quantum coding with entanglement mark m. wilde communication sciences institute, ming hsieh...
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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??.????
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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