Correlated decoding of logical algorithms with transversal gates

TL;DR

This work demonstrates that decoding logical quantum algorithms jointly across qubits, to account for error propagation during transversal gates, can substantially reduce logical error rates and the space-time cost of fault-tolerant computation. It introduces a decoding hypergraph framework and two decoders (MLE and belief-HUF) to perform correlated decoding, showing improvements for both Clifford and non-Clifford transversal gates. Numerical simulations across perfect and noisy syndrome extraction, deep Clifford circuits, and transversal CCZ establish thresholds and performance gains, including a reduction of syndrome rounds from O(d) to O(1) in Clifford circuits. The results provide a theoretical and practical foundation for leveraging correlated decoding in early fault-tolerant experiments and scalable large-scale quantum algorithms.

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.

Figures (8)

• Figure 1: Decoding logical algorithms with transversal gates. (a) Transversal CNOT gates copy physical errors between logical qubits. (b) These transferred errors flip the same stabilizers on both logical qubits after the CNOT, generating a hyperedge in the decoding hypergraph connecting the control and target check vertices. (c) Given a physical error model, the decoding hypergraph for a logical algorithm can track these transferred errors, which can appear as hyperedges connecting checks from multiple logical qubits at different points in time.
• Figure 2: Decoding the transversal CNOT. (a) We generate a logical Bell pair using a noisy transversal CNOT, with physical errors before and after the CNOT with probabilities $pf_b$ and $p(1-f_b)$, respectively. (b) When the errors occur before the CNOT ($f_b = 1$), the threshold of uncorrelated MWPM (pink) is half that of MLE (blue). (c) As $f_b$ increases, more errors are transferred between the logical qubits, and MLE increasingly outperforms MWPM.
• Figure 3: Correlated decoding of deep logical circuits. (a) Stabilizer measurement errors near transversal CNOTs can appear as hyperedges connecting the checks of both logical qubits at different times. (b) We study deep logical Clifford circuits with $n_r$ rounds of syndrome extraction between CNOTs. (c) Setting $n_r = 1$, our results are consistent with thresholds of $p_\text{th} \geq 0.56\%$ for belief-HUF (green) and $p_\text{th} \geq 0.87\%$ for MLE (blue).
• Figure 4: Reducing the space-time cost of deep logical circuits. (a) The rounds of syndrome extraction per CNOT can be reduced to $n_r\simeq \frac{1}{2}$ for MLE (bottom) and $n_r\simeq 3$ for belief-HUF (top), without increasing $P_L$. (b) The extrapolated space-time cost to reach $P_{L} = 10^{-6}$ is minimized at $n_r = 1$ for belief-HUF and $n_r = \frac{1}{4}$ for MLE.
• Figure 5: Decoding the transversal CCZ. (a) A physical $XII$ error before a CCZ propagates to $\frac{1}{2}(XII + XIZ + XZI - XZZ)$. (b) The decoding hypergraph includes a hyperedge for each Pauli error in the superposition. (c) The logical error rate is suppressed by a factor of $\simeq 1.5$ by decoding the logical qubits jointly.