Deterministic Twirling with Low Resources

TL;DR

This paper introduces an algebraic method for efficiently implementing twirling operations in quantum systems using minimal resources, enabling faster and resource-efficient quantum state averaging.

Contribution

It provides a general algebraic framework for selecting few-unitary sets to perform twirling, including a complete classification for two qubits and a generic three-unitary set for two-qudit systems.

Findings

Efficient twirling with minimal unitaries achieved exponentially quickly.

Complete classification of twirling sets for two qubits.

Constructed a universal three-unitary set for two-qudit twirling.

Abstract

Twirling operations, which average a quantum state with respect to a unitary subgroup, have become a frequently-employed tool in quantum information processing. We investigate the efficient implementation of twirling operations with minimal resources, without necessitating the ability to perform all possible unitary operations on the quantum system of interest. We present a general algebraic method allowing us to choose a set of - typically very few - unitary operators which, when applied randomly and repeatedly, produce the given twirling operation exponentially quickly. The method is applied to twirling operations for bipartite quantum systems with respect to the unitary group $U(d)\otimes U(d)$, an essential ingredient in entanglement distillation protocols. In particular, we provide a complete classification of sets of unitary operators capable of performing twirling on two qubits. Moreover, we construct a generic set containing at most three unitary operators achieving the twirling operation for a general two-qudit system.