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Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics

by Shiyu Zhou, Zhi-Cheng Yang, Alioscia Hamma, Claudio Chamon

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Authors (as registered SciPost users):

Alioscia Hamma · Shiyu Zhou

Abstract

Clifford circuits are insufficient for universal quantum computation or creating $t$-designs with $t\ge 4$. While the entanglement entropy is not a telltale of this insufficiency, the entanglement spectrum is: the entanglement levels are Poisson-distributed for circuits restricted to the Clifford gate-set, while the levels follow Wigner-Dyson statistics when universal gates are used. In this paper we show, using finite-size scaling analysis of different measures of level spacing statistics, that in the thermodynamic limit, inserting a single T $(\pi/8)$ gate in the middle of a random Clifford circuit is sufficient to alter the entanglement spectrum from a Poisson to a Wigner-Dyson distribution.

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