Mixed-state phase structure of gauge-Higgs subsystem codes under logical-preserving decoherence

TL;DR

This work studies how gauge-symmetric decoherence affects a $Z_2$ lattice gauge-Higgs model treated as a subsystem code, revealing a rich mixed-state phase diagram where the encoded logical qubit remains preserved even as bulk gauge qubits become mixed or critical ($Z_2$-LGHM). It develops a mapping to the toric code with open boundaries and employs the gauging-out framework and RBIM-based statistical mechanics to characterize the global phases, including deconfined, confined, and various mixed regimes. The key contributions include identifying a globally consistent mixed-phase structure, showing that decoherence acts on the gauge-junk sector while preserving the logical space, and demonstrating through numerics and perturbative stability analysis that critical mixed gauge states degrade logical stability. These results illuminate how subsystem codes behave in decohering environments and suggest design principles for robust quantum memories embedded in mixed-state gauges, with potential links to decoherence-free coding and strong-zero-mode concepts.

Abstract

Some of lattice-gauge-theory models, in particular gauge-Higgs model (GHM), can be regarded and work as a subsystem code. This work studies the effect of local-gauge-symmetric decoherence on the GHM from the perspective of the subsystem code. We clarify the global phase diagram of the subsystem code. In particular, the decoherence induces an unconventional critical mixed state, where the logical information is preserved but the rest of the system exhibits mixed state criticality. For a fixed point, the decohered subsystem code is understood by the ``gauging out" prescription. By mapping the GHM to the toric code subject to decoherence, we can understand the properties of the subsystem code. We further discuss and investigate the robustness of the logical space of the subsystem code. Although this kind of subsystem code can be produced by using any bulk mixed state in the GHM, its robustness is a subtle problem due to the mixed critical gauge qubits. We consider some specific unitary for examining the robustness of the stored quantum information. For dynamical unitary perturbations described by interactions between the logical qubit and gauge qubits, the deformation of the subsystem code drastically depends on the initial mixed state of the gauge qubits.

Figures (4)

• Figure 1: Schematic figures of the lattice of the LGHM with rough and smooth boundaries. The green dashed box represents the top rough boundary, where the bare logical operator $L_x$ is defined and the orange dashed box represents the left smooth boundary, where the bare logical operator $L_z$ is defined.
• Figure 2: Schematic image of the Hilbert space structure of the LGHM. The original Hilbert space $\mathcal{H}_{phys}$ is decomposed in the stabilizer group $\mathcal{Z}(\mathcal{G})$. The stabilized space $\mathcal{C}$ is also composed by $Z_2$ charge sector. Furthermore, each $Z_2$ charge sector in the stabilized space $\mathcal{C}$ is factorized with logical space $\mathcal{L}$ and gauge-junk space $\bar{\mathcal{L}}^{(P,S_Z)}$.
• Figure 3: Schematic phase structure in the bulk of the LGHM. The bulk gauge qubits in the gauge-junk space serve as the carrier of these pure and mixed phases. The name in the parentheses is on the side of the TC. The $J=0$ line can be understood by the behavior of the RBIM physics.
• Figure 4: (a) $p_x$- and $J$-dependence of the reduced entropy $S$. (b) $p_x$- and $J$-dependence for the reduced purity $P_r$. (c) Variance of the observable $\hat{O}^{TC}_{G}$. For each case, various cases of different $J$'s are investigated. $J$ determines the pure initial phase of the TC. The system size is $(L_x,L_y)=(3,2)$. The total number of spins is $N=13$.