Error Correction in Dynamical Codes
1QuICS, University of Maryland, College Park, MD 20742, USA.
2Computer Science Department, University of Maryland, College Park, MD 20742, USA
| Published: | 2025-10-20, volume 9, page 1886 |
| Editor: | Isaac Kim |
| Eprint: | arXiv:2403.04163v3 |
| Doi: | https://doi.org/10.22331/q-2025-10-20-1886 |
| Citation: | Quantum 9, 1886 (2025). |
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Abstract
We ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the distance of a dynamical code. This is a subtle concept and difficult to determine: At any given time, the system will be in a subspace which forms a quantum error-correcting code with a given distance, but the full error correction capability of that code may not be available due to the schedule of measurements associated with the code. We address this challenge by developing an algorithm that tracks information we have learned about the error syndromes through the protocol and put that together to determine the distance of a dynamical code, in a non-fault-tolerant context. We use the tools developed for the algorithm to analyze the initialization and masking properties of a generic Floquet code. Further, we look at properties of dynamical codes under the constraint of geometric locality with a view to understand whether the fundamental limitations on logical gates and code parameters imposed by geometric locality for traditional codes can be surpassed in the dynamical paradigm. We find that codes with a limited number of long range connectivity will not allow non-Clifford gates to be implemented with finite depth circuits in the 2D setting.
Popular summary
In this paper, we take a step towards laying down the theoretical framework for dynamical codes. We introduce a classification of stabilizer generators by how accessible their syndrome information is given a measurement sequence. The distance of a dynamical code is then formalized based on its syndrome accessibility. We also present a classically efficient algorithm that fully classifies the stabilizers and obtains syndrome information for a dynamical code. Our results are applied to study initialization of a generic Floquet codes, revealing the underlying structure of an arbitrary Floquet code.
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