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
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
QuickTime™ and a
decompressor
are needed to see this picture.
QuickTime™ and a
decompressor
are needed to see this picture.
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
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
X
Y
Z
0
T
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 t2
Finish
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)
S starting state
X explored states
# unexplored target state
N=4
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
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.