Performance of two different quantum annealing correction codes

Abstract

Quantum annealing is a promising approach for solving optimization problems, but like all other quantum information processing methods, it requires error correction to ensure scalability. In this work, we experimentally compare two quantum annealing correction (QAC) codes in the setting of antiferromagnetic chains, using two different quantum annealing processors. The lower-temperature processor gives rise to higher success probabilities. The two codes differ in a number of interesting and important ways, but both require four physical qubits per encoded qubit. We find significant performance differences, which we explain in terms of the effective energy boost provided by the respective redundantly encoded logical operators of the two codes. The code with the higher energy boost results in improved performance, at the expense of a lower-degree encoded graph. Therefore, we find that there exists an important trade-off between encoded connectivity and performance for quantum annealing correction codes.

Appendices

Appendix 1: Optimizing ${\pmb \gamma }$

For each chain instance, we identified the optimal penalty coupling strength $\gamma$ by varying it in increments of 0.1 in the range [0, 1]. This is shown in Figs. 17, 18, 19, and 20 where we plot the success probability as a function of $\gamma$ and $\overline{N}$. We note that for the ${[4,1,4]}_{0}$ code the optimal penalty scales with $\alpha$, i.e., $\gamma _{\text {opt}}\propto \alpha$. Lower values of $\gamma _{\text {opt}}$ are observed on the S6 device. For the ${[3,1,3]}_{1}$ code, the optimal $\gamma$ is around $\gamma \approx 0.2$–0.3 for all $\alpha$ values studied, and the optimal values are unchanged across the two devices.

Appendix 2: Comparing decoding strategies

In the main text, we compared four strategies: U, C, the ${[4,1,4]}_{0}$ code, and ${[3,1,3]}_{1}$ code. We also used different decoding strategies: EM, EP, and CT. Figures 21 and 22 show all these strategies for a few chosen values of the scaling parameter $\alpha$ for the DW2-ISI and S6 devices, respectively. The U strategy is always worst. The ${[3,1,3]}_{1}$ code can be seen to outperform all other strategies at each $\alpha$ value for sufficiently long chains. The ${[4,1,4]}_{0}$ code outperforms the C strategy below a device-dependent $\alpha$ value and for sufficiently long chains. The fact that the success probabilities of the CT and EM strategies are nearly equal suggests that there are very few tied qubits in the ${[4,1,4]}_{0}$-encoded chains, an observation that holds for both devices.

In the main text, we also presented indirect evidence for the small number of ties in the ${[4,1,4]}_{0}$ code. Figure 23 shows this directly.