1. 1 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

2. 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. 3 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

4. 4 Theoretical Approach of Quantifying Decoherence Quantum System S H S BATH H B HIHI Theoretical study of decoherence usually involves an open quantum system approach: H= H S + H B + H I All information of the system S including decoherence contains in the reduced density matrix of the reduced density matrix:  (t)=Tr B R(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. 5 Behavior of the Density Matrix Elements on Different Time Scales QC gate functions 1 2 Rest of the W orld, Bath-mode I I nteractions, Interactions with I I mpurities, Etc. S ystem B ath Quantum dynamics for short time steps, followed by Markovian approximation, etc.: ћω D →

6. 6 Short-time Approximation

7. 7 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).

8. 8 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 state’s “purity.” The fidelity can be defined as: The fidelity attains its maximal value, 1, provided

9. 9 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).

10. 10 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 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. 11 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).

12. 12 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.

13. 13 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.

14. 14 The Influence of Decoherence on Mixing Time in Quantum Walks on Cycle Graphs Alternative Approach to Quantum Information Processing: Quantum Walks

15. 15 Motivation New family of quantum computer algorithm: quantum walks based algorithms (3 rd 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. 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. 17 Hitting times: quantum vs. classical Theorem: Let G n 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. 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. 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. 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. 21 Sketch of possible realization of system considered

22. 22 Structure of each vertex

23. 23 System description

24. 24 Sketch of the graph and its density matrix evolution 0 1 2... N-1 N-2 N/2...

25. 25 Mapping of quantum walk on cycle on classical dynamics of real variable S αβ on torus 0 1 2... N-1 N-2 N/2...

26. 26 The expression for S αβ at small decoherence rates

27. 27 The probability to find particle at vertex N/2 50100150200250300 0.02 0.04 0.06 0.08 0.1 S N/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. 28 The probability to find particle at vertex N/2 50100150200250300 0.01 0.02 0.03 0.04 0.05 0.06 S N/2,N/2 N=100  =0.01 t Blue curve corresponds to the exponent with the rate  (N-2)/N

29. 29 Probability distribution along the cycle as function of time with (B) and without decoherence (A)

30. 30 Classical dynamics (high decoherence rate)

31. 31 Quantum dynamics (low decoherence rate)

32. 32 Norm of Deviation from Mixed Distribution and its the upper bound

33. 33 Mixing time vs. decoherence rate

34. 34 Upper and lower bounds for N=35 are shown Mixing time vs. decoherence rate (loglog-scale)

35. 35 N=3, n=3 Probability distribution along the hypercycle as function of time with large (B) and small decoherence (A)

36. 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. 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. 38 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.

39. 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!