A protocol called the "honeycomb code", or generically a "Floquet code", was introduced by Hastings and Haah in \cite{hastings_dynamically_2021}. The honeycomb code is a subsystem code based on the honeycomb lattice with zero logical qubits but such that there exists a schedule for measuring two-body gauge checks leaving enough room at all times for two protected logical qubits. In this work we show a way to introduce boundaries to the system which curiously presents a rotating dynamics but has constant distance and is therefore not fault-tolerant.A protocol called the "honeycomb code", or generically a "Floquet code", was introduced by Hastings and Haah in \cite{hastings_dynamically_2021}. The honeycomb code is a subsystem code based on the honeycomb lattice with zero logical qubits but such that there exists a schedule for measuring two-body gauge checks leaving enough room at all times for two protected logical qubits. In this work we show a way to introduce boundaries to the system which curiously presents a rotating dynamics but has constant distance and is therefore not fault-tolerant.

arXiv:2110.05348Oct 11, 20215 citations

Fault-tolerant quantum computation with few qubits

Ben W. Reichardt

Bulletin of the American Physical SocietyMar 5, 20190 citations

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

Yoshito Watanabe, Bianca Bannenberg, Simon Trebst

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.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.

quant-pharXiv:2602.12685Feb 16, 20260 citations

Floquet Codes without Parent Subsystem Codes

Margarita D. Davydova, Nathanan Tantivasadakarn, Shankar Balasubramanian

We propose a new class of error-correcting dynamic codes in two and three dimensions that has no explicit connection to any parent subsystem code.The two-dimensional code, which we call the CSS (Calderbank-Shor-Steane) honeycomb code, is geometrically similar to that of the honeycomb code by Hastings and Haah and also dynamically embeds an instantaneous toric code.However, unlike the honeycomb code, it possesses an explicit CSS structure and its gauge checks do not form a subsystem code.Nevertheless, we show that our dynamic protocol conserves logical information and possesses a threshold for error correction.We generalize this construction to three dimensions and obtain a code that fault tolerantly alternates between realizing two type-I fracton models, the checkerboard and the X-cube model.Finally, we show the compatibility of our CSS honeycomb-code protocol and the honeycomb code by showing the possibility of randomly switching between the two protocols without information loss while still measuring error syndromes.We call this more general aperiodic structure "dynamic tree codes," which we also generalize to three dimensions.We construct a probabilistic finite automaton prescription that generates dynamic tree codes correcting any single-qubit Pauli errors and can be viewed as a step toward the development of practical fault-tolerant random codes.We propose a new class of error-correcting dynamic codes in two and three dimensions that has no explicit connection to any parent subsystem code.The two-dimensional code, which we call the CSS (Calderbank-Shor-Steane) honeycomb code, is geometrically similar to that of the honeycomb code by Hastings and Haah and also dynamically embeds an instantaneous toric code.However, unlike the honeycomb code, it possesses an explicit CSS structure and its gauge checks do not form a subsystem code.Nevertheless, we show that our dynamic protocol conserves logical information and possesses a threshold for error correction.We generalize this construction to three dimensions and obtain a code that fault tolerantly alternates between realizing two type-I fracton models, the checkerboard and the X-cube model.Finally, we show the compatibility of our CSS honeycomb-code protocol and the honeycomb code by showing the possibility of randomly switching between the two protocols without information loss while still measuring error syndromes.We call this more general aperiodic structure "dynamic tree codes," which we also generalize to three dimensions.We construct a probabilistic finite automaton prescription that generates dynamic tree codes correcting any single-qubit Pauli errors and can be viewed as a step toward the development of practical fault-tolerant random codes.

arXiv:2210.02468PRX QuantumJun 13, 202362 citations

Floquet codes with a twist

Tyler D. Ellison, Sullivan, Joseph, Arpit Dua

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.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.

arXiv:2306.08027Jun 13, 20234 citations

\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.