Floquet implementation of a 3d fermionic toric code with full logical code space

TL;DR

This work advances three-dimensional Floquet codes by constructing a three-dimensional Kekulé-Kitaev lattice whose color-deleted subgraphs form finite loops, enabling a ten-round schedule that preserves all three logical qubits while realizing the instantaneous stabilizer group as a three-dimensional fermionic toric code. It connects Floquet code design to monitored Kitaev dynamics, providing both a concrete lattice-plus-schedule realization and a broader phase-diagram perspective for random-measurement dynamics in 3D, including numerical evidence of area-law and extended critical regimes and edge behavior. The key contributions are the explicit 3D lattice construction with finite-loop properties, a comprehensive 10-round syndrome-extraction protocol, and an exploration of measurement-induced phases that informs future decoding and gate-design efforts. Together, these results offer a geometry-driven path toward robust, fault-tolerant logical control in three dimensions and suggest routes to integrating non-Clifford operations within 3D Floquet codes and monitored dynamics.

Abstract

Floquet quantum error-correcting codes provide an operationally economical route to fault tolerance by dynamically generating stabilizer structures using only two-body Pauli measurements. But while it is well established that stabilizer codes in higher spatial dimensions gain additional levels of intrinsic robustness, higher-dimensional Floquet codes have hitherto been explored only in limited scope. Here we introduce a 3d generalization of a Floquet code whose instantaneous stabilizer group realizes a 3d fermionic toric code, while crucially preserving all three logical qubits throughout the entire measurement sequence. One central ingredient is the identification of a 3d lattice geometry that generalizes the features of the Kekulé lattice underlying the 2d Hastings-Haah code - specifically, a structure where deleting any one edge color yields a two-color subgraph that decomposes into short, closed loops rather than homologically nontrivial chains. This loop property avoids the collapse of logical information that plagues naive sequential two-color measurement schedules on many 3d lattices. Although, for our lattice geometry, a simple 3-round cycle that sequentially measures the three types of parity checks does not expose the full error syndrome set, we show that one can append a measurement sequence to extract the missing syndromes without disturbing the logical subspace. Beyond code design, 3d tricoordinated lattice geometries define a family of 3d monitored Kitaev models, in which random measurements of the non-commuting parity checks give rise to dynamically created entangled phases with nontrivial topology. In discussing the general structure of their underlying phase diagrams and, in particular, the existence of certain quantum critical points, we again make a connection to the general preservation of logical information in time-ordered Floquet protocols.

Figures (9)

• Figure 1: Three-dimensional Kekulé-Kitaev lattice. The lattice is a 3$d$ tricoordinated lattice that allows a three-edge coloring. (a) Two perspectives of the lattice: square--octagon layers are stacked along $\mathbf{a}_3$ and coupled by vertical $z$-type (blue) bonds. (b) Projection of one bilayer unit cell onto the $\mathbf{a}_1$--$\mathbf{a}_2$ plane. Each layer is based on a square--octagon lattice in which every square is decorated by additional vertices that enable three-dimensional connectivity; the unit cell contains a lower and an upper layer. The vertical ($\parallel\mathbf{a}_3$) $z$-type bonds have two inequivalent in-plane footprints within the unit cell, half-shifted along $\mathbf{a}_1$ or $\mathbf{a}_2$, corresponding to intra- and inter-unit-cell interlayer couplings (blue ovals). (c) Examples of elementary non-coplanar plaquettes involving out-of-plane bonds; plaquettes of types $p_4$, $p_5$, and $p_6$ are shown. Ignoring the edge coloring, the in-plane primitive translations are $\mathbf{a}_1$ and $\mathbf{a}_2$; each layer has a 12-site unit cell (24 sites per bilayer). The edge coloring doubles the translation period along both $\mathbf{a}_1$ and $\mathbf{a}_2$, giving 96 sites in the full (colored) unit cell (a $45^\circ$-rotated choice of lattice vectors reduces this to 64 sites, but we use the unrotated convention). We label system size by $(L_1,L_2,L_3)$, the number of unit cells along $(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3)$; a consistent coloring under periodic boundary conditions requires $L_1$ and $L_2$ to be even.
• Figure 2: Tricoordinated lattice geometries. (a) Honeycomb lattice with Kitaev edge coloring. (b) Honeycomb lattice with Kekulé edge coloring. (c) Hyperhoneycomb lattice. In (a) and (c), the shaded plaquettes are three-colored (their boundaries contain all three edge colors). In (b), the plaquettes are two-colored; we label each plaquette by the missing color (indicated by the shading), e.g., an $x$-plaquette is bounded by $y$- and $z$-colored edges. Examples of inner logical operators are shown as thick pink lines in each panel.
• Figure 3: Enclosed volumes. A volume is formed by a set of plaquettes (shaded) whose union is a closed surface enclosing a three-dimensional region. The product of the plaquette operators on the boundary of such a volume equals the identity, yielding a volume constraint. (a) A $p_5$ plaquette can be replaced by an alternative choice of surface spanning the same boundary loop: the two surface choices are equivalent because together they form a closed surface that encloses a volume. (b--e) Other examples of enclosed volumes formed from combinations of plaquette types.
• Figure 4: Logical Pauli operators for the three encoded qubits. For each logical qubit $L_i$ ($i=1,2,3$), we choose a pair of non-contractible operators: a line operator $l_i$ running along the $\mathbf{a}_i$ direction (purple) and a membrane operator $m_i$ (green) oriented transverse to $l_i$. The line operators are inner logicals: they lie in the gauge group generated by the measured checks, commute with all checks, and are therefore round-independent. The membrane operators are outer logicals and can be round-dependent; shown are representatives after a $z$ round, constructed by placing a single-site $Z$ on one endpoint of each $z$-colored bond intersected by the membrane (intersections marked by crosses).
• Figure 5: T-junction--exchange. Two point-like charges $f_1$ and $f_2$ at the endpoints $a$ and $b$ are exchanged via the three-step sequence 1-- 3 around a trivalent $(x,y,z)$ Kitaev vertex $o$. Each step is implemented by a short string operator given by the product of the two adjacent bond operators meeting at $o$. The junction contribution picks up a minus sign, diagnosing fermionic self-statistics of the charge excitation (independent of spatial dimensionality).