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arXiv:1306.0498 · Quantum Information Processing

Syndrome measurement order for the [[7,1,3]] quantum error correction code

Yaakov S. Weinstein

Published Jul 18, 2015• 3 citations
arXivHTMLPDF
DOI
10.1007/s11128-015-1068-z

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Non-Zero Syndromes and Syndrome Measurement Order for the [[7,1,3]] Quantum Error Correction Code

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In this work we explore the accuracy of quantum error correction depending of the order of the implemented syndrome measurements. CSS codes require bit-flip and phase flip-syndromes be measured separately. To comply with fault tolerant demands and to maximize accuracy this set of syndrome measurements should be repeated allowing for flexibility in the order of their implementation. We examine different possible orders of Shor state and Steane state syndrome measurements for the [[7,1,3]] quantum error correction code. We find that the best choice of syndrome order, determined by the fidelity of the state after noisy error correction, will depend on the error environment. We also compare the fidelity when syndrome measurements are done with Shor states versus Steane states and find that Steane states generally, but not always, lead to final states with higher fidelity. Together, these results allow a quantum computer programmer to choose the optimal syndrome measurement scheme based on the system's error environment.

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Fault-tolerant syndrome measurement plays an important role in the process of quantum error correction, and considerable effort had been taken for reducing the physical overhead of syndrome measurement which including the ancilla qubits, the CNOT gates and the time-slots. The two extra qubits syndrome measurement technique, known as "flag"-style syndrome measurement, cuts down the number of extra qubits to the utmost. However, it works slowly because it measures only one syndrome at a time. We extend the technique to extract all syndromes of the distance-3 quantum error-correcting code at once. We propose a new method that increases the parallelism of the syndrome measurement circuit and reduces the time overhead by allocating and adjusting the order of CNOT gates for measuring data block reasonably, which we call dynamic time-slot allocation scheme, and which is applicable to both Hamming codes and color codes. For a CSS quantum error-correcting code with the number of stabilizers m and the maximum weight w, we only need 2m extra qubits and 2 × (w+2) time-slots for one-shot measurement of all syndromes.

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In this work we explore the accuracy of quantum error correction depending of the order of the implemented syndrome measurements. CSS codes require bit-flip and phase flip-syndromes be measured separately. To comply with fault tolerant demands and to maximize accuracy this set of syndrome measurements should be repeated allowing for flexibility in the order of their implementation. We examine different possible orders of Shor state and Steane state syndrome measurements for the [[7,1,3]] quantum error correction code. We find that the best choice of syndrome order, determined by the fidelity of the state after noisy error correction, will depend on the error environment. We also compare the fidelity when syndrome measurements are done with Shor states versus Steane states and find that Steane states generally, but not always, lead to final states with higher fidelity. Together, these results allow a quantum computer programmer to choose the optimal syndrome measurement scheme based on the system's error environment.
Fault-tolerant syndrome measurement plays an important role in the process of quantum error correction, and considerable effort had been taken for reducing the physical overhead of syndrome measurement which including the ancilla qubits, the CNOT gates and the time-slots. The two extra qubits syndrome measurement technique, known as "flag"-style syndrome measurement, cuts down the number of extra qubits to the utmost. However, it works slowly because it measures only one syndrome at a time. We extend the technique to extract all syndromes of the distance-3 quantum error-correcting code at once. We propose a new method that increases the parallelism of the syndrome measurement circuit and reduces the time overhead by allocating and adjusting the order of CNOT gates for measuring data block reasonably, which we call dynamic time-slot allocation scheme, and which is applicable to both Hamming codes and color codes. For a CSS quantum error-correcting code with the number of stabilizers m and the maximum weight w, we only need 2m extra qubits and 2 × (w+2) time-slots for one-shot measurement of all syndromes.
Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.