Effects of Decoherence in Quantum Control and Computing

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Effects of Decoherence in Quantum Control and
Computing
Leonid Fedichkin in collaboration with Arkady
Fedorov, Dmitry Solenov, Christino Tamon and
Vladimir Privman Center for Quantum Device
Technology, Clarkson University, Potsdam, NY
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Center for Quantum Device Technology Clarkson
University, www.clarkson.edu/CQDT
The main objective of our program has been the
exploration of coherent quantum mechanical
processes in novel solid-state semiconductor
information processing devices, with components
of atomic dimensions quantum computers,
spintronic devices, and nanometer-scale computer
logic gates. The achievements to date include
new modeling tools for evaluating initial
decoherence and transport associated with quantum
measurement, spin polarization control, and
quantum computer design, in semiconductor device
structures. Our program has involved an
interdisciplinary team, from Physics and
Electrical Engineering to Computer Science and
Mathematics, with extensive collaborations with
leading experimental groups and with Los Alamos
National Laboratory.
Design and calculation of the reliability of
nanometer-size computer components utilizing
technology based on transport through quantum
dots.
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Definition of Decoherence
Decoherence is any deviation of the coherent
quantum system dynamics due to environmental
interactions. Decoherence can be also understood
as an error (or a probability of error) of a QC
due to environmental interaction
(noise). Application of the error-correction
codes makes stable QC be possible provided the
decoherence rate is below some threshold.
Decoherence rate (the error per elementary QC
cycle) must be below 10-6 -10-4 Proposing any
QC design one must show that the decoherence rate
is below this threshold
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Theoretical Approach of Quantifying Decoherence

• Theoretical study of decoherence usually involves
  an open quantum system approach
• H HS HB HI

• All information of the system S including
  decoherence contains in the reduced density
  matrix of the reduced density matrix
• ?(t)TrBR(t)
• To obtain ?(t) we need adopt some appropriate
  approximation schemes.

HI
BATH HB
Quantum System S HS

• However, the effect of environment onto the
  system cannot be described by ?(t) itself. Need
  some numerical measure to quantify the
  environmental impact to the dynamics of the
  quantum system.

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Behavior of the Density Matrix Elements on
Different Time Scales
QC gate functions
??D ?
Bath
System
Quantum dynamics for short time steps, followed
by Markovian approximation, etc.
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Short-time Approximation
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Spin-boson Model in Short-time Approximation

• As an instructive example, we consider a general
  model of the two-level system interacting with
  boson-modes. The Hamiltonian of the system has
  the form,

• We obtain the following expression for the
  density matrix of the spin where B(t) is a
  spectral function defined below, L. Fedichkin, A.
  Fedorov and V. Privman, Proc. SPIE 5105, 243
  (2003).

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Entropy and Fidelity

• The measure based on entropy and idempotency
  defect, also called the first order entropy, can
  be defined

• Both expressions are basis independent, have a
  minimum, 0, at pure states and measure the degree
  of the states purity.

• The fidelity can be defined as

• The fidelity attains its maximal value, 1,
  provided

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Deviation Norm
We define a deviation from the ideal (without
environment) density operator according to
As a numerical measure we use an operator norm
In case of two-level system it is
Properties
and symmetric in ?(t) and ?(i)(t).
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Measures of Decoherence at Short Times
All measures depend not only on time but also on
the initial density matrix ?(0). For spin-boson
model they are, L.Fedichkin, A. Fedorov and V.
Privman, Proc. SPIE 5105, 243 (2003).
At t0, the value of the norm is equal to 0, and
then it increases to positive values, with
superimposed modulation at the systems
energy-gap frequency.
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Maximal Deviation Norm D(t)
The effect of the bath can be better quantified
by D(t)
Provides the upper bound for decoherence which
does not depend on initial conditions. This
measure is typically increase monotonically from
0, saturating at large times at a value D(?) ? 1.
For spin-boson model it is, L. Fedichkin, A.
Fedorov and V. Privman, Proc. SPIE 5105, 243
(2003).
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The Maximal Norm and Its Properties
Averaging over the initial density matrices
removes time-dependence at the frequencies of the
system, leaving only the relaxation temporal
dynamics
The evaluation of system dynamics is complicated
for multi-qubit systems. However, we established
approximate additivity that allow us to estimate
D(t) for several-qubit systems as well.
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Additivity for Multiqubit System
Entanglement is crucial for quantum computer
D is asymptotically additive for weakly
interacting even initially entangled qubits, as
long as it is small (close to 0) for each, namely
for short times. This is similar to the
approximate additivity of relaxation rates for
weakly interacting qubits at large times, L.
Fedichkin, A. Fedorov and V. Privman,
cond-mat/0309685 (2003).
This property was established for spin-boson
model with two types of interaction. The sum of
the individual qubit error measures provides a
good estimate of the error for several-qubit
system.
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The Influence of Decoherence on Mixing Time in
Quantum Walks on Cycle Graphs
Alternative Approach to Quantum Information
Processing Quantum Walks
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Motivation

• New family of quantum computer algorithm
  quantum walks based algorithms (3rd after
  quantum Fourier transform and Grovers
  iterations)
• Quantum walks may be easier to realize in
  experiment
• What effect does decoherence produce on algorithm?

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Hitting times
How long does it take for the walk to reach a
particular vertex?
More precisely, we say the hitting time of the
walk from a to b is polynomial in n if for some
tpoly(n) there is a probability 1/poly(n) of
being at b, starting from a.
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Hitting times quantum vs. classical
Farhi, Gutmann 97
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Experimental Realizations
Electron Coupled Double-Phosphorus Impurity in Si
L.C.L. Hollenberg, A.S. Dzurak, C. Wellard, A.R.
Hamilton, D.J. Reilly, G.J. Milburn, and R.G.
Clark, Phys. Rev. B 69, 113301 (2004)
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Experimental Realizations
Gate-engineered Quantum Double-Dot in GaAs
T. Hayashi, T. Fujisawa, H.-D. Cheong, Y.-H.
Jeong, Y. Hirayama, Phys. Rev. Lett. 91, 226804
(2003)
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Experimental Realizations
Gate-engineered Quantum Double-Dot in GaAs with
QPC
M. Pioro-Ladriere, R. Abolfath, P. Zawadzki, J.
Lapointe, S.A. Studenikin, A.S. Sachrajda, P.
Hawrylak, cond-mat/0504009
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Sketch of possible realization of system
considered
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Structure of each vertex
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System description
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Sketch of the graph and its density matrix
evolution
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Mapping of quantum walk on cycle on classical
dynamics of real variable SaЯ on torus
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The expression for SaЯ at small decoherence
rates
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The probability to find particle at vertex N/2
SN/2,N/2
0.1
0.08
N10 ?0.01
0.06
0.04
0.02
t
50
100
150
200
250
300
Green and blue curves are exponents with the
rates ?(N-1)/N and ?(N-2)/N correspondingly.
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The probability to find particle at vertex N/2
0.06
SN/2,N/2
0.05
0.04
N100 ?0.01
0.03
0.02
0.01
t
50
100
150
200
250
300
Blue curve corresponds to the exponent with the
rate ?(N-2)/N
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Probability distribution along the cycle as
function of time with (B) and without decoherence
(A)
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Classical dynamics (high decoherence rate)
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Quantum dynamics (low decoherence rate)
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Norm of Deviation from Mixed Distribution and its
the upper bound
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Mixing time vs. decoherence rate
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Mixing time vs. decoherence rate (loglog-scale)
Upper and lower bounds for N35 are shown
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Probability distribution along the hypercycle as
function of time with large (B) and small
decoherence (A)
N3, n3
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References

• D. A. Meyer, On the absence of homogeneous scalar
  unitary cellular automata, quant-ph/9604011
• D. Aharonov, A. Ambainis, J. Kempe, and U.
  Vazirani, Quantum walks on graphs,
  quant-ph/00121090
• E. Farhi and S. Gutmann, Quantum computation and
  decision trees, quant-ph/9707062
• A. M. Childs, E. Farhi, and S. Gutmann, An
  example of the difference between quantum and
  classical random walks, quant-ph/0103020
• C. Moore and A. Russell, Quantum walks on the
  hypercube, quant-ph/0104137
• A. M. Childs, R. Cleve, E. Deotto, E. Farhi, S.
  Gutmann, and D. A. Spielman, Exponential
  algorithmic speedup by quantum walk,
  quant-ph/0209131
• H. Gerhardt and J. Watrous, Continuous-time
  quantum walks on the symmetric group,
  quant-ph/0305182

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References

• D. Solenov and L. Fedichkin, Phys. Rev. A, in
  press quant-ph/0506096 quant-ph/0509078.
• A. Ambainis, Quantum walks and their algorithmic
  applications, quant-ph/0403120.
• S. A. Gurvitz, L. Fedichkin, D. Mozyrsky, G. P.
  Berman, Phys. Rev. Lett. 91, 066801 (2003).
• L. Fedichkin and A. Fedorov, Phys. Rev. A 69,
  032311 (2004).
• A. Fedorov, L. Fedichkin, and V. Privman,
  cond-mat/0401248, cond-mat/0309685,
  cond-mat/0303158.
• L. Fedichkin, D. Solenov, and C. Tamon, Quantum
  Inf. Comp., in press quant-ph/0509163.

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Summary I

• We consider one possible approach to quantify
  decoherence by maximal deviation norm. The useful
  properties such as monotonic behavior were
  demonstrated explicitly on the example of
  two-level system.
• We established additivity property of this
  measure of decoherence for multiqubit system at
  short times. It allows estimation of decoherence
  for complex systems in the regime of interest for
  quantum computing applications.

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Summary II

• The concept of quantum walks can be used to build
  new family of efficient quantum algorithms
• Devices with quantum walks behavior can be
  created by using nowadays technology
• The architecture of quantum walks quantum
  computer could be simpler than that of standard
  quantum computer
• We have developed and applied a new approach to
  evaluation of the effect of decoherence on
  quantum walks.
• The density matrix is approximated by explicit
  formula asymptotically exact for small
  decoherence rates
• The dependence of mixing time vs decoherence rate
  is nontrivial small decoherence can help!