Single T gate in a Clifford circuit drives transition to universal entanglement spectrum statistics
Shiyu Zhou, Zhicheng Yang, Alioscia Hamma, Claudio Chamon
Abstract
Clifford circuits are insufficient for universal quantum computation or creating t <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>t</mml:mi> </mml:math> -designs with t\ge 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . While the entanglement entropy is not a telltale of this insufficiency, the entanglement spectrum of a time evolved random product state 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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>π</mml:mi> <mml:mi>/</mml:mi> <mml:mn>8</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> gate in the middle of a random Clifford circuit is sufficient to alter the entanglement spectrum from a Poisson to a Wigner-Dyson distribution.