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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|
|Preprint Link:||scipost_202004_00048v1 (pdf)|
|Date submitted:||2020-04-19 02:00|
|Submitted by:||Zhou, Shiyu|
|Submitted to:||SciPost Physics|
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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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:scipost_202004_00048v1, delivered 2020-05-29, doi: 10.21468/SciPost.Report.1722
The paper considers the question of how a non-universal, so-called Clifford circuit evolution crosses over to more generic dynamical behavior when a small number of additional non-Clifford gates are added to the circuit. In particular, the authors consider the behavior of the level statistics of the entanglement spectrum (ES), which has been shown previously to distinguish between Clifford dynamics (Poisson statistics) and chaotic evolution (Wigner-Dyson). They consider adding a single layer of local rotations that lie outside the Clifford group, preceded and followed by many layers of Clifford gates. They show numerical evidence that in the thermodynamic limit, even a single such rotations is sufficient to drive the transition to Wigner-Dyson statistics, showcasing an extreme sensitivity of the Clifford behavior.
I have found the basic question of the paper to be quite interesting and I believe it is of importance for both the quantum dynamics and the quantum computation communities. Moreover, I found the manuscript to be quite well written and easy to read, although slightly vague at times (in some cases it is unclear if certain statements are meant only to capture some intuition or something more rigorous). On the other hand, the actual results contained in the paper are somewhat limited. The authors consider only one particularly sensitive aspect of the dynamics, and even there they provide a relatively small amount of details from numerical calculations, while an analytical understanding, or a more thorough numerical exploration is missing. As such, I believe that the authors should elaborate on various points before the paper can be published.
In particular, there were quite a few questions that came to mind while reading the manuscript that are not addressed:
- The most striking statement, that a single T gate is enough to drive the transition, is somewhat insufficiently supported. The authors propose a scaling form as a function of the number of T gates (n_T) and the total system size (N), but it is unclear whether such a scaling form is meaningful in the limit when n_T = 1. I would find it very useful to have some data on how the level statistics changes when n_T is fixed and N is increased. Also, it would be good to see the un-scaled data from Figs. 3 and 5 for comparison.
- How long does it take for the ES statistics to saturate to its Wigner-Dyson form after the T gates are applied? What can be said about the way this limit is approached?
- How do the results depend on the time when the T gates are applied? What if they occur before the total entanglement saturates (as would be relevant in the thermodynamic limit)?
- Fig. 2 presents a color plot of the time-evolved wavefunction in various cases. While this is visually appealing, its physical meaning is quite unclear. Is there any meaningful quantitative statement one could make about this object? Has it been studied before? How much does the apparent structure in the Clifford case depend on the choice of basis?
- It would be very useful to have at least some qualitative analytical understanding of why there should be a scaling collapse of the form n_T * N.
A few other minor points:
- On p. 3 the definitions of S and T gates are the same. I believe the former is a typo.
- It would be good to state the number of Clifford layers applied in the numerical simulations.
- There is a repeated sentence in the next to last paragraph before the conclusions: "replacing the specific Z_ik..."
- Cite as: Anonymous, Report on arXiv:scipost_202004_00048v1, delivered 2020-05-09, doi: 10.21468/SciPost.Report.1671
Clifford circuits are a restricted class of quantum circuits. They are not sufficient for universal computation (their action on a restricted class of initial states can be simulated classically) but nevertheless they are important in quantum information, for example as starting points for the addition of extra gates to give universality. This paper studies an aspect of this crossover from Clifford circuits to more generic behavior on addition of a small number of non-Clifford gates.
This is an interesting question, and the paper provides some interesting results from computer simulations, showing that one quantity (r value of entanglement levels) returns to generic behavior even with only a single non-Clifford gate. They also use this to motivate a conjecture for more general quantities, albeit without much detail.
On the other hand, the paper is limited to this numerical obervation, with limited extensions. An analysis which might give understanding of the observation is lacking. The manuscript is also vague or imprecise in several places as described below. Therefore, to the extent that I can gauge the level required for a SciPost acceptance this paper in its present form is not substantial enough to meet this level.
Discussion regarding Figure 2: This is very qualitative. For example we find the extremely vague statement “adding T gates into Clifford intrinsically alters Clifford circuit’s computational power, and drives the system toward randomness”.
Similarly in the introduction “The Clifford group generates a good approximation of a 4-design” is explained as “In other words, it takes very little for the Clifford group to become something that is capable of reproducing the fluctuations of observables evolved with a universal quantum circuit.” This is very vague. The second sentence does not explain the meaning of the first.
Equation 1: It is claimed that this is a universal scaling form. The scaling variable is claimed to be n N where n is the number of T gates and N is the number of qubits. This does not sound quite correct. In order to have a finite (”order 1”) value of the scaling variable we see that N must also be finite and “order 1”. But it is unlikely a universal scaling form can apply for a finite, order 1 number of qubits. More likely, universality only holds in the limit of asymptotically large scaling variable. This is not what is usually meant by a “scaling form”. This also applies to the comment about DKL on p5.
Bottom of p4: The authors talk of a “poisson fixed point”. It is not clear in what sense this is meant. Is there an RG scheme in mind?
p6: “this example gives the noise threshold for reversibility”. It is not clear from the text whether this a conjecture or a confirmed result.
Eq 3: The objection above does not apply to this scaling form, because it is possible to take N to infinity while fixing the value of the scaling variable. However, the evidence for this scaling form from Fig 5 is not obviously 100% conclusive. For example the absence of a theoretical argument one could imagine that the right power of N is not exactly 0.5.
p7: Here we find an analytical result for the case where the two Clifford unitaries are inverse. It is nice to have a partial analytical understanding but it would be desirable to understand how many T gates are needed. Can the authors derive the sqrt N that they assert?
p8: “this result points to a new direction…” The authors suggest that their result implies a compressibility property of circuits. Why should this be true? Can the authors provide an argument, or is it a speculation?
Final paragraph of paper: A conjecture is stated but the motivation for this conjecture is compressed to a very few lines. Can the logic be made clearer?
Abstract: “the entanglement spectrum” could be more specific, e.g. “of a time-evolved random product state”
Introduction p1: “derandomization” is mentioned. Could it be explained what this means?
p2. In the initial definition of the protocol “random product states” are mentioned but further definition of these states is required. For example one type of random product state would also be a stabilizer state and this is not what the authors are talking about in this definition.
I cannot find a specification of the depth of the circuits used for the data.
Bottom of p3. Define xA and xB.
Middle of p5. “the full distribution of the ES”. The distribution of the quantity r is meant, not of the full spectrum.
Below eq 4. “essentially random Pauli strings”. During the evolution these strings retain properties of commutativity and independence.
p7 “these results indicate”. The content of this sentence and the next sentence are repeated in the following two.