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arXiv:2106.07686 · Physical Review Research

Triunitary quantum circuits

Cheryne Jonay, Vedika Khemani, Matteo Ippoliti

Published Oct 18, 2021• 52 citations
arXivHTMLPDF

Abstract

We introduce a class of quantum circuits that are unitary along three distinct ``arrows of time.'' These dynamics share some of the analytical tractability of ``dual-unitary'' circuits, while exhibiting distinctive and richer phenomenology. We find that two-point correlations in these dynamics are strictly confined to three directions in $(1+1)$-dimensional spacetime---the two light cone edges, $\ensuremath{\delta}x=\ifmmode\pm\else\textpm\fi{}v\ensuremath{\delta}t$, and the static worldline $\ensuremath{\delta}x=0$. Along these directions, correlation functions are obtained exactly in terms of quantum channels built from the individual gates that make up the circuit. We prove that, for a class of initial states, entanglement grows at the maximum allowed speed up to an entropy density of at least one half of the thermal value, at which point it becomes model-dependent. Finally, we extend our circuit construction to $2+1$ dimensions, where two-point correlation functions are confined to the one-dimensional edges of a tetrahedral light cone---a subdimensional propagation of information reminiscent of ``fractonic'' physics.

DOI
10.1103/physrevresearch.3.043046

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We introduce a class of quantum circuits that are unitary along three distinct arrows of time.'' These dynamics share some of the analytical tractability of dual-unitary'' circuits, while exhibiting distinctive and richer phenomenology. We find that two-point correlations in these dynamics are strictly confined to three directions in (1+1)(1+1)(1+1)-dimensional spacetime---the two light cone edges, \ensuremathδx=\ifmmode±\else\textpm\fiv\ensuremathδt\ensuremath{\delta}x=\ifmmode\pm\else\textpm\fi{}v\ensuremath{\delta}t\ensuremathδx=\ifmmode±\else\textpm\fiv\ensuremathδt, and the static worldline \ensuremathδx=0\ensuremath{\delta}x=0\ensuremathδx=0. Along these directions, correlation functions are obtained exactly in terms of quantum channels built from the individual gates that make up the circuit. We prove that, for a class of initial states, entanglement grows at the maximum allowed speed up to an entropy density of at least one half of the thermal value, at which point it becomes model-dependent. Finally, we extend our circuit construction to 2+12+12+1 dimensions, where two-point correlation functions are confined to the one-dimensional edges of a tetrahedral light cone---a subdimensional propagation of information reminiscent of ``fractonic'' physics.
We devise tractable models of unitary quantum many-body dynamics on tree graphs, as a first step toward a deeper understanding of dynamics in non-Euclidean spaces. To this end, we first demonstrate how to construct strictly local quantum circuits that preserve the symmetries of trees, such that their dynamical light cones grow isotropically. For trees with coordination number z , such circuits can be built from z -site gates. We then introduce a family of gates for which the dynamics are exactly solvable; these satisfy a set of constraints that we term “tree-unitarity.” Notably, tree-unitarity reduces to the previously established notion of dual-unitarity for z = 2 , when the tree reduces to a line. Among the unexpected features of tree-unitarity is a trade-off between “maximum butterfly velocity” dynamics of out-of-time-order correlators and the existence of nonvanishing correlation functions in multiple directions, a tension absent in one-dimensional dual-unitary models and their Euclidean generalizations. We connect the existence of (a wide class of) solvable dynamics with nonmaximal butterfly velocity directly to a property of the underlying circuit geometry called δ -hyperbolicity, and argue that such dynamics can only arise in non-Euclidean geometries. We give various examples of tree-unitary gates, discuss dynamical correlations, out-of-time-order correlators, and entanglement growth, and show that the kicked Ising model on a tree is a physically motivated example of maximum-velocity tree-unitary dynamics.
We present a general framework for constructing solvable lattice models of chaotic many-body quantum dynamics with multiple unitary directions using biunitary connections. We show that a network of biunitary connections on the kagome lattice naturally defines a multi-unitary circuit, where three ‘arrows of time’ directly reflect the lattice symmetry. These models unify various constructions of hierarchical dual-unitary and triunitary gates and present new families of models with solvable correlations and entanglement dynamics. Using multilayer constructions of biunitary connections, we additionally introduce multilayer circuits with monoclinic symmetry and higher level hierarchical dual-unitary solvability and discuss their (non-)ergodicity. Our work demonstrates how different classes of solvable models can be understood as arising from different geometric structures in spacetime.
Dual-unitary quantum circuits have recently attracted attention as an analytically tractable model of many-body quantum dynamics. Consisting of a 1+1D lattice of 2-qudit gates arranged in a 'brickwork' pattern, these models are defined by the constraint that each gate must remain unitary under swapping the roles of space and time. This dual-unitarity restricts the dynamics of local operators in these circuits: the support of any such operator must grow at the effective speed of light of the system, along one or both of the edges of a causal light cone set by the geometry of the circuit. Using this property, it is shown here that for 1+1D dual-unitary circuits the set of width- w conserved densities (constructed from operators supported over w consecutive sites) is in one-to-one correspondence with the set of width- w solitons – operators which, up to a multiplicative phase, are simply spatially translated at the effective speed of light by the dual-unitary dynamics. A number of ways to construct these many-body solitons (explicitly in the case where the local Hilbert space dimension d=2 ) are then demonstrated: firstly, via a simple construction involving products of smaller, constituent solitons; and secondly, via a construction which cannot be understood as simply in terms of products of smaller solitons, but which does have a neat interpretation in terms of products of fermions under a Jordan-Wigner transformation. This provides partial progress towards a characterisation of the microscopic structure of complex many-body solitons (in dual-unitary circuits on qubits), whilst also establishing a link between fermionic models and dual-unitary circuits, advancing our understanding of what kinds of physics can be explored in this framework.
Quantum dynamics with local interactions in lattice models display rich physics, but is notoriously hard to study. Dual-unitary circuits allow for exact answers to interesting physical questions in clean or disordered one- and higher-dimensional quantum systems. However, this family of models shows some non-universal features, like vanishing correlations inside the light-cone and instantaneous thermalization of local observables. In this work we propose a generalization of dual-unitary circuits where the exactly calculable spatial-temporal correlation functions display richer behavior, and have non-trivial thermalization of local observables. This is achieved by generalizing the single-gate condition to a hierarchy of multi-gate conditions, where the first level recovers dual-unitary models, and the second level exhibits these new interesting features. We also extend the discussion and provide exact solutions to correlators with few-site observables and discuss higher-orders, including the ones after a quantum quench. In addition, we provide exhaustive parametrizations for qubit cases, and propose a new family of models for local dimensions larger than two, which also provides a new family of dual-unitary models.
In recent years dual-unitary circuits and their multi-unitary generalizations have emerged as exactly solvable yet chaotic models of quantum many-body dynamics. However, a systematic picture for the solvability of multi-unitary dynamics remains missing. We present a framework encompassing a large class of such non-integrable models with exactly solvable dynamics, which we term \emph{completely reducible} circuits. In these circuits, the entanglement membrane determining operator growth and entanglement dynamics can be characterized analytically. Completely reducible circuits extend the notion of space-time symmetry to more general lattice geometries, breaking dual-unitarity globally but not locally, and allow for a rich phenomenology going beyond dual-unitarity. As example, we introduce circuits that support four and five directions of information flow. We derive a general expression for the entanglement line tension in terms of the pattern of information flow in spacetime. The solvability is shown to be related to the absence of knots of this information flow, connecting entanglement dynamics to the Kauffman bracket as knot invariant. Building on these results, we propose that in general non-integrable dynamics the curvature of the entanglement line tension can be interpreted as a density of information transport. Our results provide a new and unified framework for exactly solvable models of many-body quantum chaos, encompassing and extending known constructions.
We study the nonequilibrium dynamics of kicked Ising models in 1 + 1 dimensions which have interactions alternating between odd and even bonds in time. These models can be understood as quantum circuits tiling space-time with the generalized space-time dual properties of triunitarity (three “arrows of time”) at the global level, and also second-level dual unitarity at the local level, which constrains the behavior of pairs of local gates underlying the circuit under a space-time rotation. We identify a broad class of initial product states wherein the effect of the environment on a small subsystem can be exactly represented by influence matrices with simple Markovian structures, resulting in the subsystem's full dynamics being efficiently computable. We further find additional conditions under which the dynamics of entanglement can be solved for all times, yielding rich phenomenology ranging from linear growth at half the maximal speed allowed by locality, followed by saturation to maximum entropy (i.e., thermalization to infinite temperature), to entanglement growth with saturation to extensive but submaximal entropy. Intriguingly, for certain parameter regimes, we find a nonchaotic class of dynamics which is neither integrable nor Clifford, exemplified by nonzero operator entanglement growth but with a spectral form factor which exhibits large, apparently time-quasiperiodic revivals.