arXiv 2009.071612020-09quant-phcs.ITmath-phmath.ITmath.MP

量子通信的容错编码

Fault-tolerant Coding for Quantum Communication

Matthias Christandl, Alexander Müller-Hermes

备注：: 56 pages, 6 figures. Corrected some mistakes and restructured the article
发表信息：: IEEE Transactions on Information Theory, vol. 70, no. 1, pp. 282-317, January 2024
DOI：: 10.1109/TIT.2022.3169438

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AI总结：

本研究结合容错量子计算与量子通信技术，提出量子容错容量并证明阈值定理，确保在门误差低于阈值时，以渐近消失误差实现接近经典容量的通信速率。

摘要：

设计编码和解码电路，以便在多次使用噪声信道时可靠地发送消息，是通信理论中的一个核心问题。在研究以渐近消失误差可实现的最优传输速率时，通常假设这些电路可以使用无噪声门来实现。虽然这一假设在许多场景下对经典机器是成立的，但在不久的将来，对于量子机器而言，预计不会成立，因为退相干会导致量子门出现故障。因此，关于量子信道编码实际相关性的基本问题仍然悬而未决。通过将容错量子计算的技术与量子通信的技术相结合，我们开始了对这些问题的研究。我们引入了量子容错版本的容量，这些容量量化了当编码和解码电路受到小概率门误差影响时，以渐近消失的总误差可实现的最优通信速率。我们的主要结果是经典容量和量子容量的阈值定理：对于每个量子信道 $T$ 和每个 $ε>0$，存在一个门误差概率阈值 $p(ε,T)$，当门误差概率低于该阈值时，大于 $C-ε$ 的速率可以以消失的总通信误差容错地实现，其中 $C$ 表示通常的容量。我们的结果不仅与远距离通信相关，而且在芯片上也相关，在芯片上，量子计算机的远距离部分可能需要在比影响本地门的噪声水平更高的噪声下进行通信。

英文摘要：

Designing encoding and decoding circuits to reliably send messages over many uses of a noisy channel is a central problem in communication theory. When studying the optimal transmission rates achievable with asymptotically vanishing error it is usually assumed that these circuits can be implemented using noise-free gates. While this assumption is satisfied for classical machines in many scenarios, it is not expected to be satisfied in the near term future for quantum machines where decoherence leads to faults in the quantum gates. As a result, fundamental questions regarding the practical relevance of quantum channel coding remain open. By combining techniques from fault-tolerant quantum computation with techniques from quantum communication, we initiate the study of these questions. We introduce fault-tolerant versions of quantum capacities quantifying the optimal communication rates achievable with asymptotically vanishing total error when the encoding and decoding circuits are affected by gate errors with small probability. Our main results are threshold theorems for the classical and quantum capacity: For every quantum channel $T$ and every $ε>0$ there exists a threshold $p(ε,T)$ for the gate error probability below which rates larger than $C-ε$ are fault-tolerantly achievable with vanishing overall communication error, where $C$ denotes the usual capacity. Our results are not only relevant in communication over large distances, but also on-chip, where distant parts of a quantum computer might need to communicate under higher levels of noise than affecting the local gates.

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