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Adilet Imambekov Rice University

Many-body Spin Echo and Quantum Walks in Functional Spaces. Adilet Imambekov Rice University. Phys. Rev. A 84, 060302(R) (2011). in collaboration with. L. Jiang (Caltech, IQI). Outline. Generalization of the spin echo for arbitrary many-body quantum environments.

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Adilet Imambekov Rice University

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  1. Many-body Spin Echo and Quantum Walks in Functional Spaces Adilet Imambekov Rice University Phys. Rev. A 84, 060302(R) (2011) in collaboration with L. Jiang (Caltech, IQI)

  2. Outline Generalization of the spin echo for arbitrary many-body quantum environments Hahn spin echo (~1950s) Motivation and problem statement Uhrig dynamical decoupling (DD) (2007) Universal decoupling for quantum dephasing noise Beyond phase noise: adding relaxation, multiple qubits, ….: mapping between dynamical decoupling and quantum walks Conclusions and outlook

  3. Hahn spin echo for runners Usain Bolt Imambekov

  4. Hahn spin echo on a Bloch sphere

  5. Motivation Quantum computation: “software” to complement “hardware” for quantum error correction to work? Precision metrology Many experiments on DD: Marcus (Harvard), Yacoby (Harvard), Hanson (TU Delft), Oliver (MIT), Bollinger (NIST), Cory (Waterloo), Jianfeng Du (USTC, China), Suter (Dortmund), Davidson(Weizmann), Jelezko+Wrachtrup(Stuttgart), …

  6. Experiments with singlet-triplet qubit C. Barthel et al, Phys. Rev. Lett. 105, 266808 (2010)

  7. Problem statement How to protect an arbitrary unknown quantum state of a qubit from decoherence by using instant pulses acting on a qubit? quantum, non-commuting degrees of environment (can also be time-dependent) Spin components

  8. Hahn spin echo in the toggling frame Classical z-field B0, in the toggling frame:

  9. Uhrig Dynamical Decoupling (UDD) Slowly varying classical z-field Bz(t): N variables, N equations G.S. Uhrig, PRL 07

  10. Need to satisfy exponential in N number of equations Universality for quantum environments Slowly varying quantum operator Doesn’t have to commute with itself at different times:-(

  11. CDD and UDD: quantum universality Concatenated DD (CDD), Khodjasteh & Lidar, PRL 05 : Defined recursively by splitting intervals in half: is free evolution is a pulse along x axis Pulse number scaling ~ , but also works for quantum “dephasing” environments, kills evolution in order UDD is still universal for quantum environments!:-) Conjectured: B. Lee, W. M. Witzel, and S. Das Sarma, PRL 08 Proven: W.Yang and R.B. Liu, PRL 08

  12. Outer level Inner level t/T Beyond phase noise: adding relaxation Even for classical magnetic field, rotations do not commute! CONCATENATE! QDD: suggested by West, Fong, Lidar, PRL 10 N=2

  13. QDD: Quadratic Dynamical Decoupling Each interval is split in Uhrig ratios N=4 Y

  14. Multiple qubits, most general coupling KEEP CONCATENATING! NUDD: suggested in M.Mukhtar et al, PRA 2010, Z.-Y. Wang and R.-B. Liu, PRA 2011 N=2 t/T

  15. Finish Start Intuition behind “quantum” walks Need a natural mechanism to explain how to satisfy exponential numbers of equations “Projection” Generates a function of t2

  16. Quantum walk dictionary Basis of dimension (N+1)2: One can unleash the power of linear algebra now:-)

  17. UDD: 1D quantum walk Use block diagonal structure: (N+1)2 is reduced to(N+1) N=4 S starting state X explored states # unexplored target state

  18. Quadratic DD: 2D quantum walk Binary label Again, need to consider an exponential number of integrals …several pages of calculations…. S starting state X explored states # unexplored target state Proof generalizes for NUDD and all other known cases: e.g. CDD, CUDD + newly suggested UCDD

  19. DD vs classical interpolation? Equidistant grid is not the best for polynomial interpolation, need more information about the function close to endpoints (Runge phenomenon) 5th order 9th order

  20. In classical interpolation: suppose one needs to interpolate as a polynomial of (N-1)-th power based on values at N points. How to choose these points for best convergence of interpolation? Pick T,N, then Uhrig Ratios and Chebyshev Nodes Uhrig ratios split (0,1) in the same ratios as roots of Chebyshov polynomials T,Nsplit (-1,1).

  21. Finish Start Conclusions and Outlook Mapping between dynamical decoupling and quantum walks, universal schemes for efficient quantum memory protection Future developments: full classification of DD schemes for qubits (software meets hardware), multilevel systems (NV centers in diamond), DD to characterize “quantumness” of environments, new Suzuki-Trotter decoupling schemes (for quantum Monte Carlo), etc.

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