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

Outline Review techniques for Quantum Error Correction (Including Entanglement-Assisted Coding) Entanglement-Assisted Quantum Convolutional Coding arXiv: Unified Quantum Convolutional Coding arXiv: Hint at new directions arXiv:08??.???? arXiv:09??.???? ?

Quantum Block Code Perform measurements that learn only about errors Encode qubits with ancillas Shor, PRA 52, pp. R2493-R2496 (1995).

Example Stabilizer for a Block Code Unencoded StabilizerEncoded Stabilizer Laflamme et al., Physical Review Letters 77, (1996).

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

Example Stabilizer for an EA Code Encoded StabilizerUnencoded Stabilizer Brun, Devetak, Hsieh, Science 314, (2006).

Minimum Ebit Formulae for EA Coding Wilde and Brun, arXiv: (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 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 (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, (2003). Forney, Grassl, Guha, IEEE Trans. Inf. Theory 53, (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, (2007).

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

Example Stabilizer for an EAQCC Unencoded Stabilizer Encoded Stabilizer Wilde and Brun, arXiv: (2007).

Encoding Circuit for Example EAQCC Classical conv. code EAQCC Rate (1/2,1/2) Wilde and Brun, arXiv: (2007).

Infinite-Depth Operations Implements

Example Stabilizer for another EAQCC Unencoded Stabilizer Wilde and Brun, arXiv: (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 Coding Resources for Quantum Redundancy Ancillas (Active and Passive)Ebits (Active)Gauge qubits (Passive) Encoded Information QuantumClassical (Additional Passive)

Goal of Unified QCC Approach optimal rates in the following “grandfather” resource inequality: Forms a portion of the three-dimensional capacity region where the protocol consumes nE ebits and n channel uses to 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: , Accepted for ISIT, Toronto, July 2008.

Current Work on EAQCC Deriving 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: (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