Quantum Convolutional Coding with Entanglement Assistance

Mark M. WildeCommunication Sciences Institute,

Ming Hsieh Department of Electrical Engineering,

University of Southern California,

Los Angeles, California 90089

Communication Sciences Institute,

Ming Hsieh Department of Electrical Engineering,

University of Southern California,

Los Angeles, California 90089

QEC07, Los Angeles, California (December 2007)

Summary

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

arXiv:0708.3699arXiv:0712.2223

with Hari Krovi and Todd Brunwith Todd Brun

Classical Convolutional Coding

Convolutional Coding techniques have application in

cellular deep space communicationand

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 errorsEncode qubits with ancillas

Entanglement-Assisted Quantum Block Code

Brun, Devetak, Hsieh, Science 314, 436-439 (2006).

Quantum Convolutional Coding

Ollivier, Tillich, PRL 91, 177902 (2003).Forney, Grassl, Guha, IEEE Trans. Inf. Theory 53, 865-880 (2007).Grassl, Rötteler, In proceedings of ISIT (2005,2006,2007).

Entanglement-Assisted Quantum Convolutional Coding

Wilde and Brun, arXiv:0712.2223 (2007).

EAQCC Example 1

Infinite-Depth Operations

Implements[ 1+D-1 | 1/(1+D) ]

Implements[ 1+D-1+D-3 | 1/(1+D+D3) ]

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:0708.3699 (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 detailThere is still much to explore in these areas

(QEC07@USC)