Adaptive syndrome measurements for Shor-style error correction

Abstract

The Shor fault-tolerant error correction (FTEC) scheme uses transversal gates and ancilla qubits prepared in the cat state in syndrome extraction circuits to prevent propagation of errors caused by gate faults. For a stabilizer code of distance $d$ that can correct up to $t=\lfloor (d-1)/2\rfloor$ errors, the traditional Shor scheme handles ancilla preparation and measurement faults by performing syndrome measurements until the syndromes are repeated $t+1$ times in a row; in the worst-case scenario, $(t+1{)}^2$ rounds of measurements are required. In this work, we improve the Shor FTEC scheme using an adaptive syndrome measurement technique. The syndrome for error correction is determined based on information from the differences of syndromes obtained from consecutive rounds. Our protocols that satisfy the strong and the weak FTEC conditions require no more than $(t+3{)}^2/4-1$ rounds and $(t+3{)}^2/4-2$ rounds, respectively, and are applicable to any stabilizer code. Our simulations of FTEC protocols with the adaptive schemes on hexagonal color codes of small distances verify that our protocols preserve the code distance, can increase the pseudothreshold, and can decrease the average number of rounds compared to the traditional Shor scheme. We also find that for the code of distance $d$, our FTEC protocols with the adaptive schemes require no more than $d$ rounds on average.

arXiv comment24 pages, 3 figures, comments welcome. V2: minor correction. V3: adding numerical simulations and classical time complexity analysis. V4: minor correction

References in corpus (26)

• Good Quantum Error-Correcting Codes Exist
• Topological quantum memory
• Multiple Particle Interference and Quantum Error Correction
• A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound
• Reliable Quantum Computers
• Topological Quantum Distillation
• Stim: a fast stabilizer circuit simulator
• Overhead and noise threshold of fault-tolerant quantum error correction
• Universal fault-tolerant quantum computation with only transversal gates and error correction
• Quantum error correction with only two extra qubits
• Fault-Tolerant Quantum Computation For Local Non-Markovian Noise
• Topological and subsystem codes on low-degree graphs with flag qubits
• Fault-tolerant conversion between the Steane and Reed-Muller quantum codes
• Fault-tolerant quantum computation with cluster states
• Effective fault-tolerant quantum computation with slow measurements
• Flag fault-tolerant error correction with arbitrary distance codes
• Universal transversal gates with color codes - a simplified approach
• Triangular color codes on trivalent graphs with flag qubits
• Flag fault-tolerant error correction for any stabilizer code
• Simple proof of fault tolerance in the graph-state model
• Fault-tolerant quantum error correction for Steane's seven-qubit color code with few or no extra qubits
• Overhead analysis of universal concatenated quantum codes
• Flag fault-tolerant error correction, measurement, and quantum computation for cyclic CSS codes
• Error rates and resource overheads of encoded three-qubit gates
• Achieving fault tolerance on capped color codes with few ancillas
• Fault-tolerant quantum error correction using error weight parities

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• Minimizing readout-induced noise for early fault-tolerant quantum computers
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• Weakly Fault-Tolerant Computation in a Quantum Error-Detecting Code
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