Approximate Unitary 3-Designs from Transvection Markov Chains

TL;DR

This paper introduces a method to construct approximate unitary 3-designs using a Markov process involving Kerdock codes and symplectic transvections, with potential applications in quantum computing hardware.

Contribution

The paper presents a novel Markov process that combines Kerdock 2-designs and transvections to efficiently produce approximate unitary 3-designs, linking classical codes to quantum designs.

Findings

Constructed a Markov process mixing Kerdock 2-designs with transvections

Achieved an $oldsymbol{ extit{ ext{epsilon}}}$-approximate unitary 3-design

Process involves $oldsymbol{O( ext{log}(N^5/ ext{epsilon}))}$ transvections on $m$ qubits

Abstract

Unitary $k$-designs are probabilistic ensembles of unitary matrices whose first $k$ statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical $\mathbb{Z}_4$-linear Kerdock codes maps to a unitary $2$-design, which established a new classical-quantum connection via graph states. In this paper, we construct a Markov process that mixes this Kerdock $2$-design with symplectic transvections, and show that this process produces an $\epsilon$-approximate unitary $3$-design. We construct a graph whose vertices are Pauli matrices, and two vertices are connected by directed edges if and only if they commute. A unitary ensemble that is transitive on vertices, edges, and non-edges of this Pauli graph is an exact $3$-design, and the stationary distribution of our process possesses this property. With respect to the symmetries of Kerdock codes, the Pauli graph has two types of edges; the Kerdock $2$-design mixes edges of the same type, and the transvections mix the types. More precisely, on $m$ qubits, the process samples $O(\log(N^5/\epsilon))$ random transvections, where $N = 2^m$, followed by a random Kerdock $2$-design element and a random Pauli matrix. Hence, the simplicity of the protocol might make it attractive for several applications. From a hardware perspective, $2$-qubit transvections exactly map to the M{\o}lmer-S{\o}rensen gates that form the native $2$-qubit operations for trapped-ion quantum computers. Thus, it might be possible to extend our work to construct an approximate $3$-design that only involves such $2$-qubit transvections.