# Floquet code[1]

## Description

Also called a Hastings-Haah code. Dynamically-generated stabilizer-based code whose logical qubits are generated through a particular sequence of check-operator measurements such that the number of logical qubits is larger than when the code is viewed as a static subsystem stabilizer code.

After each measurement in the sequence, the codespace is a joint\(+1\) eigenspace of an instantaneous stabilizer group (ISG), i.e., a particular stabilizer group corresponding to the measurement. The ISG specifies the state of the system as a Pauli stabilizer state at a particular round of measurement, and it evolves into a (potentially) different ISG depending on the check operators measured in the next step in the sequence. As opposed to subsystem codes, only specific measurement sequences maintain the codespace.

A measurement can be interpreted as causing anyon condensation, thereby mapping the topoligical phase of a given code state into another condensed phase. In this way, measurements cycle logical quantum information between various condensed phases of a parent topological phase [2].

## Protection

## Parent

- Crystalline-circuit qubit code — Floquet codes are crystalline-circuit codes with non-commuting measurements in 2D.

## Children

- Floquet color code
- Honeycomb Floquet code — The honeycomb Floquet code is the first 2D Floquet code.

## Cousins

- Subsystem qubit stabilizer code — This code can be viewed as a subsystem stabilizer code, albeit one with less logical qubits.
- Monitored random-circuit code — Both Floquet and monitored random circuit codes can have an instantaneous stabilizer group which evolves through unitary evolution and measurements. However, Floquet codewords are generated via a specific sequence of measurements, while random-circuit codes maintain a stabilizer group after any measurement. Floquet codes have the additional capability of detecting errors induced during the measurement process; see Appx. A of Ref. [1].
- Majorana stabilizer code — Floquet codes are viable candidates for storage in Majorana-qubit devices [3].
- Spacetime circuit code — Spacetime circuit codes are useful for constructing fault-tolerant encoding and syndrome extraction circuits for Floquet codes.

## References

- [1]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
- [2]
- M. S. Kesselring et al., “Anyon condensation and the color code”, (2022) arXiv:2212.00042
- [3]
- A. Paetznick et al., “Performance of Planar Floquet Codes with Majorana-Based Qubits”, PRX Quantum 4, (2023) arXiv:2202.11829 DOI

## Page edit log

- Victor V. Albert (2022-07-12) — most recent
- Victor V. Albert (2022-01-01)

## Cite as:

“Floquet code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/floquet