Runtime reduction in lattice surgery utilizing time-like soft information

TL;DR

This work addresses runtime optimization for lattice-surgery-based fault-tolerant quantum computing by leveraging time-like soft information associated with logical measurement errors. It introduces the time-like complementary gap $\Delta_{\mathcal{T}}(s)$ and shows how its magnitude $g$ serves as a reliability index to decide whether to remeasure with more syndrome cycles, enabling a two-step protocol that can reduce average runtime from $d+T$ toward $L+T$. The authors compare their approach to temporally encoded lattice surgery (TELS) and demonstrate that the time-like gap method often outperforms TELS for typical $k$, and that a soft-information-augmented combination (STELS) can yield $\gtrsim 50\%$ runtime reduction in practical regimes without extra qubit overhead. The results rely on MWPM-based decoders and numerical simulations of time-like and space-like error behavior, and they are claimed to be broadly applicable to lattice-surgery architectures and compatible with sequential (Litinski-style) compilation. Overall, the work proposes a fundamental runtime-optimization component for scalable quantum computing with surface codes.

Abstract

Runtime optimization of the quantum computing within a given computational resource is important to achieve practical quantum advantage. In this paper, we propose a runtime reduction protocol for the lattice surgery, which utilizes the soft information corresponding to the logical measurement error. Our proposal is a simple two-step protocol: operating the lattice surgery with the small number of syndrome measurement cycles, and reexecuting it with full syndrome measurement cycles in cases where the time-like soft information catches logical error symptoms. We firstly discuss basic features of the time-like complementary gap as the concrete example of the time-like soft information based on numerical results. Then, we show that our protocol surpasses the existing runtime reduction protocol called temporally encoded lattice surgery (TELS) for the most cases. In addition, we confirm that the combination of our protocol and the TELS protocol can reduce the runtime further, over 50% in comparison to the naive serial execution of the lattice surgery. The proposed protocol in this paper can be applied to any quantum computing architecture based on the lattice surgery, and we expect that this will be one of the fundamental building blocks of runtime optimization to achieve practical scale quantum computing.

Figures (19)

• Figure 1: Space-time diagram of the logical $XX$ measurement by lattice surgery. Time flows vertically in this figure. We only show the $X$ boundaries of the space-time diagram (orange planes). There are two possibilities of the logical errors during this operation: time-like logical errors (denoted by ${\mathcal{T}}$) and space-like ones (denoted by ${\mathcal{S}}$). In this paper, we consider the former. The time length (= number of repeated syndrome measurements) during the merge operation is denoted as $L$ and is typically chosen as $L = d$ to sufficiently suppress the time-like logical error rate.
• Figure 2: Distribution of $\Delta_{\mathcal{T}}$. Blue and orange histograms are obtained from the logical operation described in Fig. \ref{['fig:XXmeas']} with $d = L = 9$ and the stability experiment setup with $d = 10, L = 9$, respectively.
• Figure 3: Conditional logical error probability by the time-like gap value. We use several distributions obtained by different $(d, L)$ values, as well as changing the experimental setup (H-shape or stability). The solid black line shows the global fit result using the single-parameter fit function $f(g) = \frac{1}{1 + 10^{ag}}$.
• Figure 4: Distribution of the signed time-like gap obtained from the large stability experiment with $d = 26, L = \{ 7,8,9,10,11,12,13,14,15 \}$.
• Figure 5: Overview of the proposed runtime reduction protocol. (a) Performing the lattice surgery operation with a reduced syndrome measurement cycle, $L$. The time-like gap $g$ is calculated here. (b) If $g$ is smaller than the threshold value $g_{\rm th}$, we perform the lattice surgery operation with a full syndrome measurement cycle, $d$, to ensure the resultant logical measurement value. (c) If $g$ is larger or equal to $g_{\rm th}$, we can move on to the next operation. The optimal value $g_{\rm th}$ is determined by the condition that the logical error rate of the entire protocol keeps the same value of the original lattice surgery operation with $d$ cycle syndrome measurements (details are given in Appendix \ref{['appx:g_thdet']}).