Quantum Coding withEntanglementMark M. WildeCommunication Sciences Institute,Ming Hsieh Department of Electrical Engineering,University of Southern California,Los Angeles, California 90089Quantum Lunch, Los Alamos National Lab (April 24, 2008)

Outline

Quantum Block CodePerform measurements that learn only about errorsEncode qubits with ancillasShor, PRA 52, pp. R2493-R2496 (1995).

Example Stabilizer for a Block CodeLaflamme et al., Physical Review Letters 77, 198-201 (1996).

Entanglement-Assisted Quantum Block CodeBrun, Devetak, Hsieh, Science 314, 436-439 (2006).

Example Stabilizer for an EA CodeBrun, Devetak, Hsieh, Science 314, 436-439 (2006).

Minimum Ebit Formulae for EA CodingWilde and Brun, arXiv:0804.1404 (2008).

Classical Convolutional Coding

FIR Encoding CircuitsFinite-duration input streams produce finite-duration output streams(corresponding to finite polynomials)

IIR Encoding CircuitsFinite-duration input streams can produce infinite-duration output streams(corresponding to rational polynomials)

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

Example Stabilizer for a QCCForney, Grassl, Guha, IEEE Trans. Inf. Theory 53, 865-880 (2007).

Entanglement-Assisted Quantum Convolutional CodingWilde and Brun, arXiv:0712.2223 (2007).

Example Stabilizer for an EAQCCWilde and Brun, arXiv:0712.2223 (2007).

Encoding Circuit for Example EAQCCRate (1/2,1/2)Wilde and Brun, arXiv:0712.2223 (2007).

Infinite-Depth Operations

Example Stabilizer for another EAQCCWilde and Brun, arXiv:0712.2223 (2007).

EAQCC Example 2Rate (1/2,1/2)

Classes of EAQCCs1) Finite-depth encoding and decoding circuits2) Finite-depth and infinite-depth encoding circuit, and Finite-depth decoding circuit

Advantages of EAQCCThe 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:

Unified Quantum Convolutional CodingResources for Quantum RedundancyEncoded Information

Goal of Unified QCCApproach optimal rates in the following grandfather resource inequality:Forms a portion of thethree-dimensional capacity region where the protocolconsumes nE ebits and n channel usesto send nQ noiseless qubits and nR noiseless classical bits.Devetak et al., In preparation, 2008.

Example of a [5,1,1;1,1] Unified QCCWilde and Brun, arXiv:0801.0821, Accepted for ISIT, Toronto, July 2008.

Current Work on EAQCCDeriving methods for general (non-CSS) entanglement-assisted quantum convolutional codes.Wilde and Brun, In preparation (2008).

Current Work on EAQCC

Three-Party EA Codes

Non-Additive EA CodesHave encoding circuit for classical indices j and one to encode the stabilizer

(similar to Grassl and Roetteler)Grassl and Roetteler, arXiv:0801.2144 (2008).

Conclusion and Future WorkImporting classical convolutional coding theory produces high-performance quantum codesCan convolutional quantum key distribution improve the Shor-Preskill noise threshold for BB84?Entanglement-assisted convolutional coding exploits entanglement to encode a stream of qubits