Minimal-memory realization of pearl-necklace encoders of general quantum convolutional codes

Quantum convolutional codes, like their classical counterparts, promise to offer higher error correction performance than block codes of equivalent encoding complexity, and are expected to find important applications in reliable quantum communication where a continuous stream of qubits is transmitted. Grassl and Roetteler devised an algorithm to encode a quantum convolutional code with a “pearl-necklace encoder.” Despite their theoretical significance as a neat way of representing quantum convolutional codes, they are not well-suited to practical realization. In fact, there is no straightforward way to implement any given pearl-necklace structure. This paper closes the gap between theoretical representation and practical implementation. In our previous work, we presented an efficient algorithm for finding a minimal-memory realization of a pearl-necklace encoder for Calderbank-Shor-Steane (CSS) convolutional codes. This work extends our previous work and presents an algorithm for turning a pearl-necklace encoder for a general (non-CSS) quantum convolutional code into a realizable quantum convolutional encoder. We show that a minimal-memory realization depends on the commutativity relations between the gate strings in the pearl-necklace encoder. We find a realization by means of a weighted graph which details the noncommutative paths through the pearl-necklace. The weight of the longest path in this graph is equal to the minimal amount of memory needed to implement the encoder. The algorithm has a polynomial-time complexity in the number of gate strings in the pearl-necklace encoder. Quantum error correction codes are used to protect quantum information from decoherence and operational errors [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Depending on their approach to error control, error correcting codes can be divided into two general classes: block codes and convolutional codes. In the case of a block code, the original state is first divided into a finite number of blocks of fixed length. Each block is then encoded separately and the encoding is independent of the other blocks. On the other hand, a quantum convolutional code [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] encodes an incoming stream of quantum information into an outgoing stream. Fast decoding algorithms exist for quantum convolutional codes [26] and in general, they are preferable in terms of their performance-complexity tradeoff [18]. The encoder for a quantum convolutional code has a representation as a convolutional encoder or as a pearl-necklace encoder. The convolutional encoder [12, 13], [26] consists of a single unitary repeatedly applied to a stream of quantum data (see Figure 1(a)). On the other hand, the pearl-necklace encoder (see Figure 1(b)) consists of several strings of the same unitary applied to the quantum data stream. Grassl and Rötteler [14] proposed an algorithm for encoding any quantum convolutional code with a pearl-necklace 1 ar X iv :1 00 9. 22 42 v1 [ qu an tph ] 1 2 Se p 20 10 Figure 1: Two different representations of the encoder for a quantum convolutional code. (a) Representation of the encoder as a convolutional encoder. (b) Representation of the encoder as a pearl-necklace encoder [1]. encoders. The algorithm consists of a sequence of elementary encoding operations. Each of these elementary encoding operations corresponds to a gate string in the pearl-necklace encoder. The amount of required memory plays a key role for implementation of any encoder, since this amount will result in overhead in the implementation of communication protocols. Hence any reduction in the required amount of memory will help in practical implementation of quantum computer. It is trivial to determine the amount of memory required for implementation of a convolutional encoder: it is equal to the number of qubits that are fed back into the next iteration of the unitary that acts on the stream. For example, the convolutional encoders in the Figures 1(a), 2(c) and 4(b) require two, one and four frames of memory qubits, respectively. In contrast, the practical realization of a pearl-necklace encoder is not explicitly clear. To make it realizable, one should rearrange the gate strings in the pearl-necklace encoder so that it becomes a convolutional encoder. In [1] we proposed an algorithm for finding the minimal-memory realization of a pearl necklace encoder for the CSS class of convolutional codes. This kind of encoder consists of CNOT gate strings only [27]. In this paper we extend our work to find the minimal-memory realization of a pearl-necklace encoder for a general (non-CSS) convolutional code. A general case includes all gate strings that are in the shiftinvariant Clifford group [14]: Hadamard gates, phase gates, controlled-phase gate string, finite-depth and infinite-depth [23, 25] CNOT operations. We show that there are many realization for a given pearl-necklace encoder which are obtained considering non-commutativity relations of gate strings in the pearl-necklace encoder. Then for finding the minimal-memory realization a specific graph, called non-commutativity graph is introduced. Each vertex in the non-commutativity graph, corresponds to a gate string in the pearl-necklace encoder. The graph features a directed edge from one vertex to another if the two corresponding gate strings do not commute. The weight of a directed edge depends on the degrees of the two corresponding gate strings and their type of non-commutativity. The weight of the longest path in the graph is equal to the minimal memory requirement for the pearl-necklace encoder. The complexity for constructing this graph is quadratic in the number of gate strings in the encoder. The paper is organized as follows. In Section 1, we introduce some definitions and notation that are used in the rest of paper. In Section 2, we define three different types of non-commutativity and then propose an algorithm to find the minimal memory requirements in a general case. In Section 3, we will summarize the contribution of this paper.