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

### Submission summary

 As Contributors: Alioscia Hamma · Shiyu Zhou Preprint link: scipost_202004_00048v2 Date submitted: 2020-11-03 06:07 Submitted by: Zhou, Shiyu Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approach: Theoretical

### 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.

###### Current status:
Editor-in-charge assigned

We thank the referees for their helpful comments and suggestions, and have revised our paper accordingly.

### List of changes

We made changes throughout our paper to address referees' comments, and we list the major ones below:

- To avoid any confusion, we removed the differential equation associated to Eq. 2, and the discussion of its flows to the fixed points.
- We modified the scaling exponent $\alpha$ from 0.5 to 0.6 in Eq. 6 for $U'_{\rm Cl} = U^{-1}_{\rm Cl}$ case. An exponent of $\alpha = 0.6$ is consistent with the numerical data, and we realized upon addressing the referee 1's question that this value provides a better fit than the original 0.5.
- We modified discussions related to Fig. 8 in the Conclusions to clarify our proposal of an alternative construction of quantum circuits by concatenating segments of Clifford evolutions with very few T gates inserted at largely spaced layers, upon the request of referee 1.
- We revised last paragraph in the Conclusions to clarify our conjectures upon the request of referee 1.
- We added a new study of the time dependence of $\langle \widetilde{r} \rangle$ past the insertion of the T gates, in a 1D bit array, in answering referee 2's question of how long it takes to reach the GUE limit for $\langle \widetilde{r} \rangle$. We added 1 paragraph at the end of the Introduction, 1 paragraph at Section 3.2, and a new Fig. 3.
- We added a cartoon picture in Fig. 1 (right panel) of how the ES switches from Poisson to Wigner- Dyson distributed with the spreading of the downstream effects of the inserted T gates.

### Submission & Refereeing History

Resubmission scipost_202004_00048v2 on 3 November 2020
Submission scipost_202004_00048v1 on 19 April 2020

## Reports on this Submission

### Report

The authors have significantly extended and improved their paper. However, I still have some concerns about one aspect of their interpretation of their results, brought up in their response to my previous report (and also discussed in the updated version of the paper). They claim that the time scale for saturating to WD statistics (after the insertion of the T gates) is given by the time needed for the T gates to spread over the entire system. I have various issues with this claim:
- The authors motivate this conjecture by pointing to Ref. 15 of their paper. However, in that reference, the connection to operator spreading arises when considering a subsystem with two edges; the time scale is related to when the edges cease to be independent. For a subsystem consisting of half of an open chain (which I believe is the setup considered in the present paper, although I couldn't find an explicit statement to that effect) the more relevant reference would be Ref. 16, where a saturation to RMT statistics after an O(1) time was observed.
- If the conjecture is correct, then the time needed to achieve WD statistics for n_T = O(1) T-gates is infinite in the thermodynamic limit. Does that mean that there is an issue with the order of limits taken? I.e. that the statement of the paper applies only if the long-time limit is taken first, before the thermodynamic limit? What is the value obtained in the opposite (arguably more physical) limit?

It seems to me that since these issues concern what is arguably the central claim of the paper, they should be clarified before publication. In particular, seeing data for the complementary situation compared to figure 3 (i.e. keeping n_T fixed but varying N) would be helpful in that regard.

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### Report

The authors have significantly clarified the text and I am happy to recommend publication. The paper demonstrates some interesting phenomena that may spur further studies.

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