Floquet codes with a twist

TL;DR

This work extends the $\,\mathbb{Z}_2$ Floquet code by introducing twist defects via one-dimensional fermion condensation, enabling robust, fault-tolerant storage and processing of quantum information without altering lattice connectivity or the 3-round measurement cadence. It further shows how to realize a planar variant with boundaries and generalizes the construction to $\,\mathbb{Z}_N$ Floquet codes, including twist defects that yield Abelian twisted quantum double topologies. The authors develop a coherent framework for defect insertion, boundary engineering, fault tolerance, and Clifford-type computation, including measurement-based braiding and S-gate protocols. Finally, they extend the construction to type I twisted quantum doubles and discuss practical considerations, such as hardware compatibility and decoding strategies, establishing a versatile approach to dynamic topological codes with defect-based quantum information processing.

Abstract

We describe a method for creating twist defects in the honeycomb Floquet code of Hastings and Haah. In particular, we construct twist defects at the endpoints of condensation defects, which are built by condensing emergent fermions along one-dimensional paths. We argue that the twist defects can be used to store and process quantum information fault tolerantly, and demonstrate that, by preparing twist defects on a system with a boundary, we obtain a planar variant of the $\mathbb{Z}_2$ Floquet code. Importantly, our construction of twist defects maintains the connectivity of the hexagonal lattice, requires only 2-body measurements, and preserves the three-round period of the measurement schedule. We furthermore generalize the twist defects to $\mathbb{Z}_N$ Floquet codes defined on $N$-dimensional qudits. As an aside, we use the $\mathbb{Z}_N$ Floquet codes and condensation defects to define Floquet codes whose instantaneous stabilizer groups are characterized by the topological order of certain Abelian twisted quantum doubles.

Figures (20)

• Figure 1: (a) To define the operators $K_{ij}$ in Eq. \ref{['eq: Z2 checks']}, each edge of the hexagonal lattice is labeled as an $x$-, $y$-, or $z$-edge. (b) To specify the measurement schedule, we label the plaquettes as $0$-, $1$-, and $2$-plaquettes, such that neighboring pairs of plaquettes have distinct labels. The $0$-edges (red), $1$-edges (yellow), and $2$-edges (blue) connect the $0$-, $1$-, and $2$-plaquettes, respectively. The vertices of the plaquettes are labeled 1-6 to define the plaquette stabilizer in Eq. \ref{['eq: plaquette stabilizer vertices']}.
• Figure 2: After each round of measurements, we interpret the $+1$ eigenspace of the 2-body checks as an effective qubit on the edges (gray dots). The effective qubits live on the edges of a hexagonal super-lattice (dashed gray).
• Figure 3: (a) After measuring the $r$-checks, the $e$ string operator (light blue) along a contractible path is a product of the $r$-plaquettes (gray) and $r$-checks (gray) enclosed by the path. (b) The $m$ string operator (light red) along a contractible path is a product of the $(r-2)$-plaquettes, $(r-1)$-plaquettes (both gray), and the $r$-checks (gray) enclosed by the path.
• Figure 4: Starting with an $e$ string operator (light blue) in the $r=0$ round, we multiply by $0$-checks (red) to find a representation of the logical operator that commutes with the $1$-checks (yellow). The logical operator now corresponds to an $m$ string operator (light red) in the $r=1$ round. In general, we multiply by $r$-checks to find a representation of the logical operator that commutes with the $(r+1)$-checks. The logical operator corresponding to an $e$ string operator transforms into an $m$ string operator after a single period.
• Figure 5: The procedure for inserting a defect line along an open path $\gamma$ (light gray) involves four steps. (i) We define a fermion string operator $W^\psi_\gamma$ along $\gamma$. (ii) We remove the two Pauli operators at the endpoints to define the string operator $\tilde{W}^\psi_\gamma$. (iii) We divide $\tilde{W}^\psi_\gamma$ into 2-body short string operators. (iv) We define the 2-body short string operator as defect checks (bold black) and remove the checks that fail to commute with the defect checks (dashed lines). The twist defects (bold black crosses) are hosted at the endpoints of $\gamma$.