Entanglement-Assisted Quantum Convolutional Coding (0712.2223v4)

Abstract: We show how to protect a stream of quantum information from decoherence induced by a noisy quantum communication channel. We exploit preshared entanglement and a convolutional coding structure to develop a theory of entanglement-assisted quantum convolutional coding. Our construction produces a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The rate and error-correcting properties of the classical convolutional codes directly determine the corresponding properties of the resulting entanglement-assisted quantum convolutional code. We explain how to encode our CSS entanglement-assisted quantum convolutional codes starting from a stream of information qubits, ancilla qubits, and shared entangled bits.

• The paper introduces a method for constructing entanglement-assisted quantum convolutional codes from two arbitrary classical binary convolutional codes.
• It details finite-depth and infinite-depth operations for encoding and decoding, emphasizing computational efficiency and effective error filtering.
• The study quantifies the entanglement needed by linking it to the rank of a derived matrix from classical parity-check matrices, expanding the scope of usable classical codes.

Entanglement-Assisted Quantum Convolutional Coding: A Strategic Synthesis of Classical Coding and Quantum Entanglement

The paper by Wilde and Brun presents a sophisticated theoretical framework for developing entanglement-assisted quantum convolutional codes (EAQCCs). It contributes significantly to the quantum error correction domain by leveraging classical coding theory and preshared quantum entanglement for protecting quantum information streams over noisy channels.

The authors develop a method to construct a Calderbank-Shor-Steane (CSS) entanglement-assisted quantum convolutional code using two arbitrary classical binary convolutional codes. This construction is a notable extension of previous quantum code frameworks, particularly because it circumvents the self-orthogonality limitation typical in importing classical codes, thus broadening the scope of classical codes applicable in the quantum context.

Structure and Core Contributions

1. Theoretical Construction:
   • The paper delineates a process for synthesizing quantum codes from classical counterparts using entanglement assistance. Quantum convolutional codes are crafted from two classical convolutional codes by forming a quantum check matrix incorporating the classical codes' parity-check matrices, without requiring the stringent orthogonality constraints.
   • This approach allows the use of any arbitrary classical binary convolutional code pair, significantly amplifying the potential quantum code repository.
2. Encoding and Decoding Techniques:
   • The paper introduces both finite-depth and infinite-depth operations in constructing the periodic encoding and decoding circuits. Finite-depth operations provide computationally efficient methods suitable for real-time applications.
   • Infinite-depth operations are proposed as a means to emulate filters akin to infinite-impulse-response filters in classical theories, albeit requiring operations free from noise to prevent error propagation.
3. Entanglement Utilization:
   • A major revelation is the quantification of entanglement required for encoding. It is shown that the minimum number of ebits needed corresponds to the rank of a derived matrix from the classical check matrices involved in the code construction.

Implications & Prospects

The expansion of applicable classical codes for quantum error correction materializes a novel landscape for high-performance quantum coding—all without demanding orthogonality in classical code selection—thereby harnessing the vast research in classical coding. This work addresses the ongoing need for adaptive and resilient quantum communication channels in the presence of decoherence, laying theoretical groundwork essential for future applications in quantum computing and communication systems.

The intricate mechanism involving infinite-depth operations, while theoretically robust, necessitates a noiseless operational context for practical use. However, this methodology hints at developing deeply integrated hybrid classical-quantum communication systems that exploit the synergies between classical redundancy and quantum entanglement, en route to achieving near-capacity quantum communications.

Future Avenues

Further exploration is warranted in adapting the EAQCCs for real-world settings, managing the noise in infinite-depth operations, and determining effective concatenation strategies to bridge quantum and classical communications. Additionally, opportunities lie in translating these codes for secure classical communication, leveraging the direct linkage of quantum coding with classical privacy capacities.

In summary, Wilde and Brun's paper offers a pivotal expansion in quantum error correction, integrating classical coding efficiency with quantum theory advantages, and paving the way for more versatile, high-performance quantum communication protocols.