

Findings

Some quantum algorithms maintain low-rank states enabling efficient classical simulation.

Intermediate states in many algorithms tend to increase in rank, complicating simulation.

Understanding low-rank structures helps identify quantum algorithms with potential classical simulation advantages.

Abstract

Simulating quantum algorithms on classical computers is challenging when the system size, i.e., the number of qubits used in the quantum algorithm, is moderately large. However, some quantum algorithms and the corresponding quantum circuits can be simulated efficiently on a classical computer if the input quantum state is a low-rank tensor and all intermediate states of the quantum algorithm can be represented or approximated by low-rank tensors. In this paper, we examine the possibility of simulating a few quantum algorithms by using low-rank canonical polyadic (CP) decomposition to represent the input and all intermediate states of these algorithms. Two rank reduction algorithms are used to enable efficient simulation. We show that some of the algorithms preserve the low-rank structure of the input state and can thus be efficiently simulated on a classical computer. However, the rank of the intermediate states in other quantum algorithms can increase rapidly, making efficient simulation more difficult. To some extent, such difficulty reflects the advantage or superiority of a quantum computer over a classical computer. As a result, understanding the low-rank structure of a quantum algorithm allows us to identify algorithms that can benefit significantly from quantum computers.