Efficient Simulation of Random Quantum States and Operators

TL;DR

This paper presents efficient methods for simulating random quantum states and operators, including state generation, fidelity estimation, and channel depolarization, with practical circuit constructions and insights into the Clifford group.

Contribution

It introduces efficient circuits for generating random quantum states and operators, and characterizes the Clifford group's role in depolarizing channels, including a minimal subset for this task.

Findings

Maximal sets of mutually-unbiased bases can estimate average fidelity.

Efficient circuits for arbitrary state generation from these bases.

A small subset of the Clifford group suffices for channel depolarization.

Abstract

We investigate the generation of quantum states and unitary operations that are ``random'' in certain respects. We show how to use such states to estimate the average fidelity, an important measure in the study of implementations of quantum algorithms. We re-discover the result that the states of a maximal set of mutually-unbiased bases serve this purpose. An efficient circuit is presented that generates an arbitrary state out of such a set. Later on, we consider unitary operations that can be used to turn any quantum channel into a depolarizing channel. It was known before that the Clifford group serves this and a related purpose, and we show that these are actually the same. We also show that a small subset of the Clifford group is already sufficient to accomplish this. We conclude with an efficient construction of the elements of that subset. This thesis is based on joint work with Richard Cleve, Joseph Emerson, and Etera Livine.