[Paper Review] Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps

Tamás Gombor, Balázs Pozsgay|arXiv (Cornell University)|Dec 3, 2021

Quantum many-body systemsPhysics and Astronomy71 references43 citations

TL;DR

This paper constructs superintegrable classical cellular automata using Yang-Baxter maps as local update rules, demonstrating an exponentially large set of conserved local charges that propagate ballistically without operator spreading. The key contribution is establishing a direct link between non-degenerate Yang-Baxter maps and classical dual unitary gates, revealing a new class of solvable models with rich physical behavior including coexistence of ballistic and diffusive transport.

ABSTRACT

We consider one dimensional block cellular automata, where the local update rules are given by Yang-Baxter maps, which are set theoretical solutions of the Yang-Baxter 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 non-trivial 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 Yang-Baxter maps are non-degenerate. We discuss consequences of dual unitarity, and we also discuss a family of dual unitary gates obtained by a non-integrable quantum mechanical deformation of the Yang-Baxter maps.

Motivation & Objective

• To construct classical block cellular automata (BCA) using Yang-Baxter maps as local update rules.
• To investigate the integrability and conservation laws in such systems, particularly the existence of exponentially many local conserved charges.
• To establish a correspondence between non-degenerate Yang-Baxter maps and classical dual unitary gates.
• To explore quantum deformations of these classical models and their relation to dual unitary quantum circuits.
• To characterize the dynamical complexity of the models via orbit lengths and identify constraints in integrable behavior.

Proposed method

• Use set-theoretical solutions of the Yang-Baxter equation (Yang-Baxter maps) as local update rules for one-dimensional block cellular automata.
• Demonstrate that these maps generate an exponentially large set of local conserved charges whose densities propagate ballistically without spreading.
• Define non-degenerate Yang-Baxter maps as the classical analog of dual unitary quantum gates, showing equivalence to permutation models via non-local similarity transformations.
• Construct quantum deformations of the classical models using dressed dual unitary gates via single-site unitaries and phase matrices.
• Analyze orbit lengths in finite systems to assess dynamical complexity, distinguishing between space-reflection symmetric and asymmetric models.
• Use explicit classification and enumeration of Yang-Baxter maps up to N=4 to study concrete models and their transport properties.

Experimental results

Research questions

• RQ1 Do Yang-Baxter maps as update rules in block cellular automata lead to superintegrability with exponentially many conserved local charges?
• RQ2 How is the classical dual unitary property related to non-degenerate Yang-Baxter maps, and what are its dynamical consequences?
• RQ3 Can quantum mechanical deformations of these classical models preserve integrability while breaking superintegrability?
• RQ4 What is the role of space-reflection symmetry in determining the maximal orbit length and dynamical complexity?
• RQ5 Are there integrable cellular automata that do not originate from Yang-Baxter maps, and what does this imply for the broader class of integrable systems?

Key findings

• Block cellular automata constructed from Yang-Baxter maps exhibit superintegrability with an exponentially large set of local conserved charges that propagate ballistically without operator spreading.
• Non-degenerate Yang-Baxter maps are shown to be the classical analog of dual unitary quantum gates, and their dynamics are equivalent to a permutation model via a non-local similarity transformation.
• For space-reflection symmetric models up to local dimension N=4, the maximal orbit length grows polynomially with system size, while dual unitary models exhibit maximal orbit length exactly equal to L.
• The models display rich physical behavior, including coexistence of ballistic and diffusive transport, as seen in the XXC model of type 2+2, which is proposed as a toy model for diffusive transport.
• Quantum deformations of the classical models via dressed dual unitary gates preserve integrability but break superintegrability, yielding a broader family of dual unitary gates.
• Counterexamples to polynomial orbit growth exist only when space-reflection symmetry is broken, highlighting the role of symmetry in constraining dynamics.