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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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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 (~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



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 B0,

in the toggling frame:


Uhrig Dynamical Decoupling (UDD)

Slowly varying classical z-field Bz(t):

N variables, N equations

G.S. Uhrig, PRL 07


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:-(


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


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


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


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


Quantum walk dictionary

Basis of dimension (N+1)2:

One can unleash the power of linear algebra now:-)


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


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


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


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).


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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