Minimal memory requirements for pearl necklace encoders of quantum convolutional codes

One of the major goals in quantum computer science is to reduce the overhead associated with the implementation of quantum computers, and inevitably, routines for quantum error correction will account for most of this overhead. A particular technique for quantum error correction that may be useful in the outer layers of a concatenated scheme for fault tolerance is quantum convolutional coding. The encoder for a quantum convolutional code has a representation as a convolutional encoder or as a pearl necklace encoder. In the pearl necklace representation, it has not been particularly clear in the research literature how much quantum memory such an encoder would require for implementation. Here, we offer an algorithm that answers this question. The algorithm first constructs a weighted, directed acyclic graph where each vertex of the graph corresponds to a gate string in the pearl necklace encoder, and each path through the graph represents a non-commutative path through gates in the encoder. We show that the longest path through the graph corresponds to the minimal amount of memory needed to implement the encoder. A dynamic programming search through this graph determines the longest path. The running time for the construction of the graph and search through it is quadratic in the number of gate strings in the pearl necklace encoder.