[Paper Review] Matching and maximum likelihood decoding of a multi-round subsystem quantum error correction experiment

Neereja Sundaresan, Theodore J. Yoder+8 more|arXiv (Cornell University)|Mar 14, 2022

Quantum Computing Algorithms and Architecture26 citations

TL;DR

The work demonstrates fault-tolerant operation of a distance-3 heavy-hexagon subsystem code on superconducting qubits, comparing perfect-matching and maximum-likelihood decoders, achieving logical error per syndrome round as low as ~0.04 (matching) and ~0.035 (ML).

ABSTRACT

Quantum error correction offers a promising path for performing quantum computations with low errors. Although a fully fault-tolerant execution of a quantum algorithm remains unrealized, recent experimental developments, along with improvements in control electronics, are enabling increasingly advanced demonstrations of the necessary operations for applying quantum error correction. Here, we perform quantum error correction on superconducting qubits connected in a heavy-hexagon lattice. The full processor can encode a logical qubit with distance three and perform several rounds of fault-tolerant syndrome measurements that allow the correction of any single fault in the circuitry. Furthermore, by using dynamic circuits and classical computation as part of our syndrome extraction protocols, we can exploit real-time feedback to reduce the impact of energy relaxation error in the syndrome and flag qubits. We show that the logical error varies depending on the use of a perfect matching decoder compared to a maximum likelihood decoder. We observe a logical error per syndrome measurement round as low as $\sim0.04$ for the matching decoder and as low as $\sim0.03$ for the maximum likelihood decoder. Our results suggest that more significant improvements to decoders are likely on the horizon as quantum hardware has reached a new stage of development towards fully fault-tolerant operations.

Motivation & Objective

• Motivate and test quantum error correction on a heavy-hexagon subsystem code with real-time feedback.
• Evaluate how decoder choice (perfect matching vs maximum likelihood) affects logical error under circuit-level noise.
• Show the impact of deflagging and leakage-aware post-processing on logical failure rates.
• Demonstrate multi-round syndrome extraction with mid-circuit measurements on a superconducting platform.

Proposed method

• Model the decoding problem with a decoding hypergraph that captures error-sensitive events and hyperedges.
• Implement two decoding strategies: (i) perfect matching on X- and Z-error graphs with either uniform or log-likelihood edge weights, and (ii) maximum likelihood decoding updating Pr[βγ] across hyperedges.
• Use deflagging to simplify the hypergraph by applying virtual Z corrections based on flag outcomes.
• Incorporate circuit-level noise models with fault probabilities pC for cx, h, id, idm, x, y, z, measure, initialize, reset.
• Utilize leakage-aware post-selection to mitigate non-Pauli errors and improve decoding performance.
• Present experimental results from a 27-qubit IBM Quantum Falcon processor implementing the heavy-hexagon distance-3 code.

Experimental results

Research questions

• RQ1 How does decoder choice (perfect matching vs maximum likelihood) influence logical error rates in a multi-round subsystem code experiment?
• RQ2 Can real-time classical processing and feedback, including flag qubits and leakage mitigation, reduce logical errors in a heavy-hexagon code?
• RQ3 What are the practical benefits and trade-offs of deflagging and post-selection in decoder performance?
• RQ4 How do multi-round syndrome measurements (r rounds) affect the observed logical error rates for different logical states?
• RQ5 To what extent can circuit-level noise models capture the dominant error processes in this FT subsystem code experiment?

Key findings

• Logical error per syndrome-round can be as low as ~0.04 with a perfect matching decoder and ~0.035 with maximum likelihood decoding.
• Leakage-aware post-selection further reduces observed logical errors, with notable improvements when conditioning on leakage-free runs.
• Across r=0 to 4 rounds, ML decoding consistently outperforms matching decoders in the reported experiments.
• Deflagging flag information and performing real-time feedback help mitigate energy-relaxation-related errors in syndrome measurements.
• The results indicate decoders’ performance improvements are accessible with current hardware, suggesting further gains as hardware and decoding methods advance.