Fault-tolerant Coding for Quantum Communication

Abstract

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 $\epsilon >0$ there exists a threshold $p(\epsilon ,T)$ for the gate error probability below which rates larger than $C-\epsilon$ 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.

56 pages, 6 figures. Corrected some mistakes and restructured the article

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• Scalable Networking of Neutral-Atom Qubits: Nanofiber-Based Approach for Multiprocessor Fault-Tolerant Quantum Computer
• Mitigating errors in logical qubits
• Fundamental Limit on the Power of Entanglement Assistance in Quantum Communication
• Fault-tolerant Coding for Entanglement-Assisted Communication
• Concatenate codes, save qubits
• Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically
• Memory-corrected quantum repeaters with adaptive syndrome identification
• Explicit decoders using fixed-point amplitude amplification based on QSVT
• Information storage and transmission under Markovian noise