Hardness results for decoding the surface code with Pauli noise

IF 5.1 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY Quantum Pub Date : 2024-10-28 DOI:10.22331/q-2024-10-28-1511
Alex Fischer, Akimasa Miyake
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Abstract

Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit.

In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent $X$ and $Z$ noise.
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利用保利噪声解码面码的难度结果
真正的量子计算机将受到复杂的、依赖于量子比特的噪声的影响,而不是简单的噪声,如所有量子比特强度相同的去极化噪声。如果我们的解码算法能考虑到有关特定噪声的先验信息,我们就能更有效地进行量子纠错。这促使我们考虑表面代码解码的复杂性,在这种情况下,解码问题的输入不仅是综合征测量结果,还有以每个量子比特的单量子比特保利误差概率为形式的噪声模型。在这种情况下,我们证明表面码的量子极大似然解码(QMLD)和退化量子极大似然解码(DQMLD)分别是 NP 难和 #P 难。我们通过展示如何将布尔公式转化为依赖于量子比特的保利噪声模型和编码该公式的可满足性属性的综合征集,直接从 SAT 对 QMLD 进行了简化,并从 #SAT 对 DQMLD 进行了简化。我们还给出了 QMLD 和 DQMLD 的近似硬度结果。这些都是最坏情况下的硬度结果,与许多高效表面代码解码器在平均情况下(即对于大多数联合集和大多数合理噪声模型)都是正确的这一经验事实并不矛盾。这些硬度结果与已知的 QMLD 和 DQMLD 的硬度结果非常相似,它们适用于具有独立 $X$ 和 $Z$ 噪声的任意稳定器代码。
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来源期刊
Quantum
Quantum Physics and Astronomy-Physics and Astronomy (miscellaneous)
CiteScore
9.20
自引率
10.90%
发文量
241
审稿时长
16 weeks
期刊介绍: Quantum is an open-access peer-reviewed journal for quantum science and related fields. Quantum is non-profit and community-run: an effort by researchers and for researchers to make science more open and publishing more transparent and efficient.
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