Fault-tolerant Coding for Entanglement-Assisted Communication

TL;DR

This work addresses fault-tolerant entanglement-assisted communication by proving that the fault-tolerant capacity $C^{ea}_{\mathcal{F}(p)}(T)$ remains near the ideal capacity $C^{ea}(T)$ when gate error $p$ is small, using a modular construction based on the concatenated 7-qubit Steane code and fault-tolerant interfaces. It introduces fault-tolerant entanglement distillation to recover high-quality entanglement inside the code space and develops an information-theoretic framework for effective channels and arbitrarily varying perturbations (AVP), culminating in a threshold-type coding theorem that combines distillation with AVP-based coding. The results are built around an effective-channel reduction and a structured decomposition into distillation, classical communication, and fault-tolerant AVP coding, enabling near-capacity performance despite faults. This approach provides a practical, adaptable toolkit for fault-tolerant quantum communication, with explicit bounds and a clear path to applying these techniques to other fault-tolerant communication tasks and potentially to quantum repeaters and chip-scale quantum networks.

Abstract

Channel capacities quantify the optimal rates of sending information reliably over noisy channels. Usually, the study of capacities assumes that the circuits which sender and receiver use for encoding and decoding consist of perfectly noiseless gates. In the case of communication over quantum channels, however, this assumption is widely believed to be unrealistic, even in the long-term, due to the fragility of quantum information, which is affected by the process of decoherence. Christandl and Müller-Hermes have therefore initiated the study of fault-tolerant channel coding for quantum channels, i.e. coding schemes where encoder and decoder circuits are affected by noise, and have used techniques from fault-tolerant quantum computing to establish coding theorems for sending classical and quantum information in this scenario. Here, we extend these methods to the case of entanglement-assisted communication, in particular proving that the fault-tolerant capacity approaches the usual capacity when the gate error approaches zero. A main tool, which might be of independent interest, is the introduction of fault-tolerant entanglement distillation. We furthermore focus on the modularity of the techniques used, so that they can be easily adopted in other fault-tolerant communication scenarios.

Figures (7)

• Figure 1: Sketch of the setup for the effective channel. The fault-tolerantly implemented encoder $\Gamma_{\mathcal{C}_l}^{\mathcal{E}}$ takes input in the form of $m$ classical bits $x$ and $r$ physical qubits which are encoded in the code space. The resulting codewords are sent through $n$ copies of the quantum channel $T$, preceeded by the decoding interface. The output of the channel is fed into the encoding interface, whose output serves as input to the fault-tolerantly implemented decoder $\Gamma_{\mathcal{C}_l}^{\mathcal{D}}$, which also receives additional quantum input in the form of $s$ qubits in the code space. Theorem \ref{['thm-eff-channel']} shows that this setup is very close to a faultless setup with an effective channel, where the quantum systems are transformed by the perfect decoding operation $\mathop{\mathrm{Dec}}\nolimits^*$, represented by a triangle marked with a star. The effective channel $T_{p,N_l}$ receives input in the form of data qubits and a potentially correlated syndrome state.
• Figure 2: Basic setup for entanglement-assisted communication. The encoding map $\mathcal{E}$ maps a bit string $x$ of length $m$ and one part of each entangled state $\varphi$ to a quantum state in $\mathcal{M}_{d_A}^{\otimes n}$. The quantum channel $T$ acts on each of the $n$ subsystems, and the decoder $\mathcal{D}$ uses the other part of each maximally entangled state to decode the received quantum state to a bit string $x'$, which should be identical to the input bit string $x$. Note that classical information transfer is indicated by double lines (input into encoder, output of decoder), while the transfer of quantum states is indicated by single lines.
• Figure 3: Basic setup for our coding scheme for fault-tolerant entanglement-assisted communication, see Definition \ref{['defn:FTEACS']}. The encoding map $\mathcal{E}$ (yellow) encodes a bit string of length $m$ into a quantum state that serves as input into $n$ copies of the quantum channel $T$, and the decoding map $\mathcal{D}$ (blue) decodes the received quantum state back to a bit string. In the entanglement-assisted scenario, $\mathcal{E}$ and $\mathcal{D}$ are connected by $\sim nR_{ea}$ maximally entangled states. To make the communication fault-tolerant, the encoding and decoding circuits are implemented fault-tolerantly in an error correcting code $\mathcal{C}_l$ as $\Gamma_{\mathcal{C}_l}^{\mathcal{E}}$ and $\Gamma_{\mathcal{C}_l}^{\mathcal{D}}$, and combined with interfaces $\mathop{\mathrm{Enc}}\nolimits$ and $\mathop{\mathrm{Dec}}\nolimits$ mapping between the quantum states serving as input and output for $T$, and the quantum states being transformed in the fault-tolerantly implemented encoding and decoding circuits.
• Figure 4: Setup for entanglement distillation based on the protocol in DW03. Two parties each have access to one part of $k$ noisy entangled states $\phi_q$. One party performs local operations ${\mathcal{E}}^{\mathop{\mathrm{Dist}}\nolimits}$ and sends one-way classical communication to the other, who performs local operations ${\mathcal{D}}^{\mathop{\mathrm{Dist}}\nolimits}$. The output state of this scheme is close in fidelity to $(1-H(\phi_q))k$ copies of the maximally entangled state (cf. Theorem \ref{['thm-ent-dist-original']}).
• Figure 5: Setup for fault-tolerant entanglement distillation. The local operations performed in Figure \ref{['fig-ent-dist-normally']} are implemented in an error correcting code $\mathcal{C}_l$, and interfaces map the $k$ logical states into effective mixed states in the code-space (cf. Theorem \ref{['thm-ft-ent-dist-normally']}).

Theorems & Definitions (28)

• Definition 2.1: Interfaces, CMH20
• Theorem 2.2: Correctness of interfaces for the concatenated 7-qubit Steane code, CMH20
• Lemma 2.3: Effective encoding interface
• Lemma 2.4: Effective decoding interface
• Theorem 2.5: Effective channel with quantum input
• Definition 3.1: Entanglement-assisted coding scheme
• Remark 3.2
• Definition 3.3: Entanglement-assisted capacity
• Definition 3.4: Fault-tolerant entanglement-assisted coding scheme
• Remark 3.5