Error Correction in Dynamical Codes

Error Correction in Dynamical Codes

7 Mar 2024 | Xiaozhen Fu and Daniel Gottesman
This paper introduces a framework for determining the distance of dynamical codes, which are quantum error-correcting codes defined by a sequence of measurements. Unlike static codes, dynamical codes have a more complex distance concept due to the schedule of measurements affecting error correction. The authors develop an algorithm to track syndrome information through the protocol and determine the distance of a dynamical code in a non-fault-tolerant context. They apply this algorithm to analyze the initialization and masking properties of Floquet codes and explore whether geometric locality limitations can be surpassed in the dynamical paradigm. They find that codes with limited long-range connectivity cannot implement non-Clifford gates with finite depth circuits in 2D settings. The paper also discusses the concept of distance for dynamical codes, including unmasked and permanently masked stabilizers, and presents a general method to determine the unmasked distance. The results show that the distance of a dynamical code is upper bounded by the minimum distance of its instantaneous stabilizer groups. The paper also explores the logical operators and syndrome information in dynamical codes, showing how errors and measurements affect the logical outcome. The authors conclude that the distance algorithm has applications in code construction and analysis, and that dynamical codes can potentially circumvent the no-go theorem for transversal gates. The paper also discusses the implications of measurement errors and the challenges of decoding in the presence of such errors.This paper introduces a framework for determining the distance of dynamical codes, which are quantum error-correcting codes defined by a sequence of measurements. Unlike static codes, dynamical codes have a more complex distance concept due to the schedule of measurements affecting error correction. The authors develop an algorithm to track syndrome information through the protocol and determine the distance of a dynamical code in a non-fault-tolerant context. They apply this algorithm to analyze the initialization and masking properties of Floquet codes and explore whether geometric locality limitations can be surpassed in the dynamical paradigm. They find that codes with limited long-range connectivity cannot implement non-Clifford gates with finite depth circuits in 2D settings. The paper also discusses the concept of distance for dynamical codes, including unmasked and permanently masked stabilizers, and presents a general method to determine the unmasked distance. The results show that the distance of a dynamical code is upper bounded by the minimum distance of its instantaneous stabilizer groups. The paper also explores the logical operators and syndrome information in dynamical codes, showing how errors and measurements affect the logical outcome. The authors conclude that the distance algorithm has applications in code construction and analysis, and that dynamical codes can potentially circumvent the no-go theorem for transversal gates. The paper also discusses the implications of measurement errors and the challenges of decoding in the presence of such errors.
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