[Paper Review] Correlated decoding of logical algorithms with transversal gates

Madelyn Cain, Zhao Chen+6 more|arXiv (Cornell University)|Mar 5, 2024

DNA and Biological ComputingBiochemistry, Genetics and Molecular Biology3 citations

TL;DR

This paper proposes correlated decoding to improve fault-tolerant quantum computation by jointly decoding logical qubits during transversal gates, leveraging deterministic error propagation across code blocks. By constructing a space-time decoding hypergraph that captures cross-qubit error correlations, the method reduces logical error rates and cuts syndrome extraction rounds by up to 50% in deep Clifford circuits, significantly lowering space-time overhead.

ABSTRACT

Quantum error correction is believed to be essential for scalable quantum computation, but its implementation is challenging due to its considerable space-time overhead. Motivated by recent experiments demonstrating efficient manipulation of logical qubits using transversal gates (Bluvstein et al., Nature 626, 58-65 (2024)), we show that the performance of logical algorithms can be substantially improved by decoding the qubits jointly to account for error propagation during transversal entangling gates. We find that such correlated decoding improves the performance of both Clifford and non-Clifford transversal entangling gates, and explore two decoders offering different computational runtimes and accuracies. In particular, by leveraging the deterministic propagation of stabilizer measurement errors through transversal Clifford gates, we find that correlated decoding enables the number of noisy syndrome extraction rounds between these gates to be reduced from $O(d)$ to $O(1)$ in Clifford circuits, where $d$ is the code distance. We verify numerically that this approach substantially reduces the space-time cost of deep logical Clifford circuits. These results demonstrate that correlated decoding provides a major advantage in early fault-tolerant computation, as realized in recent experiments, and further indicate it has considerable potential to reduce the space-time cost in large-scale logical algorithms.

Motivation & Objective

• To address the high space-time overhead of quantum error correction in large-scale logical algorithms.
• To improve decoder accuracy by exploiting structured error propagation during transversal entangling gates.
• To reduce the number of syndrome extraction rounds required in deep logical Clifford circuits.
• To demonstrate that correlated decoding outperforms independent decoding in both Clifford and non-Clifford gate scenarios.
• To provide a theoretical foundation for recent experimental results using reconfigurable neutral atom arrays.

Proposed method

• Construct a space-time decoding hypergraph that models error propagation across logical qubits during transversal gates.
• Use a maximum likelihood estimator (MLE) on the hypergraph to jointly decode errors across multiple logical qubits.
• Model error propagation from physical Pauli errors before transversal gates as superpositions of Pauli errors in the hypergraph.
• Include hyperedges for all terms in the Pauli error superposition after gate propagation to capture correlated error effects.
• Compare correlated decoding (using full hypergraph) against independent decoding (using single-qubit error sources only).
• Simulate noisy circuits using efficient Clifford circuit techniques, with Pauli noise pre-CCZ gate propagated through the CCZ operation.

Experimental results

Research questions

• RQ1 Can joint decoding of logical qubits during transversal gates reduce logical error rates compared to independent decoding?
• RQ2 To what extent can correlated decoding reduce the number of syndrome extraction rounds in deep Clifford circuits?
• RQ3 How does correlated decoding perform on non-Clifford transversal gates such as CCZ?
• RQ4 What is the impact of error propagation structure on decoder performance in transversal gate operations?
• RQ5 Can approximate decoders with efficient runtime still achieve high accuracy by leveraging error correlations?

Key findings

• Correlated decoding reduces the number of syndrome extraction rounds by up to 50% in deep logical Clifford circuits, significantly lowering space-time cost.
• The method improves logical error rate performance for both Clifford and non-Clifford transversal entangling gates, including the CCZ gate.
• A highly accurate MLE decoder using the full decoding hypergraph outperforms independent decoding, especially under high error rates.
• An approximate decoder with guaranteed efficient runtime achieves near-optimal performance by modeling error superpositions in the hypergraph.
• Error correlations from transversal gates—such as X errors copied from control to target in CNOTs—can be exploited to halve effective error density in decoding.
• The approach provides a theoretical foundation for experimental results in reconfigurable neutral atom quantum processors using transversal gates.