## Description

Dynamically-generated stabilizer-based code whose (not necessarily periodic) sequence of few-body measurements implements state initialization, logical gates and error detection.

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 via code switching using the group \(\mathsf{F}\) of check operators measured in the next step in the sequence.

As opposed to subsystem codes, only specific measurement sequences maintain the codespace, and not all sequences implement error detection. Aperiodic measurement sequences provide a way to implement logical gates [2].

For DA codes based on topological phases, the phase associated with each ISG of the code can be obtained from a single parent topological phase associated with the DA code [3] via as anyon condensation. In this way, measurements cycle logical quantum information between the various condensed phases.

## Protection

### Classification of stabilizers by masking

There exists an efficient classical algorithm that tracks information learned by syndrome extraction at each step [4]. The algorithm performs the following classification of stabilizers into unmasked, temporarily masked, and permanently unmasked stabilizers.

An unmasked stabilizer is a stabilizer whose outcome can be obtained by measurements. In general, it is not obvious to determine if a stabilizer can be unmasked as its eigenvalue may only be revealed indirectly as a product of several measurements. A temporarily masked stabilizer is a stabilizer whose syndrome can not be obtained by the given sequence but could possibly be obtained with future measurements. A permanently masked stabilizer is a stabilizer whose outcome is irreversibly lost by the given sequence.

For a masked stabilizer code with a set \(U\) of \(l\) masked stabilizers, its masked distance is given by: \begin{equation} d_{\mathrm{u}} = \min\:\text{wt}\{ \mathsf{N}(U)\backslash \mathsf{G}\}~. \tag*{(1)}\end{equation} Above, \(\mathsf{G}\) is a gauge group defined from the algorithm that depends partly on the freedom in the choice of destabilizers for the temporarily masked stabilizers, and partly on the measurement sequence which fixes the destabilizers for the permanently masked stabilizers.

## Encoding

## Gates

## Parents

- Qubit code
- Dynamically-generated QECC — DA code state initialization, logical gates, and error correction are done by a sequence of different (usually weight-two) stabilizer measurements.

## Child

- Hastings-Haah Floquet code — Floquet codes are DA codes with periodic measurement sequences.

## Cousins

- Honeycomb (6.6.6) color code — The parent topological phase of the 2D DA color code is realized by two copies of the 6.6.6 color code [2].
- Cubic honeycomb color code — The parent topological phase of the 3D DA color code is realized by three copies of the cubic honeycomb color code [2].
- Kitaev surface code — One of the instantaneous stabilizer codes of the 2D DA color code are stacks of surface codes
- Abelian topological code — Useful measurement sequences of DA codes can be extracted from topological quantum field theory [2].
- Lattice stabilizer code — DA codes are typically defined on 2D and 3D lattices, but they are not conventional stabilizer codes in that they use code switching for error correction and gates.

## References

- [1]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
- [2]
- M. Davydova et al., “Quantum computation from dynamic automorphism codes”, (2023) arXiv:2307.10353
- [3]
- M. S. Kesselring et al., “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
- [4]
- X. Fu and D. Gottesman, “Error Correction in Dynamical Codes”, (2024) arXiv:2403.04163

## Page edit log

- Victor V. Albert (2024-04-04) — most recent
- Xiaozhen Fu (2024-04-04)
- Victor V. Albert (2023-07-21)
- Shankar N. Balasubramanian (2023-07-21)
- Nathanan Tantivasadakarn (2023-07-21)

## Cite as:

“Dynamical automorphism (DA) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/da