Error Correction in Dynamical Codes

TL;DR

This work develops a rigorous framework for quantum error correction defined by sequences of Pauli measurements, i.e., dynamical or Floquet codes. It introduces the instantaneous stabilizer group (ISG) and a distance notion that accounts for information that may be unrecoverable (masked) by the measurement schedule, with an efficient distance algorithm to identify unmasked syndrome information and the corresponding distance $d_{ ext{u}}$. The approach is applied to Floquet codes to analyze initialization and masking dynamics, and it yields a no-go theorem that geometrically local, 2D dynamical codes cannot implement transversal non-Clifford gates unless sufficient long-range connectivity is present, with extensions to higher dimensions. Overall, the framework enables practical assessment of distance, syndrome extraction, and logical-gate limitations for dynamical codes, providing a foundation for designing spacetime codes and understanding fault-tolerance in measurement-driven quantum error correction.

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.

Figures (8)

• Figure 1: The figure illustrates an example of a Floquet code which is a special case of dynamical codes consisting of periodic measurements (In this case, 4 rounds of measurements). Each round of measurements consists of commuting Paulis. The $i^{\mathrm{th}}$ round of measurements brings the code from the ISG defined by $S_{i-1}$ to the next ISG defined by $S_{i(\mathrm{mod} \: 4)}$.
• Figure 2: The figure illustrates an example of a code with 7 stabilizers, out of which 4 are unmasked stabilizers (pink online) and 3 are masked stabilizers. The destabilizers are chosen to anti-commute with their respective masked stabilizers. The entire set (blue online) forms the generators of a gauge group $\mathcal{G}$. The distance of this subsystem code is given by $d_{\mathrm{subsystem}} = \min\:\mathrm{wt\:} \{\mathcal{N}(U)\backslash \mathcal{G}\}$.
• Figure 3: The figure illustrates the first three rounds of a sequence of measurements. $S_i$ is the set of stabilizer generators of the ISG in the $i^{\mathrm{th}}$ round and $L_i$ is the updated logical operator from a given logical representative $L_0$ in the $0^{\mathrm{th}}$ round.
• Figure 4: An illustration of the first three rounds of a sequence of measurements that result in the updated sets of stabilizer generators. The union of $C_i$ and $V_i$ gives an overcomplete set of generators for $\langle S_i \rangle$ which is the ISG at round $i$ or after the $i^{\mathrm{th}}$ round of measurements. $M_i$ is the set of measurements made in the $i^{\mathrm{th}}$ round of measurements.
• Figure 5: An illustration of the three colorable honeycomb lattice. The plaquettes are labelled R for red, B for blue and G for green. The vertices of the center blue plaquette are labelled from 1 to 6.

Theorems & Definitions (89)

• Definition 1
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