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Superintegrable cellular automata and dual unitary gates from YangBaxter maps
by Tamás Gombor, Balázs Pozsgay
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Submission summary
Authors (as registered SciPost users):  Tamás Gombor · Balázs Pozsgay 
Submission information  

Preprint Link:  https://arxiv.org/abs/2112.01854v2 (pdf) 
Date accepted:  20220314 
Date submitted:  20220114 08:30 
Submitted by:  Pozsgay, Balázs 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We consider one dimensional block cellular automata, where the local update rules are given by YangBaxter maps, which are set theoretical solutions of the YangBaxter equations. We show that such systems are superintegrable: they possess an exponentially large set of conserved local charges, such that the charge densities propagate ballistically on the chain. For these quantities we observe a complete absence of "operator spreading". In addition, the models can also have other local charges which are conserved only additively. We discuss concrete models up to local dimensions $N\le 4$, and show that they give rise to rich physical behaviour, including nontrivial scattering of particles and the coexistence of ballistic and diffusive transport. We find that the local update rules are classical versions of the "dual unitary gates" if the YangBaxter maps are nondegenerate. We discuss consequences of dual unitarity, and we also discuss a family of dual unitary gates obtained by a nonintegrable quantum mechanical deformation of the YangBaxter maps.
Published as SciPost Phys. 12, 102 (2022)
Reports on this Submission
Anonymous Report 2 on 2022221 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2112.01854v2, delivered 20220221, doi: 10.21468/SciPost.Report.4491
Report
The authors study deterministic circuits that arise from YangBaxter maps: set theoretical solutions of the YangBaxter equation. They show that this structure implies the existence of ballistically propagating local operators that experience no broadening, which in turn imply a superexponentiallygrowing number of local conservation laws, of which an exponential number appears to be linearly independent. This is in contrast with usual integrable systems, where conservation laws are expected to grow linearly with their support. The authors also discuss the ergodicity properties of these systems, and the existence of higher commuting update rules. The next part of the paper is mostly devoted to constructions and examples of these cellular automata. Out of the full classification of integrable YangBaxter maps up to local configuration space N=4, they pick a few representative examples, and discuss the obvious mapping between the models. The authors also briefly show how the previous ideas can be applied to 3>1 models, and to quantum models. They conclude with a brief discussion of nonintegrable dualunitary gates.
The paper presents a collection of interesting results and represents an impressive step towards classification and construction of solvable circuits, which have been in the past mostly studied on the casebycase basis. Furthermore, the authors bring some more specialised mathematical literature to the attention of the wider community, and the discussion is clearly structured, which makes the article easy to follow. Therefore I strongly recommend this paper for publication in SciPost. In particular, I believe this article matches points 3 and 4 of the journal's acceptance criteria.
Requested changes
I have a few minor remarks:
1 Line after eq. (39): If I understand correctly $P_{j,j+1}$ is the operation in $X^L$ and not $S_L$.
2 Line before eq. (74): "The we" > "Then we"
3 Sec. 7: In a few places $\mathcal{V}$ is used instead of (what I think should be) $\mathbb{C}^N$, e.g. first line after eq. (83) and (84).
4 Paragraph after eq. (88), "We stress that this holds...": I suggest to make this sentence a bit clearer also for a more sloppy reader. Maybe removing "there" could help. To me it was not immediately clear that "but there..." refers to dual unitary models (i.e. the cases in which it holds).
Report 1 by Pieter W. Claeys on 2022216 (Invited Report)
 Cite as: Pieter W. Claeys, Report on arXiv:2112.01854v2, delivered 20220216, doi: 10.21468/SciPost.Report.4436
Strengths
Detailed analysis of the dynamics in cellular automata with local updates given by YangBaxter maps.
 Illustrative examples with nontrivial physics.
 New classes of cellular automata and dualunitary gates are presented.
Weaknesses
 Consequences of classical dualunitarity are only briefly discussed.
 Section 9, while interesting, is largely unrelated to the bulk of the paper.
Report
In this work the authors investigate cellular automata where the local update rules are given by YangBaxter maps satisfying the braid relation. For such cellular automata the local update rules can be mapped to elementary permutations and the resulting dynamics can be analyzed for the corresponding 'permutation system'. While this homomorphism can not be used to extract global information about the dynamics due to its breakdown at (periodic) boundaries, local information can be directly obtained starting from this simpler permutation system. The authors use this mapping to illustrate a lack of "operator spreading" and construct ballistically spreading local charge densities as well as commuting update rules. After a discussion of the notions of ergodicity/complexity and orbit lengths in classical cellular automata the authors discuss various constructions and present several illustrative and nontrivial examples. These examples are superintegrable and the orbit lengths are classified and shown to grow polynomially with system size. The authors also make clear which properties are expected for general YangBaxter cellular automata and which are specific to the presented examples. While the bulk of the paper focuses on classical cellular automata, it is also shown how YangBaxter maps give rise to quantum circuits acting as "integrable Trotterizations" of the corresponding integrable quantum spin chains.
A second aim of this paper is to relate the classical YangBaxter maps to (quantum) dualunitary gates. It is shown that 'nondegenerate' YangBaxter maps can be used to construct classical dualunitary gates, for which the mapping to the permutation system is compatible with the periodic boundary conditions — leading to a maximal orbital length that equals the system size. Furthermore, a new class of quantum dualunitary gates is constructed by an appropriate dressing of the classical gate.
As pointed out by the authors, the idea of combining YangBaxter maps with cellular automata is natural and relates to various topics of interest. The presented analysis and results are technically sound and sure to be of interest to anyone working in related fields. Furthermore, the paper is written in an extremely clear and accessible way, including various illustrative and pedagogical examples. While various ideas in this work already appeared in the literature, the authors consistently make clear which ideas are new and provide appropriate references and context otherwise.
Considering SciPost Physics' acceptance criteria, I believe this paper satisfies criteria 3) Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work; and 4) Provide a novel and synergetic link between different research areas. Criterion 3) is evidenced by the new cellular automata introduced in the second half of the paper, and criterion 4) by the aim of the paper to combine YangBaxter maps with cellular automata.
I very much enjoyed reading this paper and am happy to recommend it for publication in SciPost Physics. Some minor comments can be found below, but these are only meant for clarification and are optional.
Requested changes
As mentioned in my report, these are mainly meant for clarification.
1 In Section 3.3 it is argued that nondegenerate YangBaxter maps give rise to maps that are also deterministic when acting in the space direction — the classical equivalent of dualunitarity. Does the map acting in the space direction, i.e. the 'dual' of $U$, also satisfy the YangBaxter equation? There also exist dualunitary gates parametrized in terms of Latin squares, as discussed by one of the authors in another work. Are these related to the nondegenerate YangBaxter maps? It also seems as if the 'nondegeneracy' suffices to return a dualunitary map, even without the YangBaxter properties, but then the resulting dynamics no longer map to the permutation system. Can the authors comment on this? This would help to clarify the difference between the physical consequences of 'classical dualunitarity' and 'classical dualunitarity for YangBaxter maps'.
As a minor detail, in Eq. (38) the notation $R(x,y)$ is introduced, whereas otherwise the authors use $U(x,y)$.
2 When introducing the permutation system in Eq. (27), it might be useful to recall the initial definition of $\mathcal{V}=\mathcal{V}_2\mathcal{V}_1$. Even if $\Lambda(V)\neq \mathcal{V}$, this makes the motivation for Eq. (27) more explicit, which was initially not clear to me.
3 In Section 3.3. it is discussed how classical dualunitary maps can be mapped to the permutation system. Does this property relate to the 'exact solvability' of dualunitary gates in the quantum case? Is there any trace of the dualunitarity lightcone dynamics of onepoint correlation functions in the classical case?