Quantum Information Science: Algorithms, ErrorCorrection,

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Quantum Information Science Algorithms,
Error-Correction, and Control
CBA Lunch, Oct. 28 2002
Andrew Landahl
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Quantum measurement algorithms
A.M. Childs, E. Deotto, E. Farhi, J. Goldstone,
S. Gutmann, AJL
Phys. Rev. A 66, 032314 (2002) quant-ph/0204013
Can measurement backaction be harnessed to do
useful quantum computations?
Quantum Zeno effect
Adiabatic quantum algorithms
Quantum measurement algorithms
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Computation is Physical
Measurement circuit
Unitary circuit
Adiabatic
Topological field theory
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Adiabatic algorithms Farhi et al. 00
Search problem
Find the minimum of the function
Adiabatic solution

• Encode in the Hamiltonian .

• Prepare the ground state of
  .

• Evolve adiabatically from to .

• Measure the final energy .

N.B. By Lloyd 96 this can be simulated by a
quantum circuit.
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Quantum Zeno Effect
Pictorial analogy
Malus Law
http//230nsc1.phy-astr.gsu.edu/hbase/phyopt/polcr
oss.html
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Quantum measurement algorithms
Can we simulate adiabatic algorithms by the Zeno
effect? How well?
Two questions
How many measurements? How long does it take to
measure?
Perturbation theory
Phase estimation
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Continuous-time quantum error correction
C. S. Ahn, A. C. Doherty, AJL
Phys. Rev. A 66, 042301 (2002) quant-ph/0110111
Can unknown quantum states be protected from
noise using only continuous weak measurements and
Hamiltonian feedback?
Quantum error correction theory
Quantum feedback control theory
Continuous-time quantum error correction
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Passive Control

• System Isolation

• Fast dynamics

• Decoherence- Free Subspaces

Active Control
Fault-tolerant quantum computing

• Quantum Feedback Control

• Quantum Error Correction

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Quantum Error Correction
Measure the error, not the data!
Example Bit-flip code
Measure parity of neighboring qubits
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Quantum Feedback Control
Fully Quantum Loop
Backaction
Adaptive Measurement
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Features of Continuous QEC

• Target of control is a code space, not a state.
• Measurement, noise, and correction are
  simultaneous.
• Control solution is bang-bang.

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Bit-flip code model
Noise
Measurement
Feedback
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Optimizing Quantum Control
General control Hamiltonian
Projector onto code space
Cost function Overlap of control with code
space
Maximize subject to
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Fidelity of Performance
Delayed strong QEC
Fidelity
One qubit, no correction
Three qubits, no correction
Decoherence times
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Fixed Measurement Strength
Fidelity
Increasing feedback strength is better
Decoherence times
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Fixed Feedback Strength
Fidelity
Increasing measurement strength is not always
better
Decoherence times
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Quantum observers
AJL, S. Lloyd, J. J. Slotine
Research in progress
How well can one quantum system (the observer)
make a model of another quantum system?
Quantum mechanics
Classical observer theory
Quantum observer theory
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Issues to consider

• Quantum states cant be cloned Approximate
  cloning.
• Classically observable Initial state can be
  perfectly reconstructed from output history.
• Quantum mechanically observable 1 Quantum
  state can be reconstructed.
• Quantum mechanically observable 2 Set of
  observables can be reconstructed.
• Efficiency of process tomography?
• Semiclassical observers Nonlinear feedback
  induced.

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Quantum Information Science Algorithms,
Error-Correction, and Control
CBA Lunch, Oct. 28 2002
Andrew Landahl