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  Hahn spin echo (~1950s)  Motivation and problem statement 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

Hahn spin echo for runners Usain Bolt Imambekov 3

Hahn spin echo on a Bloch sphere 4

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), … 5

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

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 7

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

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

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

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 11

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

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

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     14

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

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

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 17

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 18

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 19

Uhrig Ratios and Chebyshev Nodes Uhrig ratios split (0,1) in the same ratios as roots of Chebyshov polynomials T,N split (-1,1). 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 20

Conclusions and Outlook Mapping between dynamical decoupling and quantum walks, universal schemes for efficient quantum memory protection Start Finish 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. 21