Quantum State Compression with Polar Codes
In the quantum compression scheme proposed by Schumacher, Alice compresses a message that Bob decompresses. In that approach, there is some probability of failure and, even when successful, some distortion of the state. For sufficiently large blocklengths, both of these imperfections can be made arbitrarily small while achieving a compression rate that asymptotically approaches the source coding bound. However, direct implementation of Schumacher compression suffers from poor circuit complexity. In this paper, we consider a slightly different approach based on classical syndrome source coding. The idea is to use a linear error-correcting code and treat the state to be compressed as a superposition of error patterns. Then, Alice can use quantum gates to apply the parity-check matrix to her message state. This will convert it into a superposition of syndromes. If the original superposition was supported on correctable errors (e.g., coset leaders), then this process can be reversed by decoding. An implementation of this based on polar codes is described and simulated. As in classical source coding based on polar codes, Alice maps the information into the 'frozen' qubits that constitute the syndrome. To decompress, Bob utilizes a quantum version of successive cancellation coding.
