entanglement boosts quantum turbo codes mark m. wilde school of computer science mcgill university...
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Entanglement BoostsQuantum Turbo Codes
Mark M. WildeSchool of Computer Science
McGill University
Seminar for the Quantum Computing Group at McGillMontreal, Quebec, Canada
Wednesday, September 29, 2010
Joint work with Min-Hsiu HsieharXiv:1010.????
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Overview
Brief review of quantum error correction (entanglement-assisted as well)
Review of quantum convolutional codes and their properties
Adding entanglement assistance to quantum convolutional encoders
Review of quantum turbo codes and their decoding algorithm
Results of simulating entanglement-assisted turbo codes
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Quantum Error Correction
Shor, PRA 52, pp. R2493-R2496 (1995).
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Stabilizer Formalism
Laflamme et al., Physical Review Letters 77, 198-201 (1996).
Unencoded Stabilizer
UnencodedLogical Operators
Encoded Stabilizer
EncodedLogical Operators
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Distance of a Quantum Code
Distance is one indicator of a code's error correcting ability
To determine distance, feed in X,Y, Z acting on logical qubits and I, Z acting on ancillas:
It is the minimum weight of a logical operator that changes the encoded quantum information in the code
Distance is the minimum weight of all resulting operators
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Maximum Likelihood DecodingFind the most likely error
consistent with the channel model and the syndrome
MLD decision is
where
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Entanglement-Assisted Quantum Error Correction
Brun, Devetak, Hsieh. Science (2006)
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Entanglement-Assisted Stabilizer Formalism
Unencoded Stabilizer Encoded Stabilizer
UnencodedLogical Operators
EncodedLogical Operators
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Distance of an EA Quantum Code
Distance definition is nearly the same
It is the minimum weight of a logical operator that changes the encoded quantum information in the code
To determine distance, feed in X,Y, Z acting on logical qubits, I, Z acting on ancillas, and I acting on half of ebits:
Distance is the minimum weight of all resulting operators
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EA Maximum Likelihood DecodingFind the most likely error
consistent with the channel model and the syndrome
MLD decision is
where
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Quantum Convolutional Codes
H. Ollivier and J.-P. Tillich, “Description of a quantum convolutional code,” PRL (2003)
Memory
Memory
Example:
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State DiagramUseful for analyzing the properties of a quantum convolutional code
How to construct? Add an edge from one memory state to anotherif a logical operator and ancilla operator connects them:
State diagramfor our example encoder
Tracks the flow of logical operatorsthrough the convolutional encoder
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Catastrophicity
I
YI
II
II
II
II
II
II
I
Z
ZZ
YX
Z
triggered
Z
ZZ
not triggered
Z
ZZ
not triggered
Z
ZZ
not triggered
Z
ZZ
not triggered
Z
ZZ
not triggered
Catastrophic error propagation!
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Catastrophicity (ctd.)
Check state diagram for cycles of zero physical weightwith non-zero logical weight
(same as classical condition)
Viterbi. Convolutional codes and their performance in communication systems.IEEE Trans. Comm. Tech. (1971)
The culprit!
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Recursiveness
A recursive encoder has an infinite responseto a weight-one logical input
{X,Y,Z}
{I}
{I}
{I,Z}
{I,Z}
{I,Z}
{I,Z}
Responseshould beinfinite
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No-Go Theorem
Both recursiveness and non-catastrophicity aredesirable properties for a quantum convolutional encoder
when used in a quantum turbo code
But a quantum convolutional encoder cannot have both!(Theorem 1 of PTO)
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
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Idea: Add Entanglement
Memory
Memory
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
Example:
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State DiagramAdd an edge from one memory state to another
if a logical operator and identity on ebit connects them:
State diagramfor EA example encoder
Tracks the flow of logical operatorsthrough the convolutional encoder
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
Ebit removes half the edges!
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Catastrophicity
I
YI
II
II
II
II
II
II
I
Z
ZZ
YX
Z
triggered
Z
ZZ
triggered
Z
ZZ
triggered
Z
ZZ
triggered
Z
ZZ
triggered
Z
ZZ
triggered
Catastrophic error propagation eliminated!(Bell measurements detect Z errors)
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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Catastrophicity (ctd.)
Check state diagram for cycles of zero physical weightwith non-zero logical weight
Culprit gone!
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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Recursiveness
A recursive encoder has an infinite responseto a weight-one logical input
{X,Y,Z}
{I}
{I}
{I}
{I}
{I}
{I}
Responseshould beinfinite
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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Non-Catastrophic and Recursive Encoder
Entanglement-assisted encoders can satisfy both properties simultaneously!
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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Quantum Turbo Codes
A quantum turbo code consists of two interleaved andserially concatenated quantum convolutional encoders
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
Performance appears to be goodfrom the results of numerical simulations
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How to decode a Quantum Turbo Code?
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
Pauli error Detectsyndromes
Detectsyndromes
Estimateerrors
Do this last part with aniterative decoding algorithm
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Iterative Decoding
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
Three steps for each convolutional code:
1) backward recursion
2) forward recursion
1) local update
Decoders feed probabilistic estimates back and forthto each other until they converge on an estimate of the error
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Iterative Decoding (Backward Recursion)
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
Use probabilistic estimates of “next” memory and logical operators,and the channel model and syndrome,
to give soft estimate of “previous memory”:
Channel model
“Next memory”prob. estimate
syndrome
Log. op. estimate
Estimate“Previous memory”
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Iterative Decoding (Forward Recursion)
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
Use probabilistic estimates of “previous” memory and logical operators,and the channel model and syndrome,to give soft estimate of “next memory”:
Channel model
“Next memory”prob. estimate
syndrome
Log. op. estimate
Estimate“Previous memory”
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Iterative Decoding (Local Update)
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
Use probabilistic estimates of“previous” memory, “next memory”, and syndrome
to give soft estimate of logical ops and channel:
Channel model
“Next memory”prob. estimate
syndrome
Log. op. estimate
Estimate“Previous memory”
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Iterative Decoding of a Quantum Turbo Code
D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Transactions on Information Theory, vol. 55, no. 6, pp. 2776– 2798, June 2009.
“Exhaust”of outer code
Iterate this procedureuntil convergence or
some maximum number of iterations
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Simulations
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
Selected an encoder randomlywith one information qubit, two ancillas, and three memory qubits
Serial concatenation with itself givesa rate 1/9 quantum turbo code
Non-catastrophic and quasi-recursive
Distance spectrum:
Replacing both ancillas with ebits gives EA encoder
Non-catastrophic and recursive
Serial concatenation with itself givesa rate 1/9 quantum turbo code
with 8/9 entanglement consumption rate
Distance spectrum improves dramatically:
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Compare with the Hashing Bounds
Bennett et al., “Entanglement-assisted classical capacity,” (2002)Devetak et al., “Resource Framework for Quantum Shannon Theory (2005)
Rate 1/9 at ~0.16 EA rate 1/9 at ~0.49
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Unassisted Turbo Code
Pseudothreshold at ~0.098(comparable with PTO)
Within 2.1 dB of hashing bound
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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Fully Assisted Turbo Code
True Threshold at ~0.345
Within 1.53 dB of EA hashing bound(operating in a regime whereunassisted codes cannot!)
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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“Inner” Entanglement Assisted Turbo Code
True Threshold at ~0.279
Within 2.45 dB of EA hashing bound(operating 4.5 dB beyond
the unassisted codeand 1 dB past the midpoint!)
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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“Outer” Entanglement Assisted Turbo Code
Pseudo Threshold at ~0.145
Within 5.3 dB of EA hashing bound(far away from EA hashing bound)
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
AnotherPseudothreshold?
Quasi-recursiveness does not explain good performance of unassisted code!
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Adding Noise to Bob's Share of the Ebits
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
Could be best compromisein practice
(performance does not decreasemuch even with ebit noise)
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No-Go Theorem for Subsystem or Classically-Enhanced Codes
Encoder of the above form cannot be recursive and non-catastrophic
Proof: Consider recursive encoder.Change gauge qubits and cbits to ancillas (preserves recursiveness)Must be catastrophic (by PTO)Change ancillas back to gauge qubits and cbits (preserves catastrophicity).
M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” In preparation.
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Conclusion
Entanglement gives both a theoretical and practical boost to quantum turbo codes
Recursiveness is essential to good performance of the assisted code (not mere quasi-recursiveness)
No-Go Theorem for subsystem and classically-enhanced encoders
Open question: Find an EA turbo code with positive catalytic rate that outperforms a PTO encoder
Open question: Can turbo encoders with logical qubits, cbits, and ebits come close to achieving trade-off capacity rates?