1. Approximate unitary 3-designs from transvection Markov chains.
- Author
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Tan, Xinyu, Rengaswamy, Narayanan, and Calderbank, Robert
- Subjects
MARKOV processes ,QUANTUM gates ,PAULI matrices ,UNITARY groups ,HAAR integral ,AUTOMORPHISM groups ,QUANTUM computers - 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 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 ϵ -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 / ϵ)) 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. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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