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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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Submission summary

Authors (as registered SciPost users): Alioscia Hamma · Shiyu Zhou
Submission information
Preprint Link: scipost_202004_00048v2  (pdf)
Date submitted: 2020-11-03 06:07
Submitted by: Zhou, Shiyu
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Quantum Physics
Approach: Theoretical


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.

Author comments upon resubmission

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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2020-11-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202004_00048v2, delivered 2020-11-15, doi: 10.21468/SciPost.Report.2191


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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Author:  Shiyu Zhou  on 2020-12-02  [id 1068]

(in reply to Report 2 on 2020-11-15)
answer to question
pointer to related literature

1). We point out that Ref. 16 finds that the level repulsion develops between consecutive non-zero eigenvalues of the reduced density matrix after $\mathcal{O}(1)$ time, when the reduced density matrix still has very low rank. The timescale for the full distribution of the entanglement spectrum to saturate to random matrix theory is set by the operator spreading, consistent with Ref. 15. In our work, past the first battery of Clifford gates but prior to the insertion of T gates, we have a maximally entangled state, with a full-ranked reduced density matrix and a Poisson-distributed entanglement spectrum, and then we look at the level spacing ratios of the full entanglement spectrum following the subsequent time evolution. Therefore, the average $\langle r \rangle$ will only reach that of a Wigner-Dyson distribution when a significant fraction of the entire entanglement spectrum has settled to random matrix theory. Hence, the relevant timescale in our setup should be controlled by the operator spreading, in agreement with Refs. 15 and 16.

2). We concur with the referee that there is indeed an issue with order of limits of time and system size; but the issue is no different from that encountered in phase ordering kinetics. For example, a ferromagnet quenched below its critical temperature will never equilibrate to a uniform magnetization state if the thermodynamic limit is taken first (see Ref. 1 below). The domain sizes of different broken symmetry states will grow with time, but would never reach the system size if the latter is infinite. As long as the domain size grows as a power law in time, one reaches a symmetry broken state in large but finite systems in experimentally observable times (polynomial in the system size). (Once the system settles into a symmetry broken state, reversing the magnetization requires time exponential in the system size). Therefore, as in phase ordering kinetics, to reach the conclusions in our paper one must take the limit of both time and system size to infinity but keeping a power law relation between time and size.

3). ** Figure is in the attached file**
As requested by the referee, we provide the data for $\langle \widetilde{r}(\tau) \rangle$ for the case in which we fix $n_T = 8$ and vary $N = 12, ~14, ~16, ~18$ (see the attached figure). We see that all four curves collapse to a single one when plotted against $\tau \, N^{-1}$, complementing our argument that the action of the inserted T gates is contained within a light-cone, such that each T gate covers a spatial region of size $\xi \sim \tau$ at time $\tau$. At fixed $n_T$, the larger the system size is, the longer time it needs for the effect of $n_T$ T gates to spread over the whole system.

[1] A. J. Bray, “Theory of phase-ordering kinetics,” Advances in Physics 43, 357–459 (1994).



Anonymous Report 1 on 2020-11-3 (Invited 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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