Entanglement-Assisted Quantum Convolutional Coding

Mermin proclaims it a “miracle” that quantum error correction is even possible [15]. Various obstacles such as the no-cloning theorem [16], measurement destroying a quantum state, and continuous quantum errors seem to pose an insurmountable barrier to a protocol for quantum error correction. Despite these obstacles, Shor demonstrated the first quantum error-correcting code that reduces the negative effects of decoherence on a quantum bit [1]. Shor’s code overcame all of the above difficulties and established the basic principles for constructing a general theory of quantum error correction [4][5][6].

Gottesman formalized the theory of quantum block coding by establishing the stabilizer formalism [4]. The stabilizer formalism allows one to import self-orthogonal classical block codes for use in quantum error correction [6]. This technique has the benefit of exploiting the large body of research on classical coding theory [17] for use in quantum error correction, but the self-orthogonality constraint limits the classical block codes that we can import.

Bowen was the first to extend the stabilizer formalism by providing an example of a code that exploits entanglement shared between a sender and a receiver [18]. The underlying assumption of Bowen’s code is that the sender and receiver share a set of noiseless ebits (entangled qubits) before quantum communication begins. Many quantum protocols such as teleportation [19] and superdense coding [20] are “entanglement-assisted” protocols because they assume that noiseless ebits are available.

Brun, Devetak, and Hsieh generalized Bowen’s example by constructing a theory of stabilizer codes that employs ancilla qubits and shared ebits for encoding a quantum error-correcting code [21,22]. The so-called entanglement-assisted stabilizer formalism subsumes the stabilizer formalism as the theory of active quantum error correction.

The major benefit of the entanglement-assisted stabilizer formalism is that we can construct an entanglementassisted quantum code from two arbitrary classical binary block codes or from an arbitrary classical quaternary block code. The rates and error-correcting properties of the classical codes translate to the resulting quantum codes. The entanglement-assisted stabilizer formalism may be able to reduce the problem of finding highperformance quantum codes approaching the quantum capacity [23][24][25][26][27] to the problem of finding good classical linear codes approaching the classical capacity [28].

Another extension of the theory of quantum error correction protects a potentially-infinite stream of quantum information against the corruption induced by a noisy quantum communication channel [29][30][31][32][33][34][35]. These quantum convolutional codes possess several advantages over quantum block codes. A quantum convolutional code typically has lower encoding and decoding complexity and superior code rate when compared to a block code that protects the same number of information qubits [35].

Forney et al. have determined a method for importing an arbitrary classical self-orthogonal quaternary code for use as a quantum convolutional code [34,35]. The technique is similar to that for importing a classical block code as a quantum block code [6]. One limitation of this technique is that the self-orthogonality constraint is more Typeset by REVT E X arXiv:0712.2223v4 [quant-ph] 2 Apr 2010 restrictive in the convolutional setting. Each generator for the quantum convolutional code must commute not only with the other generators, but it must commute also with any arbitrary shift of itself and any arbitrary shift of the other generators. Forney et al. performed specialized searches to determine classical quaternary codes that satisfy the restrictive self-orthogonality constraint [35].

In this paper, we develop a theory of entanglementassisted quantum convolutional coding for a broad class of codes. Our major result is that we can produce an entanglement-assisted quantum convolutional code from two arbitrary classical binary convolutional codes. The resulting quantum convolutional codes admit a Calderbank-Shor-Steane (CSS) structure [2,3,36]. The rates and error-correcting properties of the two binary classical convolutional codes directly determine the corresponding properties of the entanglement-assisted quantum convolutional code.

Our techniques for encoding and decoding are also an expansion of previous techniques from quantum convolutional coding theory. Previous techniques for encoding and decoding include finite-depth operations

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