Low-rank tensor decompositions of quantum circuits

TL;DR

This paper explores how low-rank tensor decompositions, specifically matrix product states and operators, can efficiently represent and simulate complex quantum circuits on classical computers, revealing their structure and reducing computational complexity.

Contribution

It introduces a tensor-based framework for representing quantum circuits using MPSs and MPOs, improving simulation efficiency over traditional methods.

Findings

MPO representations can simulate certain quantum circuits more efficiently.

Low bond dimensions maintain computational tractability.

Tensor decompositions reveal underlying quantum circuit structures.

Abstract

Quantum computing is arguably one of the most revolutionary and disruptive technologies of this century. Due to the ever-increasing number of potential applications as well as the continuing rise in complexity, the development, simulation, optimization, and physical realization of quantum circuits is of utmost importance for designing novel algorithms. We show how matrix product states (MPSs) and matrix product operators (MPOs) can be used to express certain quantum states, quantum gates, and entire quantum circuits as low-rank tensors. This enables the analysis and simulation of complex quantum circuits on classical computers and to gain insight into the underlying structure of the system. We present different examples to demonstrate the advantages of MPO formulations and show that they are more efficient than conventional techniques if the bond dimensions of the wave function representation can be kept small throughout the simulation.