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Bulletin of the American Physical Society

Optimizing the frequency of post-selected quantum error correction in the [[7,1,3]] Steane code

Ali Abu-Nada, Ben Fortescue, Mark Byrd

Published Mar 3, 2015• 0 citations
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Post-selected stabilizer state preparation is a necessary subroutine in fault-tolerant quantum computation, both for initialization of logical qubits, and for logical-ancilla-based error correction gadgets (e.g. Steane and Knill). Therefore, reducing the number of gates needed to prepare a stabilizer state fault-tolerantly can simultaneously reduce time-to-solution and increase reliability. For small, low-distance codes such as the [[7, 1, 3]] Steane code, circuits with low gate counts can be found by inspection. This becomes impractical for larger codes, necessitating automation. There are two state-of-the-art methods for automated fault-tolerant state preparation, SAT-based stabilizer measurement and flag-at-origin. In this work, we optimize state preparation circuits using the circuit gauge operator formalism to express the construction of flag circuits as an integer linear program. This allows the construction of circuits with equal or lower gate count than the state of the art, while detecting up to three errors. We use this technique to derive a Steane error correction gadget for the [[24, 10, 4]] two-block group algebra code, and test it on Quantinuum's System Model H2 quantum computer with 10,000 shots, resulting in a logical block error rate ~0.00014 (~0.000014 per logical qubit), with ~1.6% of the shots post-selected due to weight-two errors.
Selecting the optimal QEC code requires expertise in both hardware noise and multiple code families. We present an automated engine that takes raw error rates, classifies noise, detects bias (Z/X asymmetry ratio) and correlated burst errors, and recommends the optimal code with minimum distance and resource estimate. The decision tree maps Z-biased noise to repetition codes, correlated burst noise to color codes, very low rates to Steane codes, and general noise to surface codes. Integration with FTQC resource estimation provides total physical overhead. We demonstrate up to 52.5x reduction in physical qubit count compared to default surface code selection when noise exhibits strong bias.
Since quantum states are limited in their ease of manipulation, quantum computing has great advantages when used for computation. These interactions, even if infinitesimal, are the very reason why errors arise, be it bit-flip, phase-flip, or composite, thus diminishing the accuracy in computation. Quantum error correction (QEC) intervenes systematically to guard against the loss of coherence from quantum information. Amongst the various QEC protocols developed so far, Steane code [ [7], [1], [3]] is a very famous CSS stabilizer code that was derived from a classical Hamming code. It will encode a single logical qubit into seven physical qubits, which corrects all single-qubit errors by separating X and Z syndrome measurements. The Steane code enables the implementation of transversal Clifford gates, making it an ideal playground to study fault-tolerant quantum computation. Although surface codes remain the codes of choice for scaling implementations, the Steane code will forever remain one of the cornerstone codes due to its clear structure of error correction and ease in teaching.