Berkeley QC Seminar Feb' 14, 2002

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Continuous-TimeQuantum Error Correction
Andrew Landahl
?quant-ph/0110111?To appear in Phys. Rev. A
CollaboratorsCharlene AhnAndrew Doherty
Berkeley QC SeminarFeb. 14, 2002
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What is continuous-time quantum error correction?
I LoveYou
OliveYo
Every 5 sec.
Passive control
Active control
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Where is the noise coming from?
Quantum error correction
Fault-tolerant quantum computing
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How Quantum Error Correction Works
Measure the error, not the data!
Example Bit-flip code
Measure parity of neighboring qubits
Can protect any superposition from bit-flips
e.g.
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Strengths Weaknesses of QEC
Strengths

• Protects an unknown quantum state.

?
Weaknesses

• Uses projective measurements. (Too strong.)
• Uses unitary corrections. (Too fast.)

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The 12.5 Introduction to Continuous Measurement
Theory
Generalized measurement (POVM)
Continuous POVM
Stochastic Schrödinger equation
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Fully Quantum Loop
Backaction
Adaptive Measurement
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Strengths Weaknesses of QFC
Strengths

• Allows weak measurements.
• Allows Hamiltonian corrections.

Weaknesses

• Designed to protect known (target) quantum
  states.

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Stochastic Master Equations
Measure weakly, disturb weakly!
Direct measurement
Conditioned density matrix
Stochastic jump (localization)
Decoherence (backaction)
Hamiltonian evolution
Diffusive measurement
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Quantum Feedback Control
Current feedback Wiseman Milburn 1993
Extra feedback terms
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Quantum Feedback Control
Estimate feedback Doherty Jacobs 1999
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Continuous-time Quantum Error Correction
Only requires continuous weak Bell (parity)
measurement!
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Optimizing Quantum Control
General control Hamiltonian
Projector onto code space
Overlap of control with code space
Maximize subject to
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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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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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Measurement/Correction Tradeoff
Optimal measurement strength
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Future Directions

• Controlling dynamics of unknown quantum states.
• Designing new quantum algorithms using feedback
  control.
• Dual to quantum controllability quantum
  observability.
• Robustness and fault-tolerance using quantum
  control.
• Putting continuous quantum error correction to
  beneficial use in the laboratory.

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Continuous-TimeQuantum Error Correction
Andrew Landahl
?quant-ph/0110111?To appear in Phys. Rev. A
CollaboratorsCharlene AhnAndrew Doherty
Berkeley QC SeminarFeb. 14, 2002
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Quantum Feedback Control

• Goal
• Protect a known quantum state
• Tools
• Weak measurements
• Hamiltonians

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Quantum Error Correction

• Goal
• Protect an unknown quantum state
• Tools
• Projective measurements
• Unitary gates

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Continuous Quantum Measurement
Quantum Feedback Control
Current feedback Wiseman Milburn 1993
Estimate feedback Doherty Jacobs 1999
A. C. Doherty and K. Jacobs, Phys. Rev. A 60,
2700 (1999), quant-ph/9812004.
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Bit-flip QEFC results
Increasing measurement strength, no feedback
No Zeno effect!
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Bit-flip QEFC results
Increasing feedback strength
Theres an optimal feedback strength!
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Bit-flip QEFC results
Decreasing measurement strength
Overly strong feedback generates ringing
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Bit-flip QEFC results
Fixed measurement/feedback ratio (4)
Arbitrarily good as measurement strength
increases. (?)
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Bit-flip QEFC results
Fixed measurement/feedback ratio (1)
Does the crossing point persist?