Phases of Floquet code under local decoherence

TL;DR

This work analyzes the Hastings-Haah Floquet code under local decoherence with perfect measurements, establishing a finite threshold below which the code retains an $e$-$m$ automorphism and thus encodes quantum information in a robust Floquet phase. The authors derive a 3D statistical-mechanics model for maximum-likelihood decoding and identify a class of simple two-qubit errors that decouple into 2D RBIMs on a honeycomb lattice, yielding the threshold $p_c=0.0119$; they also introduce information-theoretic diagnostics based on quantum relative entropy $D_{em}^{(n)}$ and coherent information $I_c^{(n)}$, which undergo a concurrent phase transition and distinguish the Floquet code from the toric code. A stabilizer-expansion framework maps these diagnostics to $(n-1)$-flavor Ising models, enabling an Ising-mapping description of the decohered dynamics and the automorphism. Together, these results establish the Floquet code below threshold as a distinct dynamical phase with robust automorphism, and they pave the way for understanding intrinsic thresholds in dynamical quantum memories via statistical-mechanics mappings.

Abstract

Floquet code is a dynamical quantum memory with a periodically evolving logical space. As a defining feature, the code exhibits an anyon automorphism after each period, giving rise to a non-trivial evolution of each logical state. In this paper, we study the Floquet code under local decoherence and perfect measurements and demonstrate that below the decoherence threshold, the code is in a robust phase characterized by the anyon automorphism. We first derive the 3D statistical mechanics model for the maximum likelihood decoder of the 2D Floquet code under local Pauli decoherence. We identify a class of two-qubit Pauli channels under which the 3D statistical mechanics model becomes decoupled 2D models and obtain the threshold for such decoherence channels. We then propose a diagnostic of the anyon automorphism in the presence of local decoherence. We analytically show that this diagnostic distinguishes the Floquet code from the toric code under repeated syndrome measurements and undergoes a phase transition at the threshold.

Figures (8)

• Figure 1: Hastings-Haah code on a 3-colored triangular lattice. The physical qubits are defined on the triangular plaquettes. The logical operators $L_R^Y$ and $L_B^X$ in round R are generated by $E_R^Y$ and $E_B^X$ operators along homologically non-trivial loops.
• Figure 2: Toric code on the R-superlattice after round R measurements. The qubits live on edges in the superlattices. The vertex stabilizers $\Tilde{V}_R^{\Tilde{X}}$ involves $\Tilde{X}$ on the six edges emanating from a red vertex. They correspond to six-body vertex operators $V_R^X$ associated with red vertices on the original lattice. The plaquette stabilizers $\Tilde{P}_G^{\Tilde{Z}}$ and $\Tilde{P}_B^{\Tilde{Z}}$ are defined as $\Tilde{Z}^{\otimes 3}$ on the edges of the plaquettes on the superlattice. In the original lattice, they are six-qubit vertex operators $V_G^Y$ and $V_{B}^{Z}$ associated with green and blue vertices, respectively.
• Figure 3: Detection of simple errors in the Floquet code. Syndromes of $V_R^X$, $V_G^Y$ and $V_B^Z$ are read out in rounds $B$, $R$, and $G$, respectively. Each type of syndrome change is caused by two types of simple errors in the previous three rounds, enclosed by dashed lines.
• Figure 4: Error strings of simple errors $E_B^X$ and $E_B^Y$ on the B-superlattice. The syndrome changes $s$ (highlighted in yellow) live on the vertices of the blue superlattice. The solid and dashed lines $\mathcal{E}_{1,2}$ (highlighted in light blue) represent two equivalent error operators that differ by a product of stabilizers $V_R^X$, $V_G^Y$, and check operators $E_B^Z$.
• Figure 5: Random bond Ising model on the honeycomb lattice. The Ising spins $\sigma_i = \pm 1$ live on the vertices. The domain wall between highlighted spins and the rest represents the cycle $C = \mathcal{E}_{0,\kappa}+\mathcal{E}$ with $\mathcal{E}_{0,\kappa}$ and $\mathcal{E}$ marked by dashed and solid lines, respectively. The Ising couplings $J\eta_\ell$ on the edges perpendicular to $\mathcal{E}_{0,\kappa}$ are anti-ferromagnetic, while the rest are ferromagnetic.