Quantum Feedback Control

1
Quantum Feedback Control

• John Sun
• Julian Gibbons

2
What is Feedback Control?

• A measurement and correction to disturbed
  information systems
• Used in
• Signal processing
• Telescopes
• As technology gets smaller, quantum effects must
  be considered

3
Classical vs. Quantum Information

• Classical information bits
• 0 or 1
• Quantum information qubits
• 0gt or 1gt
• Also allow linear superposition of 0gt and 1gt
• where
• We can only measure phase difference, so absolute
  phase is irrelevant

4
Bloch Sphere Picture

• Two degrees of freedom, so model with angles on a
  sphere (the Bloch sphere)
• Two orthogonal states are diametrically opposite
  each other
• Analogous to electron spin

5
Density Matrices

• Often more useful to use density matrices to
  represent a qubit
• For point on the Bloch sphere, we can
  write the qubit as
• where

6
Quantum Gates and Measurement

• Two ways to transform a state.
• A quantum gate, in which the action of the gate
  is described by conjugation by a matrix
• A measurement collapses the state into one of the
  states 0gt or 1gt
• Information is lost in the measurement
• Very different to classical systems

7
Example Thermostat

• Suppose we want an oven to maintain a fixed
  temperature
• Thermostat periodically checks whether the oven
  is too hot or too cold

• Temperature is then corrected based on this
  information
• Note that measurement is crucial, and does not
  disturb the system

8
Quantum Control
Noise
Quantum Control

• We send one of two non-orthogonal states, and
  correct for noise
• The state of a qubit cannot be cloned (no
  cloning theorem)
• We need a different approach to classical system

9
Weak Measurement

• Rather than measuring a qubit directly, we
  entangle with a meter qubit
• We do a rotation and then measurement on the
  meter state
• The amount of rotation governs
• How much information we gain
• How little we disturb the qubit
• This gives us a parameter to optimise

10
Our Control Scheme

• We want an easily implemented control scheme.
• We light in different polarisation states
• 0gt for horizontal polarisation
• 1gt for vertical polarisation
• We choose initial state of
• We then act on it with noise, and correct

z
x
11
Mixed States and Noise

• We consider noise that has a probability p of
  flipping the state, and (1 p) of doing nothing
• The result is called a mixed state, being a
  combination of quantum and classical
  uncertainties
• This brings our qubit inside the Bloch sphere
• We model this action on initial state
• ? by rotation around the x-axis

z
x
12
Diagrammatic Representation
Detector
Noise
13
The Beamsplitter

• The beamsplitter
• Perfectly transmits horizontally polarised light
• Partially reflects vertically polarised light
• Acts to entangle the meter with the input
• The meter state is taken to be
• 0gt if no detection is made
• 1gt if a detection is made
• Our parameter, ?, for measurement strength
  corresponds to the reflectivity parameter
• If we detect at the meter, the qubit has
  collapsed
• We therefore choose to replace it with 0gt

14
Fidelity

• We need to analyse the accuracy of our scheme
• The fidelity measures the similarity between the
  initial density state ? and the final state
  ?final (after noise, measurement, and correction)
• Returns a value between 0 (anti-correlation) and
  1 (perfect correlation)

15
Fidelity (our scheme)

• We adjust the reflectivity of the beamsplitter to
  optimise fidelity
• We try to solve (in ?) the equation ?F/?? 0,
  and then calculate the locally optimal fidelity

16
Fidelity (our scheme)

• But, ?F/?? 0 is gives solution
• Hence, our optimal fidelity only valid in a
  restricted domain in (p, ?)-space
• Outside this domain, we have no local maxima, so
  we use the boundary value (the do nothing
  scheme)

17
Fidelity (our scheme)
18
Fidelity (optimal scheme)

• There is an optimal control scheme, which
  produces fidelity
• We adjust the measurement signal strength to
  optimise fidelity
• Solve (in ?) the equation ?F/?? 0
• This gives

19
Fidelity (optimal scheme)
20
Fidelity (comparison)
21
Conclusion

• Fidelity is greatest for low noise
• Maximal fidelity for
• No noise
• Horizontal polarisation (perfectly correctible)
• Orthogonal states (noise has no effect)
• Our scheme can easily be implemented in a
  laboratory
• Our scheme is not optimal, but deviates in
  fidelity by at most 0.06 (approximately 6
  greater failure chance)

22
Acknowledgements

• We should like to thank the following people for
  this project
• Dr. Stephen Bartlett, project supervisor
• Prof. Dick Hunstead, TSP co-ordinator
• Agata Branczyk, Quantum Control of a Single Qubit
  (2005)
• The Bloch sphere diagram was taken from
• http//en.wikipedia.org/wiki/Bloch_sphere
• All plots generated with Mathematica 5.0