Dynamical automorphism (DA) code[1] 


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 [1].

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 [2] via as anyon condensation. In this way, measurements cycle logical quantum information between the various condensed phases.


The measurement sequences of the 2D DA color code on a \(N\) layers of triangular patches (which encodes \(N\) logical qubits) with a Pauli boundary can be used to implement all Clifford logical gates via a sequence of two- and three-qubit measurements [1].The 3D DA color code allows for a non-Clifford logical gate via adaptive two-qubit measurements [1].


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



  • Two-dimensional color code — The parent topological phase of the 2D DA color code is realized by two copies of the 2D color code.
  • Three-dimensional color code — The parent topological phase of the 3D DA color code is realized by three copies of the 3D color code.
  • Kitaev surface code — One of the instantaneous stabilizer codes of the 2D DA color code are stacks of toric/surface codes
  • Abelian topological code — Useful measurement sequences of DA Floquet codes can be extracted from topological quantum field theory [1].
  • Lattice stabilizer code — DA codes are defined on 2D and 3D lattices, but they are not conventional stabilizer codes in that they use code switching for error correction and gates.


M. Davydova et al., “Quantum computation from dynamic automorphism codes”, (2023) arXiv:2307.10353
M. S. Kesselring et al., “Anyon condensation and the color code”, (2022) arXiv:2212.00042
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Zoo Code ID: da

Cite as:
“Dynamical automorphism (DA) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/da
@incollection{eczoo_da, title={Dynamical automorphism (DA) code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/da} }
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Cite as:

“Dynamical automorphism (DA) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/da

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/dynamic_gen/da/da.yml.