Manybody Spin Echo and Quantum Walks in Functional

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 Hahn spin echo (~1950 s) 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

Hahn spin echo on a Bloch sphere

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), …

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

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

Hahn spin echo in the toggling frame Classical z-field B 0, in the toggling frame:

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

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

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

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

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

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

Intuition behind “quantum” walks Need a natural mechanism to explain how to satisfy exponential numbers of equations “Projection” Start Generates a function of t 2 Finish

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

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

Quadratic DD: 2 D 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

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

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

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.