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 Mølmer–Sørensen 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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Acknowledgements
We are grateful to Tanmay Singal and Min-Hsiu Hsieh for pointing out that we needed to define the Pauli graph as a directed graph in order to accurately analyze the Transvection Markov Chain. They also communicated that it is possible to improve the convergence analysis to \(O(\log (N^3/\epsilon ))\) [27]. The work of N. Rengaswamy and R. Calderbank was supported in part by the National Science Foundation (NSF) under Grant No. 1908730.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: The Art of Combinatorics—A Volume in Honour of Aart Blokhuis”.
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Tan, X., Rengaswamy, N. & Calderbank, R. Approximate unitary 3-designs from transvection Markov chains. Des. Codes Cryptogr. 90, 2181–2204 (2022). https://doi.org/10.1007/s10623-021-01000-4
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DOI: https://doi.org/10.1007/s10623-021-01000-4