Property testing of unitary operators

TL;DR

This paper develops efficient property testing algorithms for specific classes of unitary operators, such as the Clifford group and quantum juntas, using a new distance measure, with applications to permutation groups.

Contribution

Introduces a new distance measure for unitary operators and provides efficient testing algorithms for the Clifford group, quantum juntas, and finite subsets of the unitary group.

Findings

Testing algorithms have query complexities independent of system size.

Clifford group and quantum juntas can be efficiently tested with one-sided error.

An algorithm for testing finite subsets of the unitary group with polynomial query complexity.

Abstract

In this paper, we systematically study property testing of unitary operators. We first introduce a distance measure that reflects the average difference between unitary operators. Then we show that, with respect to this distance measure, the orthogonal group, quantum juntas (i.e. unitary operators that only nontrivially act on a few qubits of the system) and Clifford group can be all efficiently tested. In fact, their testing algorithms have query complexities independent of the system's size and have only one-sided error. Then we give an algorithm that tests any finite subset of the unitary group, and demonstrate an application of this algorithm to the permutation group. This algorithm also has one-sided error and polynomial query complexity, but it is unknown whether it can be efficiently implemented in general.