1

Effects of Decoherence in Quantum Control and

Computing

Effects of Decoherence in Quantum Control and

Computing

Leonid Fedichkinin collaboration with Arkady Fedorov, Dmitry Solenov, Christino Tamon and Vladimir Privman

Center for Quantum Device Technology, Clarkson University, Potsdam, NY

2

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.

Center for Quantum Device Technology

Clarkson University, www.clarkson.edu/CQDT

Design and calculation of the reliability of nanometer-size computer components utilizing technology based on transport through quantum dots.

3

Definition of DecoherenceDefinition 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

4

Theoretical Approach of Quantifying Decoherence Theoretical Approach of

Quantifying Decoherence

Quantum System S

HS

BATH

HB

HI

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. 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.

5

Behavior of the Density Matrix Elements on Different

Time Scales

Behavior of the Density Matrix Elements on Different

Time Scales

t

( )n m t 1T

2T

/( ) E kTnnn e

( ) 0n m QC gatefunctions

1/ / kT1/ C

( )nn t

1

2

Rest of the World,

Bath-mode I

Interactions,

Interactions with I

Impurities, Etc.

SystemBath

Quantum dynamics for short time steps, followed by Markovian

approximation, etc :.

12

pure decoh1 12

erenceTT

ћωD →

6

Short-time ApproximationShort-time ApproximationIn the short-time approximation V. Privman, J. Stat. Phys. 110, 957 (2003) the time dependence of the overall density matrix ( )R t of the system and bath, is given by

( ) ( )( ) (0)S B I S B Ii H H H t i H H H tR t e R e .

As our short-time approximation, we utilize the following approximate relation, expressing the exponent in the previous equation as products of unitary operators,

3( ) ( ) 2 2( )S B I S SB Ii H H H t O t iH t iH ti H H te e e e .

We now consider the approximation to the matrix element,

2 2 2 2( ) ( )( ) (0)Tr S S S SB I B IiH t iH t iH t iH ti H H t i H H tmn Bt m e e e R e e e n .

7

Spin-boson Model in Short-time Approximation

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,

† †( )2S B I z k k k x k k k k

k k

H H H H a a g a g a

2 2

2 2

( ) ( )11 11 00

( ) ( )10 10 01

1 1( ) 1 (0) 1 (0)

2 21 1

( ) 1 (0) 1 (0).2 2

B t B t

i t B t B t

t e e

t e e e

•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).

8

Entropy and FidelityEntropy and Fidelity

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

2( ) ln , ( ) 1 TrS t s t •Both expressions are basis independent, have a minimum, 0, at pure states and measure the degree of the state’s “purity.”

( ) ( ) 0 ( ) ( ) ( )S t s t t t t

•The fidelity can be defined as:

•The fidelity attains its maximal value, 1, provided ( )( ) 1 ( ) ( ) ( ) ( )iF t t t t t

( ) ( )( ) Tr ( ) ( ) ( ) (0)S SiH t iH ti iF t t t t e e

9

Deviation NormDeviation Norm

We define a deviation from the ideal (without environment) density operator according to

ideal( ) ( ) ( );t t t

As a numerical measure we use an operator norm

max ii

In case of two-level system it is 2 2

00 01 .

ideal ( ) (0)S SiH t iH tt e e

Properties: ( )0 ( ) ( )it t

and symmetric in (t) and (i)(t).

10

Measures of Decoherence at Short Times

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). :

2

2

2

2 22 ( ) 201 011 00

2 2( ) 201 011 00

1 22 2( ) 201 011 00

1( ) 1 4 (0) sin [( 2) ](0) (0)

21

1 ( ) 1 4 (0) sin [( 2) ] ,(0) (0)2

1( ) 1 4 (0) sin [( 2) ] .(0) (0)

2

B t

B t

B t

s t e t

F t e t

t e t

At t=0, the value of the norm is equal to 0, and then it increases to positive values, with superimposed modulation at the system’s energy-gap frequency.

11

Maximal Deviation Norm D(t)

Maximal Deviation Norm D(t)

The effect of the bath can be better quantified by D(t)

(0)( ) sup( ( (0)) )D t t

2

1

( )

/1 2

1( ) 1 , for short times

2

( ) 1 , for large times assuming T =T /2

B t

t T

D t e

D t e

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).

12

The Maximal Norm and Its Properties

The Maximal Norm and Its Properties

0.0

0.1

0.2

0.3

0.4

0.5

t

D(t)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.

13

Additivity for Multiqubit System

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).

1 2(0) (0) (0) ... (0)!N

2 ( )1( ) ( ) ( ) , ( ) 1 .

2q

N NB t

S q q qq q

D t D t o D t D t e

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.

14

The Influence of Decoherence on Mixing Time in Quantum Walks

on Cycle Graphs

Alternative Approach to Quantum Information Processing: Quantum Walks

Alternative Approach to Quantum Information Processing: Quantum Walks

15

Motivation

New family of quantum computer algorithm: quantum walks based algorithms (3rd after quantum Fourier transform and Grover’s iterations)

Quantum walks may be easier to realize in experiment

What effect does decoherence produce on algorithm?

16

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 t=poly(n) there is a probability 1/poly(n) of being at b, starting from a.

17

Hitting times: quantum vs. classical

Theorem: Let Gn be a family of graphs with designated ENTRANCE and EXIT vertices.

Suppose the hitting time of the classical random walk from ENTRANCE to EXIT is polynomial in n.

Then the hitting time of the quantum walk from ENTRANCE to EXIT is also polynomial in n (for a closely related graph).

Farhi, Gutmann 97

18

Experimental Realizations

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)

Electron Coupled Double-Phosphorus Impurity in Si

19

Experimental Realizations

T. Hayashi, T. Fujisawa, H.-D. Cheong, Y.-H. Jeong, Y. Hirayama, Phys. Rev. Lett. 91, 226804 (2003)

Gate-engineered Quantum Double-Dot in GaAs

20

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

21

Sketch of possible realization of system considered

22

Structure of each vertex

23

System description

1 1

, ,0 0

N N

Graph Det j Int jj j

H H H H

, , , , , , , , , , , , Det j l j l j l j r j r j r j lr j l j r j r j l jl r lr

H E a a E a a a a a a

1

1 1 00

1;

4

N

Graph j j j j Nj

H c c c c c c

, , , , , ,Int j lr j j j l j r j r j l jlr

H c c a a a a

24

Sketch of the graph and its density matrix evolution

, , 1 1, 1, , 1 , ,14

d i

dt

0

1

2

...

N-1

N-2

N/2

...

25

α

β Sα,β+1Sα-1,β

Sα+1,β

Sα,β-1

Mapping of quantum walk on cycle on classical dynamics of real variable Sαβ on torus

, , 1 1, 1, , 1 , ,

11

4

dS S S S S S

dt

1 0

1 0

N

N

1 1 1

1 1 1

N

N

0

1

2

..

.

N-1

N-2

N/2

..

.

, ,aS i

26

The expression for Sαβ at small decoherence rates

,, 2

12

,0 2 ,0

1 1

, ,20 0

2

1exp 1

( )

221 exp sin 1

2exp 1

1 1

2exp cos sin 1

N

k k Nk

N N

k m k m Nk m

Nt O

NS t

N N

ikkit O

N N

Nt O

NN

k m k m i k mit O

N N N

27

The probability to find particle at vertex N/2

50 100 150 200 250 300

0.02

0.04

0.06

0.08

0.1 SN/2,N/2

N=10=0.01

t

Green and blue curves are exponents with the rates (N-1)/N and (N-2)/N correspondingly.

28

The probability to find particle at vertex N/2

50 100 150 200 250 300

0.01

0.02

0.03

0.04

0.05

0.06

SN/2,N/2

N=100=0.01

t

Blue curve corresponds to the exponent with the rate (N-2)/N

29

Probability distribution along the cycle as function of time with (B)

and without decoherence (A)

30

Classical dynamics (high decoherence rate)

0 4 8 12 16 190

250

5000 4 8 12 16 19

0

250

500

#

t

31

0 4 8 12 16 190

25

500 4 8 12 16 19

0

25

50

#

t

Quantum dynamics (low decoherence rate)

32

Norm of Deviation from Mixed Distribution and its the upper bound

33

Mixing time vs. decoherence rate

34

Upper and lower bounds for N=35 are shown

Mixing time vs. decoherence rate (loglog-scale)

35

N=3, n=3

Probability distribution along the hypercycle as function of time with large (B) and small decoherence (A)

Probability distribution along the hypercycle as function of time with large (B) and small decoherence (A)

36

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

37

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.

38

Summary ISummary 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.

39

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!