Fault-tolerant Coding for Quantum Communication

TL;DR

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 are introduced.

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. 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.

Figures (5)

• Figure 1: Basic communication setting.
• Figure 2: Our approach: Encoder and decoder are implemented in error correcting codes and interfaces are used to convert logical to physical states and vice-versa.
• Figure 3: Transforming faulty but well-behaved extended rectangles using Definition \ref{['defn:ExRecCorr']}.
• Figure 5: The circuit $\mathop{\mathrm{Enc}}\nolimits_{0\rightarrow 1}$ encoding the unknown state $| \psi \rangle$ into the first level of the $7$-qubit Steane code. Note that the logical CNOT gate is implemented in this code by applying elementary CNOT gates to each physical qubit. Here, $U_i$ denotes a certain Pauli gate (implemented in the $7$-qubit Steane code) depending on the outcome $i\in\lbrace 0,1,2,3\rbrace$ of the Bell measurement, and $\mathop{\mathrm{EC}}\nolimits$ denotes the error correction of the code.
• Figure 6: The circuit $\mathop{\mathrm{Dec}}\nolimits_{0\rightarrow 1}$ decoding the unknown state $| \overline{\psi} \rangle$ from the first level of the $7$-qubit Steane code to a physical qubit. Here, $U_i$ denotes a certain Pauli gate (applied to the physical qubit) depending on the outcome $i\in\lbrace 0,1,2,3\rbrace$ of the Bell measurement.

Theorems & Definitions (74)

• Definition 2.1: Pauli fault patterns and faulty circuits
• Definition 2.2: I.i.d. Pauli noise model
• Definition 2.3: Implementation of a quantum circuit
• Definition 2.4: Extended rectangles
• Definition 2.5: Well-behaved extended rectangles and quantum circuits
• Lemma 2.6: Transformation of well-behaved implementations
• proof
• Lemma 2.7: Threshold lemma
• Theorem 2.8: Threshold theorem II.
• Definition 2.9: Coding schemes for cq-channels