Hybrid quantum error prevention, reduction, and correction methods

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Hybrid quantum error prevention, reduction, and
correction methods
Principles and Applications of Control in Quantum
Systems Caltech, Aug. 12, 2005
Phys. Rev. Lett. 91, 217904 (2003)

• Daniel Lidar
• University of
• Southern California

DFS-encoded
no encoding
Science 291, 1013 (2001)

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Symmetry Protects Q. Information
Symmetry ? conserved quantity quantum info.

• DFS Error Prevention
• use an existing exact symmetry to
• perfectly hide q. info. from bath

BB Decoupling Open-loop control dynamically
generate a symmetry
strong
time
bath correlation time
BANG free evolution
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Symmetric coin flipping noise
How to reliably store a single bit?
logical 0
logical 1
A noiseless subspace.
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Decoherence from System-Bath Interaction
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Decoherence-Free Subspaces
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Generalization Decoherence-free/noiseless
subsystemsE. Knill, R. Laflamme, L. Viola, PRL
84, 2525 (2000)
1D irreps condition not needed.
A theorem from C algebras
Model of decoherence
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Simplest DFS Example Collective Dephasing
DFS idea
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Why it Works
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Isotropic Quantum Errors Collective Decoherence
Model
Describes, e.g., low-T decoherence due to phonons
in wide range of solid state QC proposals
Do irreps analysis of n copies of su(2)
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All Decoherence-Free Subspaces/Subsystems for
Collective Decoherence
n
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How likely is a DFS?
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In the beginning
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What about symmetry breaking?D.L., I.L. Chuang,
K.B. Whaley, PRL 81, 2594 (1998) D. Bacon, D.L.,
K.B. Whaley, PRA 60, 1944 (1999)

• Symmetry breaking
• unequal coupling constants, lowering of symmetry
  by a perturbation, etc.

Mixed-state memory fidelity
Introduce perturbation
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Robustness of DFS to symmetry breaking
perturbations
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Strong Symmetry Breaking
Bare qubit two hyperfine
states of trapped 9Be ion

• Chief decoherence sources
• fluctuating long-wavelength ambient magnetic
  fields
• heating of ion CM motion during computation a
  symmetry-breaking process
• DFS encoding into pair of ions

DFS-encoded
Bare qubits
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Dealing with Symmetry Breaking
Classification of all decoherence processes on
two qubits
Enforce DFS conditions by BB pulses
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Bang-Bang DecouplingRefocusing method in NMR
Viola Lloyd Phys. Rev. A 58, 2733 (1998) Byrd
Lidar, Q. Inf. Proc. 1, 19 (2002)
More general symmetrization
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Eliminating all Leakage Errors Using Bang-Bang
Sorensen-Molmer Pulses
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Other Hybrid DFS-X Methods
PRL, in press (2005)
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Boulant et al., PRL 94, 130501 (2005)
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Computation Inside a DFS
Universal quantum computation fully compatible
with BB pulses DFS encoding, for a wide range
of systems governed by effective exchange
Hamiltonians
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Computation Inside a DFS
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Can we combine DFS, BB and universal QC?Yes
e.g., SM Pulses are Universal on 01gt,10gt Code
D.L., L.-A. Wu, Phys. Rev. Lett. 88, 017905
(2002) Phys. Rev. A 67, 032313 (2003).
All ingredients present for universal QC
decoherence suppression
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Heisenberg Systems

• Same method works, e.g., for spin-coupled
  quantum dots QC

Earlier DFS work showed universal QC with
Heisenberg interaction alone possible Bacon,
Kempe, D.A.L., Whaley, Phys. Rev. Lett. 85, 1758
(2000)
All ingredients available for Heisenberg-only QC
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Other applications of DFS/NS concept

• Overcoming reference frame alignment
• Fault-tolerant quantum cryptography
• Black holes

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Overcoming reference frame alignment
If Alice and Bob are communicating via spin-12
systems, it is generally presumed that they must
share a reference frame for spatial orientation
so that they may prepare and measure spin
components relative to this frame. But do they?
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Fault-tolerant quantum cryptography
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Hybrid Q. Error Correction The Big Picture
- symmetry not for free
Composite pulse method
- systematic (unknown) gate errors
- random gate errors
BB pulses (time-concatenated)
QECC (space-concatenated) also used for
Markovian part

• Universal fault tolerant QC with
• fewer qubits, fewer gates
• lower threshold

universal QC with naturally available
interactions