Quantum Coding withEntanglement

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

Quantum Lunch, Los Alamos National Lab (April 24, 2008)

Outline•Review techniques for Quantum Error Correction

(Including Entanglement-Assisted Coding)

•Entanglement-Assisted Quantum Convolutional CodingarXiv:0712.2223

•Unified Quantum Convolutional CodingarXiv:0801.0821

•Hint at new directionsarXiv:08??.????arXiv:09??.????

??

Quantum Block Code

Perform measurements that learn only about errorsEncode qubits with ancillas

Shor, PRA 52, pp. R2493-R2496 (1995).

Example Stabilizer for a Block Code

Unencoded Stabilizer Encoded Stabilizer

Laflamme et al., Physical Review Letters 77, 198-201 (1996).

Entanglement-Assisted Quantum Block Code

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

Example Stabilizer for an EA Code

Encoded StabilizerUnencoded Stabilizer

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

Minimum Ebit Formulae for EA Coding

Wilde and Brun, arXiv:0804.1404 (2008).

Given a set of generators H with good error-correcting properties

The minimum number of ebits the quantum code needs is

CSS code imported from 2 classical codes

Quantum code imported from classical GF(4) code

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(corresponding to finite polynomials)

IIR Encoding Circuits

Finite-duration input streams can produce infinite-duration output streams(corresponding to rational polynomials)

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).

Example Stabilizer for a QCC

Unencoded Stabilizer

Encoded Stabilizer

Forney, Grassl, Guha, IEEE Trans. Inf. Theory 53, 865-880 (2007).

Entanglement-Assisted Quantum Convolutional Coding

Wilde and Brun, arXiv:0712.2223 (2007).

Example Stabilizer for an EAQCC

Unencoded Stabilizer

Encoded Stabilizer

Wilde and Brun, arXiv:0712.2223 (2007).

Encoding Circuit for Example EAQCC

Classical conv. code

EAQCC

Rate (1/2,1/2)

Wilde and Brun, arXiv:0712.2223 (2007).

Infinite-Depth Operations

Implements Implements

Example Stabilizer for another EAQCC

Unencoded Stabilizer

Wilde and Brun, arXiv:0712.2223 (2007).

Encoded Stabilizer

EAQCC Example 2

Rate (1/2,1/2)

Classical conv. code

EAQCC

Classes of EAQCCs

1) Finite-depth encoding and decoding circuits

2) Finite-depth and infinite-depth encoding circuit, and Finite-depth decoding circuit

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:

Unified Quantum Convolutional CodingResources for Quantum Redundancy

Ancillas (Active and Passive)

Ebits (Active)

Gauge qubits (Passive)

Encoded Information

Quantum

Classical (Additional Passive)

Goal of Unified QCC

Approach 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 QCC

Wilde 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.

Important Technique

Equivalent Code

Wilde and Brun, In preparation (2008).

Current Work on EAQCC

•Have finished Alice’s encoding for a general EAQCC

•Have finished Bob’s decoding circuit method.

Quantum Check Matrix

Shifted Symplectic Product Matrix

(special form)

Three-Party EA Codes

Non-Additive EA Codes

Unencoded Subspaces

Ground Subspace

Have 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 Work

•Importing classical convolutional coding theory produces high-performance quantum codes

•Can convolutional quantum key distribution improve the Shor-Preskill noise threshold for BB84?

•Entanglement-assisted convolutional coding exploits entanglement to encode a stream of qubits