Approximate Unitary 3-Designs from Transvection Markov Chains

• URL: http://arxiv.org/abs/2011.00128v2
• Date: Tue, 25 May 2021 20:55:48 GMT
• Title: Approximate Unitary 3-Designs from Transvection Markov Chains
• Authors: Xinyu Tan, Narayanan Rengaswamy, and Robert Calderbank
• Abstract summary: We construct a Markov process that mixes a Kerdock $2$-design with symplectic transvections, and show that this process produces an $epsilon$-approximate unitary $3$-design. From a hardware perspective, $2$-qubit transvections exactly map to the Molmer-Sorensen gates that form the native $2$-qubit operations for trapped-ion quantum computers.
• Score: 3.0586855806896045
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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