Unitary $2$-designs from random $X$- and $Z$-diagonal unitaries

TL;DR

This paper demonstrates that approximate unitary 2-designs can be efficiently generated using alternating random diagonal unitaries in X and Z bases, with implications for quantum circuits and Hamiltonian dynamics.

Contribution

It introduces a novel method to implement approximate unitary 2-designs via alternating random diagonal unitaries and provides efficient quantum circuits with commuting gates.

Findings

Achieves a $ heta(d^{- ext{l}})$-approximate 2-design after $ ext{l}$ repetitions.

Provides quantum circuits with a constant number of commuting parts.

Shows that time-dependent Hamiltonians can generate 2-designs after few switchings.

Abstract

Unitary $2$-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols, and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary $2$-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-$Z$ basis and that in the Pauli-$X$ basis. We also provide a converse about the number of repetitions needed to achieve unitary $2$-designs. These results imply that the process after $\ell$ repetitions achieves a $\Theta(d^{-\ell})$-approximate unitary $2$-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary $2$-designs. Although a more efficient implementation of unitary $2$-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary $2$-design after switching interactions only a few times.