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arXiv:2304.05718 · Physical review. B./Physical review. B

Interplay between lattice gauge theory and subsystem codes

Yoshihito Kuno, Ikuo Ichinose

Published Jul 31, 2023• 6 citations
arXivHTMLPDF

Abstract

It is now widely recognized that the toric code is a pure gauge-theory model governed by a projective Hamiltonian with topological orders. In this paper, we extend the interplay between quantum information system and gauge-theory model from the viewpoint of subsystem code, which is suitable for gauge systems including matter fields. As an example, we show that ${Z}_{2}$ lattice gauge-Higgs model in ($2+1$)-dimensions with specific open boundary conditions is nothing but a kind of subsystem code. In the system, Gauss-law constraints are stabilizers, and order parameters identifying Higgs and confinement phases exist and they are nothing but logical operators in subsystem codes residing on the boundaries. Mixed anomaly of them dictates the existence of boundary-zero modes, which is a direct consequence of symmetry-protected topological order in Higgs and confinement phases. After identifying phase diagram, subsystem codes are embedded in the Higgs and confinement phases. As our main findings, we give an explicit description of the code (encoded qubit) in the Higgs and confinement phases, which clarifies duality between Higgs and confinement phases. The degenerate structure of subsystem code in the Higgs and confinement phases remains even in very high-energy levels, which is analogous to notion of strong-zero modes observed in some interesting condensed-matter systems. Numerical methods are used to corroborate analytically-obtained results and the obtained spectrum structure supports the analytical description of various subsystem codes in the gauge theory phases.

DOI
10.1103/physrevb.108.045150

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It is now widely recognized that the toric code is a pure gauge-theory model governed by a projective Hamiltonian with topological orders. In this paper, we extend the interplay between quantum information system and gauge-theory model from the viewpoint of subsystem code, which is suitable for gauge systems including matter fields. As an example, we show that Z2{Z}_{2}Z2​ lattice gauge-Higgs model in (2+12+12+1)-dimensions with specific open boundary conditions is nothing but a kind of subsystem code. In the system, Gauss-law constraints are stabilizers, and order parameters identifying Higgs and confinement phases exist and they are nothing but logical operators in subsystem codes residing on the boundaries. Mixed anomaly of them dictates the existence of boundary-zero modes, which is a direct consequence of symmetry-protected topological order in Higgs and confinement phases. After identifying phase diagram, subsystem codes are embedded in the Higgs and confinement phases. As our main findings, we give an explicit description of the code (encoded qubit) in the Higgs and confinement phases, which clarifies duality between Higgs and confinement phases. The degenerate structure of subsystem code in the Higgs and confinement phases remains even in very high-energy levels, which is analogous to notion of strong-zero modes observed in some interesting condensed-matter systems. Numerical methods are used to corroborate analytically-obtained results and the obtained spectrum structure supports the analytical description of various subsystem codes in the gauge theory phases.
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This note extends the record-classification framework into a two-dimensional lattice gauge theory by studying pure Z2 gauge theory on finite tori. Using exact diagonalization in winding sectors, the work tests how quantum information is stored, distributed, and destroyed across different gauge phases. The classification survives and becomes sharper. The constraint record becomes a topological redundancy: any enclosing loop can recover the enclosed charge parity exactly at every coupling and in every phase, as a direct consequence of Gauss's law. String records appear only where confinement localizes flux into physical strings. The toric-code sector qubit forms a matter-blind record class, preserved in the topological phase but destroyed by confinement. Across the phase diagram, different records migrate or disappear, but the constraint record remains invariant. The study also identifies an important boundary case: antipodally placed charges do not produce a missing record, but an undefined local record because degenerate winding routes prevent a unique assignment. The results show that quantum records in gauge theories are determined by the underlying constraint structure and topology. Some information becomes redundantly accessible through global constraints, some becomes localized through confinement, and some survives only as nonlocal topological information. All results are obtained through exact finite-lattice calculations with documented verification checks and reproducibility data.
We extensively study long-time dynamics and the fate of a topologically ordered state in the toric code model evolving through a projective measurement-only circuit. The circuit is composed of several measurement operators corresponding to each term of the toric code Hamiltonian with magnetic-field perturbations, which is a gauge-fixed version of a (2+12+12+1)-dimensional gauge Higgs model. We employ a cylinder geometry with distinct upper and lower boundaries to classify stationary states after long-time measurement dynamics. The appearing stationary states depend on measurement probabilities for each measurement operator. The Higgs, confined, and deconfined phases emerge in the time evolution by the circuit. We find that both Higgs and confined phases are separated from the deconfined phase by topological entanglement entropy. We numerically clarify that both Higgs and confined phases are characterized by a long-range order on the boundaries accompanying edge modes, and they shift with each other by a crossover reflecting properties in the bulk phase diagram.
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Simulating quantum-mechanical systems is a very challenging issue due to the high computational costs. The main reason for this is related to the dimension of the Hilbert space that grows exponentially with the number of degrees of freedom, making an exact implementation impossible even for the most powerful today’s supercomputers. \nOn this ground it appears clear the need of a new simulation method, i.e. quantum simulation. The general idea is very simple: using a controllable quantum system, called quantum simulator, to emulate and to analyze another quantum system that usually results less controllable or accessible. Quantum simulation is currently a growing and multidisciplinary physical area that involves theoretical and experimental research. From the theoretical side, the main focus is on the study of simulation-models that create a mapping between the target physics and the simulator. \nIn the first two parts of this thesis, we explore these ideas by studying the ground-state properties and the real-time dynamics of a class of Zn lattice gauge theories in 1+1 dimensions, in which the gauge fields are coupled to fermionic matter. These models can be considered as quantum simulation-models of lattice 1+1 Quantum Electrodynamics (QED), which is the simplest gauge theory that shows non-trivial phenomena, like confinement, that are also observed in more complicated gauge theories, such as 3+1 Quantum Chromodynamics (QCD). \nIn the third part of this thesis, we unveil an interesting interplay of symmetry and topology by demonstrating the existence of symmetry-protected topological (SPT) phases in the presence of gauge interactions. More specifically, we introduce an alternative discretization of the Schwinger model leading to a new type of topological-QED. We use bosonization and Density-Matrix Renormalization Group (DMRG) techniques to analyze the phase diagram of the model, opening an interesting route to study topological phases of matter in the context of gauge theories.
Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic Z2\mathbb Z_2Z2​ topological order in (2+1)(2{+}1)(2+1) dimensions, using Z8\mathbb Z_8Z8​ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for ZDF\mathbb Z_D^FZDF​ symmetries for DDD even. Using this, we realize a stabilizer model for a nontrivial Z8F\mathbb Z_8^FZ8F​ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.