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 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. Joint work with Matthias Christandl.
Bio: Alexander Müller-Hermes received the Ph.D. degree in 2015 from the Technical University of Munich. After being a postdoc at the Centre of Mathematics in Quantum Theory (QMATH) at the University of Copenhagen, he obtained a Marie SkłodowskaCurie fellowship at Institute Camille Jordan at the Université Claude Bernard Lyon 1. Since 2021, he is associate professor at the Department of Mathematics at the University of Oslo. His research interests include the mathematical aspects of quantum information theory, quantum Shannon theory, mathematical cake cutting, and entanglement theory.
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