Quantum Annealing and the Schrödinger-Langevin-Kostin equation

Transcription

1 Quantum Annealing and the Schrödinger-Langevin-Kostin equation Diego de Falco Dario Tamascelli Dipartimento di Scienze dell Informazione Università degli Studi di Milano IQIS Camerino, October 28th 2008

2 Outline 1 Introduction 2 The SLK Equation Toy Models 3 4

3 References G. Jona-Lasinio, F. Martinelli, and E. Scoppola. New approach to the semiclassical limit of quantum mechanics. i. multiple tunnelig in one dimension. Comm. Math. Phys., 80: , B. Apolloni, C. Carvalho, and D. de Falco. Quantum stochastic optimization. Stoc. Proc. and Appl., 33: , G.E. Santoro and E. Tosatti. Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A: Math. Gen., 39:R393 R431, M.D. Kostin. On the Scrödinger-Langevin equation. J. Chem. Phys., 57(9): , R.P. Feynman. Quantum mechanical computers. Found. Phys., 16(6):507 31, D. de Falco and D. T. Speed and entropy of an interacting continuous time quantum walk. J. Phys. A: Math. Gen., 39: , F. Dominiguez-Adame and V. Malyshev. A simple approach to in one-dimensional disordered lattices. Am. J. Phys., 72: , H. Perets et al.. Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett., 100:170506, 2008.

4 Optimization problems and heuristic approach Combinatorial optimization: find good approximations to solution(s) of hard minimization problems. Thermal simulated annealing 1 : temperature dependent random walk on the space of all admissible solutions, endowed with a potential profile dependent on the cost function. 1 Kirkpatrick et al., Science, 220, , 1983

5 Quantum annealing V(x) SA Thermal jumps vs. Tunnelling QA x

6 Quantum annealing Use a Quantum Walk to explore the solution space: suggested by the behaviour of the stochastic process q ν(t) associated (G. Jona-Lasinio et al. 1981) to the ground state φ ν of the Hamiltonian H ν = ν2 + V (x) (1) 2 x 2 where V (x) encodes the cost function to be minimized. 2 x x Markov chain with state space determined by stable configurations t Long sojourns around minima of V (x) Rare large fluctuations from one minimum to another. 4

7 QA: Imaginary time evolution Hovever: The ground state of the Hamiltonian is seldom known. Approximations are required. Idea: Quantum Annealing (de Falco et al., 1989) 1 Start from a suitable initial condition φ trial (x) let it evolve under the Hamiltonian semigroup e thν, d 2 H ν = ν 2 dx + V (x) 2 for some time get an (unnormalized) approximation of φ ν decrease ν go back to step 2 until ν 0 (slowly enough).

8 Adiabatic computation vs. Dissipative dynamics Can we use viscous friction instead of an adiabatic change of the Hamiltonian to reach the ground state?

9 Outline Introduction The SLK Equation Toy Models 1 Introduction 2 The SLK Equation Toy Models 3 4

10 The SLK Equation The SLK Equation Toy Models SLK (Kostin, 1972) derives from Heisenberg-Langevin equation (Ford, Kac, Mazur, 1965). Represents the quantum analogue of classical motion with frictional force proportional to velocity (rough Drude-Lorentz model of Ohmic friction). If ψ(t, x) = p ρ(t, x)e ts(t,x) is a solution of the nonlinear, norm preserving, SLK equation ψ(t, x) i = ν2 2 ψ(t, x) + V (x)ψ(t, x) + βs(t, x) (2) t 2 x 2 then it satisfies d ψ(t) Hν ψ(t) 0, β 0. dt In the following we will show how this dissipative evolution can drive a suitable initial condition ψ(0, x) to the ground state φ ν(t).

11 Outline Introduction The SLK Equation Toy Models 1 Introduction 2 The SLK Equation Toy Models 3 4

12 The SLK Equation Toy Models Toy model 1: warm up and numerical check Require φ ν(x) = c + exp to be «(x a)2 + c 4σ+ 2 exp 1 the ground state of an Hamiltonian H ν 2 belonging to the eigenvalue 0. Those requirements determine the potential Set the initial condition ψ(0, x) = ν2 d 2 V (x) = 2φ ν(x) dx φν(x). 2 exp! (x+a)2 4σ 2 (2πσ 2 ) 1 /4 «(x + a)2 + c 4σ 2 0 exp x «2 4σ0 2

13 Toy model 1 The SLK Equation Toy Models E Ψt Ψt ΦΝ V x t 0 t tmax t t ψ(t) H ν ψ(t) decreases with time the vacuum overlap ψ(t) φ ν 2 approaches the value 1 some probability mass tunnels from the leftmost (local) to the rightmost (global) minimum. Since we know the ground state we can use this class of examples to check properties of the dynamics (convergence, convergence speed) and of the numerical methods used.

14 Toy model 2 The SLK Equation Toy Models Double-well potential: 8 (x >< V 2 a+) δx, for x 0 a+ V (x) = 4 (x >: V 2 a ) δx, for x < 0. a 4 (3) (parametrization as in 2 ) The local minimum of the potential is wider than the global one. In this case the gound state is unknown and we use SLK dynamics to find it. 2 G.E. Santoro and E. Tosatti. Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A: Math. Gen., 39:R393-R431, 2006.

15 Toy model 2 The SLK Equation Toy Models V x t 0 t tmax Wt x E Ψt x t ψ(t) H ν ψ(t) decreases with time ψ(t max, x) is a good approximation of the unknown ground state: compare V (x) with the potential W (t, x) = 1 2 ψ(t, x) + 2ψ(t,x) x 2 ψ(t) H ν ψ(t) t t max, of which ψ(t max, x) is the ground state belonging to the eigenvalue 0.

16 Outline Introduction 1 Introduction 2 The SLK Equation Toy Models 3 4

17 Quantum Optimization: General framework To make the intuition developed so far available on the context of quantum optimization (quantum annealing or adiabatic computation) was the main objective of the seminal paper on QA. 3 V : Q R: cost function; underlying graph: G = (Q, E), E possible moves in the search: e.g. Q = Q n = { 1, 1} n : search on the hypercube with edges between vertices at 1 Hamming distance. 3 B. Apolloni, C. Carvalho, and D. de Falco. Quantum stochastic optimization. Stoc. Proc. and Appl., 33: , 1989.

18 A simple case Here, we take the much simpler following example: Q = Λ s = {1, 2,..., s}; E = {{i, j} : (i, j) Λ s Λ s i j = 1}. Search by P an interacting continuous time quantum walk governed by h = 1 s 1 2 j=1 j + 1 j + j j P s j=1 V (j) j j. Quantum search of this kind can suffer from two quantum effects: but...

19 Viscosity as a resource... both problems can benefit of the introduction of a certain amount of viscous friction.

20 Outline Introduction 1 Introduction 2 The SLK Equation Toy Models 3 4

21 Introduction 400 t x 100 t x 100 The effect on ballistic (in the sense of defalco, D.T 2006) evolution of a linear potential V (x) = gx: energy-momentum relation E(p) = 1 cos p determines Bloch oscillations; this prevents the wave packet from approaching the point x = s where the minimum of V (x) is located; greedy (gradient) optimization hindered by the fact that, on a lattice, v (p) = sin p.

22 IDEA Introduction Idea: add some friction to prevent first Brillouin zone crossing.

23 Discrete Kostin equation Given the discrete equation ψ(t, x) i = (h ψ)(t, x) + K (t, x)ψ(t, x) = (4) t = 1 (ψ(t, x + 1) + ψ(t, x 1)) + V (x)ψ(t, x) + K (t, x)ψ(t, x). (5) 2 {z } {z } dissipation h the requirement s 1 d dt ψ(t) h ψ(t) = X (K (t, x + 1) K (t, x)) sin (S(t, x + 1) S(t, x)), x=1 is saftisfied, for example, if xx K (t, x) = β sin (S(t, y) S(t, y 1)) (7) y=2 (6)

24 Introduction Linear Potential + Viscous force: 400 Linear potential: 400 t t t x x g = 0, β = 0; s = 100, = 17 g0 = 2/s, 0 t 4s. 80 x t g = 3g0, β = 2g0 t g = 3g0, β = 4g0 :

25 Outline Introduction 1 Introduction 2 The SLK Equation Toy Models 3 4

26 t Random Gaussian noise: 2σ 0 = 2 (10/s) 3/ t t x x Linear potential: g = 3g 0. Kostin friction: β = 4g 0. pseudo-ballistic motion is much more stable than the truly inertial one with respect to the onset of Anderson localization. x t

27 Conclusions and can hinder a complete search of the solution space of the optimization problem. Viscous friction of Kostin suppresses both these effects. are suppressed by preventing the wavepacket momentum from crossing the first Brillouin zone. is avoided by probability percolation through the irregular potential profile.

28 Outlook The same framework developed in this note for the combinatorial optimization metaphor can be used, with minor changes, to describe an excitation travelling along a spin chain or a light pulse propagating through a waveguide lattice. 4 SLK dynamics can be exploited also in these fields. Increase the fidelity of state transmission, in presence of imperfections along a spin chain, by applying a tension at both ends of it. 5 The sole convergence toward the ground state could, instead, find applications in all-optical switching of light in waveguide arrays 6 : the injected light pulse can be steered toward a given position by a suitable tuning of the thermal gradient which determines the potential profile of the lattice. So far we considered the dissipative part. Inclusion of fluctuations is required. Extension to general graphs. First step: Hypercube. 4 H. Perets et al.. Phys. Rev. Lett., 100:170506, R.P. Feynman. Quantum mechanical computers. Found.Phys., 16(6):507-31, D.N. Christodoulides, F. Lederer, and Y. Silberberg. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature, 424: , 2003

29 Thank You!