Universal Leakage Elimination Operator (LEO)

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ONE COMPONENT (or 1-d) QUANTUM MECHANICS AND A
UNIVERSAL LEAKAGE ELIMINATION OPERATOR

• Lian-Ao Wu
• Department of Theoretical Physics, The Basque
  Country University (EHU/UPV), Bilbao, Spain
• Ikerbasque, Basque Foundation for Science

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BILBAO, THE BASQUE COUNTRYONE-DIMENSIONAL CITY

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OUTLINE

• Feshbach P-Q partitioning technique and
  One-Component Exact Dynamical Equation (different
  perspective to look into finite dimensional QM)
• Introduce Leakage Elimination Operator (LEO)
• General condition for a type of quantum control
  Keeping a (quantum) system walking on a given
  state A(t)gt in the presence of environment.
• Examples
• Fast pulses and noise preserved quantum memory
• Noise or fast pulses induced Adiabaticity
• Application to the 3SAT algorithm
• Summary

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FESHBACH P-Q PARTITIONINGConsider a linear
equation of motion
We can write down the solution matrix as
We can also set , otherwise, rotate it
to
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LEAKAGE ELIMINATION OPERATOR ( LEO)

• Consider a Hamiltonian (L.-A. Wu, et al. PRL 89,
  127901)

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SUCH THAT (ideally)
Leakage from P space to Q is eliminated.
Examples Qubit
Approximate LEO
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ONE-COMPONENT DYNAMICAL EQUATION (P(0)1,Q(0)0)
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DERIVATION OF THE EQUATION
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Classical Harmonic oscillator Schrodingers equation Liouville's equation for open quantum system Quantum state diffusion equation for open system

Generalized coordinates and momenta H Hamiltonian wave function. L super-operator, Liouvillian. density matrix of the system. h non-Hermitan effective hamiltonian z is a state of bath
Examples of linear equations of motion
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REFERENCES

• L. -A. Wu, et al. Master Equation and Control an
  Open System with Leakage, Phys. Rev. Lett. 102,
  080405 (2009).
• J. Jing and L. -A. Wu, Control of decoherence
  with no control, Sci. Rep. 3, 2746 (2013).
• J.Jing, A. Bishop and L.-A. Wu, Nonperturbative
  dynamical decoupling with random control, Sci.
  Rep. 4,6299 (2014).
• J. Jing, L.-A. Wu, T. Yu, J. Q. You, Z.-M. Wang,
  L. Garcia, One-component dynamical equation and
  noise-induced adiabaticity, Phys. Rev. A 89,
  032110 (2014).
• H. F. Wang and L.-A. Wu, Fast quantum algorithm
  for EC3 problem with trapped ions.
  arXiv1412.1722
• J. Jing, L.-A. Wu, M. Byrd, J. Q. You, T. Yu,
  Z.-M. Wang, Nonperturbative Leakage Elimination
  Operators and Control of a Three-Level System, ,
  Phys. Rev. Lett. 114, 190502 (2015).

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ONE-COMPONENT EQUATION. tracing footprint of a
target state
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GENERAL CONDITION FOR FULL CONTROL
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ONE CAN HAVE DIFFERENT CONTROL FUNCTION OR PULSE
SEQUENCES
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BANG-BANG PULSES AND NOISE PROTECTED MEMORY
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Regular Fast Pulse Control
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CONTROL WITH RANDOMNESS AND EVEN BY NOISE
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Noise-Induced Adiabaticity SCHRÖDINGER EQUATION
AND EIGEN-EQUATION
Instantaneous Eigenstate
Dynamical Phase
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STANDARD ADIABATIC CONDITION
THE HAMILTONIAN in ROTATING REPRESENTATION
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ONE-COMPONENT DYNAMICAL EQUATION. tracing
footprint of target state
General Adiabatic (necessary) Condition
TWO-LEVEL SYSTEM
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Adiabatic and non-adiabatic passages
The fast-varying factor. It can be noise
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NOISE INDUCED ADIABATICITY
Continuous biased Poissonian white shot noise
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EXAMPLE OF STRENGTH NOISE
First s in rotating framework, frequency w
Second noise in rotating framework.
In lab framework, the model could be
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NOISE-INDUCED ADIABATICITY
Slow varying function
Fast varying Noise function
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APPLICATION SPEED UP QUANTUM ADIABATIC
ALGORITHM, 3SAT example
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CONCLUSION

• We first derive a one-component
  integro-differential equation by P-Q
  partitioning.
• We introduce an LEO to separate P-space from
  Q-space, and approximate LEO
• All given quantum paths may be realized in terms
  of fast-varying signals, even noise signals.
• We find that the effectiveness of LEOs
  exclusively depends on the integral of the pulse
  sequence or control function in the time domain,
  which has been missing for a long time.

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• Thanks for attentions
• Many thanks to organizers
• One joint postdoc opening