Improved QLDPC Surgery: Logical Measurements and Bridging Codes

Improved QLDPC Surgery: Logical Measurements and Bridging Codes

November 4, 2024 | Andrew Cross, Zhiyang He (Sunny), Patrick Rall, and Theodore Yoder
This paper introduces an improved logical measurement scheme for quantum error-correcting codes, specifically for the gauge-fixed QLDPC surgery. The scheme is based on the construction of Cohen et al. (Sci. Adv. 8, eabn1717) and leverages the expansion properties of the Tanner graph to reduce the space overhead of QLDPC surgery. The scheme requires only Θ(w) ancilla qubits to fault-tolerantly measure a weight w logical operator. The paper provides rigorous analysis of the code distance and fault distance of the scheme and presents a modular decoding algorithm that achieves maximal fault-distance. A bridge system is introduced to facilitate fault-tolerant joint measurements of logical operators. The scheme can connect different families of QLDPC codes into one universal computing architecture. The paper applies the toolbox to perform all logical Clifford gates on the [[144,12,12]] bivariate bicycle code. The scheme adds 103 ancilla qubits into the connectivity graph, and one of the twelve logical qubits is used as an ancilla for gate synthesis. Logical measurements are combined with the automorphism gates studied by Bravyi et al. (Nature 627, 778-782) to implement 288 Pauli product measurements. The practicality of the scheme is demonstrated through circuit-level noise simulations, leveraging a modular decoder that combines BPOSD with matching. The paper discusses the theoretical and practical contributions of the gauge-fixed QLDPC surgery scheme. The scheme reduces the space overhead of the CKBB scheme by gauge-fixing the ancilla system so that the merged code contains no gauge qubits. The resulting gauge-fixed ancilla system ensures that the merged code has distance at least d, while using much fewer than 2d - 1 layers if the Tanner graph induced by the logical operator is expanding. The scheme also introduces new, distance-preserving constructions of Y and joint measurement systems, utilizing a set of bridge qubits. The scheme can be used as a code adapter to connect two logical operators from two arbitrary code blocks. The paper presents a rigorous analysis of fault distance and proves that d rounds of syndrome extraction ensures fault distance d for all proposed measurement schemes. The scheme also introduces a modular decoding algorithm that decomposes the decoding graph into two parts, one supported on the original code, the other supported on the gauge-fixed ancilla system, and decodes them separately. The modular decoder achieves fault distance d. The paper also discusses the theoretical and practical implications of the gauge-fixed QLDPC surgery scheme. The scheme has a noteworthy caveat: without additional assumptions on the measured operator or the code, we do not have upper bounds on the weight of the gauge-fixed operators. However, the paper shows theoretical evidence suggesting that for many important families of QLDPC codes, these gauge checks may not break the LDPC property. The paper also discusses the practical implications of the scheme, including the demonstration thatThis paper introduces an improved logical measurement scheme for quantum error-correcting codes, specifically for the gauge-fixed QLDPC surgery. The scheme is based on the construction of Cohen et al. (Sci. Adv. 8, eabn1717) and leverages the expansion properties of the Tanner graph to reduce the space overhead of QLDPC surgery. The scheme requires only Θ(w) ancilla qubits to fault-tolerantly measure a weight w logical operator. The paper provides rigorous analysis of the code distance and fault distance of the scheme and presents a modular decoding algorithm that achieves maximal fault-distance. A bridge system is introduced to facilitate fault-tolerant joint measurements of logical operators. The scheme can connect different families of QLDPC codes into one universal computing architecture. The paper applies the toolbox to perform all logical Clifford gates on the [[144,12,12]] bivariate bicycle code. The scheme adds 103 ancilla qubits into the connectivity graph, and one of the twelve logical qubits is used as an ancilla for gate synthesis. Logical measurements are combined with the automorphism gates studied by Bravyi et al. (Nature 627, 778-782) to implement 288 Pauli product measurements. The practicality of the scheme is demonstrated through circuit-level noise simulations, leveraging a modular decoder that combines BPOSD with matching. The paper discusses the theoretical and practical contributions of the gauge-fixed QLDPC surgery scheme. The scheme reduces the space overhead of the CKBB scheme by gauge-fixing the ancilla system so that the merged code contains no gauge qubits. The resulting gauge-fixed ancilla system ensures that the merged code has distance at least d, while using much fewer than 2d - 1 layers if the Tanner graph induced by the logical operator is expanding. The scheme also introduces new, distance-preserving constructions of Y and joint measurement systems, utilizing a set of bridge qubits. The scheme can be used as a code adapter to connect two logical operators from two arbitrary code blocks. The paper presents a rigorous analysis of fault distance and proves that d rounds of syndrome extraction ensures fault distance d for all proposed measurement schemes. The scheme also introduces a modular decoding algorithm that decomposes the decoding graph into two parts, one supported on the original code, the other supported on the gauge-fixed ancilla system, and decodes them separately. The modular decoder achieves fault distance d. The paper also discusses the theoretical and practical implications of the gauge-fixed QLDPC surgery scheme. The scheme has a noteworthy caveat: without additional assumptions on the measured operator or the code, we do not have upper bounds on the weight of the gauge-fixed operators. However, the paper shows theoretical evidence suggesting that for many important families of QLDPC codes, these gauge checks may not break the LDPC property. The paper also discusses the practical implications of the scheme, including the demonstration that
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