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

2. 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

3. Hahn spin echo for runners
Usain Bolt
Imambekov 3

4. Hahn spin echo on a Bloch sphere4

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), … 5

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

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 7

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

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

10. 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

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 11

12. 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

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

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 14

15. 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

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

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 17

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 18

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 19

20. 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

21. 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