Quantum Convolutional Coding with Entanglement Assistance Mark M. Wilde Communication Sciences Institute, Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, California Communication Sciences Institute, Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, California QEC07, Los Angeles, California (December 2007)

Summary arXiv: arXiv: with Hari Krovi and Todd Brunwith Todd Brun

Classical Convolutional Coding Convolutional Coding techniques have application in cellulardeep space communicationan d Viterbi Algorithm is most popular technique for determining errors

FIR Encoding Circuits Finite-duration input streams produce finite-duration output streams

IIR Encoding Circuits Finite-duration input streams can produce infinite-duration output streams

Quantum Block Code Perform measurements that learn only about errors Encode qubits with ancillas

Entanglement-Assisted Quantum Block Code Brun, Devetak, Hsieh, Science 314, (2006).

Quantum Convolutional Coding Ollivier, Tillich, PRL 91, (2003). Forney, Grassl, Guha, IEEE Trans. Inf. Theory 53, (2007). Grassl, Rötteler, In proceedings of ISIT (2005,2006,2007).

Entanglement-Assisted Quantum Convolutional Coding Wilde and Brun, arXiv: (2007).

EAQCC Example 1

Infinite-Depth Operations Implements [ 1+D -1 | 1/(1+D) ] Implements [ 1+D -1 +D -3 | 1/(1+D+D 3 ) ]

EAQCC Example 2

Classes of EAQCCs 1) Finite-depth encoding and decoding circuits 2) Finite-depth and infinite-depth encoding circuit Finite-depth decoding circuit 3) Finite-depth and infinite-depth encoding circuit Finite-depth and infinite-depth decoding circuit (infinite-depth operations only on Bob’s half of the ebits)

Advantages of EAQCC The rate and error-correcting properties of the classical codes translate to the EAQCC. (high-performance classical codes => high-performance quantum codes) Produce an EAQCC from two arbitrary classical binary convolutional codes:

Block Entanglement Distillation

Convolutional Entanglement Distillation Wilde, Krovi, Brun, arXiv: (2007).

Conclusion Entanglement-assisted convolutional coding exploits entanglement to encode a stream of qubits Importing classical convolutional coding theory should produce high-performance quantum codes Explore the connection to quantum key distribution in more detail There is still much to explore in these areas