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arXiv:2401.12305 · Physical Review Letters

Strong Zero Modes in Integrable Quantum Circuits

Eric Vernier, Hsiu-Chung Yeh, Lorenzo Piroli, Aditi Mitra

Published Aug 2, 2024• 19 citations
arXivHTMLPDF

Abstract

It is a classic result that certain interacting integrable spin chains host robust edge modes known as strong zero modes (SZMs). In this Letter, we extend this result to the Floquet setting of local quantum circuits, focusing on a prototypical model providing an integrable Trotterization for the evolution of the XXZ Heisenberg spin chain. By exploiting the algebraic structures of integrability, we show that an exact SZM operator can be constructed for these integrable quantum circuits in certain regions of parameter space. Our construction, which recovers a well-known result by Paul Fendley in the continuous-time limit, relies on a set of commuting transfer matrices known from integrability, and allows us to easily prove important properties of the SZM, including normalizabilty. Our approach is different from previous methods and could be of independent interest even in the Hamiltonian setting. Our predictions, which are corroborated by numerical simulations of infinite-temperature autocorrelation functions, are potentially interesting for implementations of the XXZ quantum circuit on available quantum platforms.

DOI
10.1103/physrevlett.133.050606

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It is a classic result that certain interacting integrable spin chains host robust edge modes known as strong zero modes (SZMs). In this Letter, we extend this result to the Floquet setting of local quantum circuits, focusing on a prototypical model providing an integrable Trotterization for the evolution of the XXZ Heisenberg spin chain. By exploiting the algebraic structures of integrability, we show that an exact SZM operator can be constructed for these integrable quantum circuits in certain regions of parameter space. Our construction, which recovers a well-known result by Paul Fendley in the continuous-time limit, relies on a set of commuting transfer matrices known from integrability, and allows us to easily prove important properties of the SZM, including normalizabilty. Our approach is different from previous methods and could be of independent interest even in the Hamiltonian setting. Our predictions, which are corroborated by numerical simulations of infinite-temperature autocorrelation functions, are potentially interesting for implementations of the XXZ quantum circuit on available quantum platforms.
Strong Zero Modes (SZMs) are (approximately) conserved quantities that result in (approximate) double degeneracies in the entire spectra of certain Hamiltonians, with the Majorana zero mode of the transverse-field Ising chain being a primary example. In this work, we discover via a systematic search that many examples of SZMs can be understood as symmetries in the commutant algebra framework, which reveals novel algebraic structures hidden in Hamiltonians with well-known SZMs, including the transverse-field Ising chain. Our findings unify the understanding of different examples of SZMs in the literature, demystify their connections to ground state phases of matter, and reveal novel symmetries in simple models, such as exact quasilocal U(1)U(1)U(1) symmetries that sometimes accompany the SZMs such as in the spin-1/2 XY model for certain parameter values. Moreover, while analytically tractable SZMs have mostly been demonstrated only for non-interacting or integrable models, the algebraic structures revealed in this work can be exploited to construct integrability-breaking interactions that exactly preserve these SZMs. Such non-integrable models are expected to show more clear dynamical signatures of SZMs without the interference of other conserved quantities that appear in integrable models, and we discuss many examples, including those of novel hydrodynamic modes associated with such symmetries for some special parameter values. We also show that while this commutant understanding extends to the non-interacting limit of the celebrated Fendley SZM in the spin-1/2 XYZ chain, the SZM in the interacting case cannot be understood in this framework. This suggests that there are two types of SZMs -- those that survive integrability breaking and those that do not. We close by using this commutant understanding to construct an alternate proof of the Fendley SZM, which might be of independent interest.
Strong zero modes (SZMs) are edge-localized operators that commute with the Hamiltonian up to corrections exponentially small in system size, yielding anomalously long edge coherence times. In some settings, notably certain integrable models, this commutator can be made to vanish exactly at finite size, defining an exact SZM (ESZM). Existing ESZM constructions in the integrable setting, however, have proceeded model by model and have not been unified into a common framework. Here, I show that ESZMs arise generically in a broad family of integrable spin models with anisotropic interactions. Their existence follows from two algebraic properties of the underlying R- and K-matrices -- quasi-periodicity in the spectral parameter and tracelessness, respectively -- providing a uniform, model-independent mechanism. The framework recovers the known ESZM in XXZ chain and its higher-spin generalizations as special cases and predicts ESZMs in previously unrecognized models.
We construct exact strong zero mode operators (ESZM) in integrable quantum circuits and the spin-1/2 XXZ chain for general open boundary conditions, which break the bulk U(1) symmetry of the time evolution operators. We show that the ESZM is localized around one of the boundaries and induces infinite boundary coherence times. Finally, we prove that the ESZM becomes spatially non-local under the map that relates the spin-1/2 XXZ chain to the asymmetric simple exclusion process, which suggests that it does not play a significant role in the dynamics of the latter.
We investigate discrete dynamics in quantum circuits with gates corresponding to the S-matrix of a supersymmetric 1+1D quantum field theory. We show that for a brick-wall configuration such circuits support both localized and delocalized dynamically conserved operators known as strong zero modes (SZM), the number of which depends on the parameter regime. We demonstrate that, while some of the SZM remain localized at boundaries, other SZM propagate ballistically, guided by a choice of circuit parameters. Such propagation can be explained by a strong Dzyaloshinskii Moriya term appearing in the dynamics. We describe how to exploit propagating SZM for quantum information transport and discuss the robustness to various types of noise.
Strong zero modes (SZMs) are conserved operators localized at the edges of certain quantum spin chains, which give rise to long coherence times of edge spins. Here we define and analyze analogous operators in one-dimensional classical stochastic systems. For concreteness, we focus on chains with single occupancy and nearest-neighbor transitions, in particular particle hopping and pair creation and annihilation. For integrable choices of parameters we find the exact form of the SZM operators. Being in general nondiagonal in the classical basis, the dynamical consequences of stochastic SZMs are very different from those of their quantum counterparts. We show that the presence of a stochastic SZM is manifested through a class of exact relations between time-correlation functions, absent in the same system with periodic boundaries.
We derive exact strong zero mode (ESZM) operators for integrable spin-S chains with open boundary conditions and a boundary field. Their locality properties are generally weaker than in the previously known cases, but they still imply infinite coherence times in the vicinity of the edges. We explain how such integrable chains possess multiple ground states describing a first-order quantum phase transition, and that the odd number of such states for integer S makes the weaker locality properties necessary. We make contact with more traditional approaches by showing how the ESZM for S=1/2 acts on energy eigenstates given by solutions of the Bethe equations.