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

## TL;DR

Introducing a single T gate into a Clifford circuit transitions its entanglement spectrum from Poisson to Wigner-Dyson statistics, indicating a move towards universal quantum behavior.

## Contribution

Demonstrates that adding one T gate in a Clifford circuit suffices to change its entanglement spectrum to universal statistics in the thermodynamic limit.

## Key findings

- Single T gate induces spectrum transition in large systems.
- Entanglement spectrum shifts from Poisson to Wigner-Dyson.
- Finite-size scaling confirms the transition.

## 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.