The rapid progress in quantum hardware is expected to make them viable tools for the study of quantum algorithms in the near term. The timeline to useful algorithmic experimentation can be accelerated by techniques that use many noisy shots to produce an accurate estimate of the observable of interest. One such technique is to encode the quantum circuit using an error detection code and discard the samples for which an error has been detected. An underexplored property of error-detecting codes is the flexibility in the circuit encoding and fault-tolerant gadgets, which enables their co-optimization with the algorthmic circuit. However, standard circuit optimization tools cannot be used to exploit this flexibility as optimization must preserve the fault-tolerance of the gadget. In this work, we focus on the $[[k+2, k, 2]]$ Iceberg quantum error detection code, which is tailored to trapped-ion quantum processors. We design new flexible fault-tolerant gadgets for the Iceberg code, which we then co-optimize with the algorithmic circuit for the quantum approximate optimization algorithm (QAOA) using tree search. By co-optimizing the QAOA circuit and the Iceberg gadgets, we achieve an improvement in QAOA success probability from $44\%$ to $65\%$ and an increase in post-selection rate from $4\%$ to $33\%$ at 22 algorithmic qubits, utilizing 330 algorithmic two-qubit gates and 744 physical two-qubit gates on the Quantinuum H2-1 quantum computer, compared to the previous state-of-the-art hardware demonstration. Furthermore, we demonstrate better-than-unencoded performance for up to 34 algorithmic qubits, employing 510 algorithmic two-qubit gates and 1140 physical two-qubit gates.

翻译：量子硬體的快速進展預計將使其在近期內成為研究量子算法的可行工具。通過利用大量有噪聲的量子比特樣本對感興趣的可觀測量進行精確估計的技術，可以加速實現實用算法實驗的時間表。其中一種技術是使用檢錯碼對量子電路進行編碼，並丟棄檢測到錯誤的樣本。檢錯碼的一個未被充分探索的特性是電路編碼和容錯邏輯門的靈活性，這使得它們能夠與算法電路進行協同優化。然而，標準的電路優化工具無法利用這種靈活性，因為優化必須保留邏輯門的容錯性。在本工作中，我們專注於$[[k+2, k, 2]]$ Iceberg量子檢錯碼，該編碼針對離子阱量子處理器進行了量身定製。我們為Iceberg碼設計了新的靈活容錯邏輯門，然後使用樹搜索將其與量子近似優化算法（QAOA）的算法電路進行協同優化。通過協同優化QAOA電路和Iceberg邏輯門，我們在Quantinuum H2-1量子計算機上、使用330個算法雙量子比特門和744個物理雙量子比特門、針對22個算法量子比特，將QAOA成功概率從$44\%$提升至$65\%$，後選擇率從$4\%$提升至$33\%$，相比於先前的先進硬件演示。此外，我們在最多34個算法量子比特上，使用510個算法雙量子比特門和1140個物理雙量子比特門，展示了優於未編碼的性能。