IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum Annealing and the Schrödinger-Langevin-Kostinequation
Diego de Falco Dario Tamascelli
Dipartimento di Scienze dell’InformazioneUniversità degli Studi di Milano
IQIS Camerino, October 28th 2008
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Outline
1 Introduction
2 The Schrödinger-Langevin-Kostin equation: Continuous caseThe SLK EquationToy Models
3 SLK Equation: Discrete CaseQuantum optimizationBloch oscillationsAnderson localization
4 Conclusions and Outlook
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
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., 80:223 – 254, 1981.
B. Apolloni, C. Carvalho, and D. de Falco.
Quantum stochastic optimization.Stoc. Proc. and Appl., 33:223–244, 1989.
G.E. Santoro and E. Tosatti.
Optimization using quantum mechanics: quantum annealing through adiabatic evolution.J. Phys. A: Math. Gen., 39:R393–R431, 2006.
M.D. Kostin.
On the Scrödinger-Langevin equation.J. Chem. Phys., 57(9):3589–3591, 1972.
R.P. Feynman.
Quantum mechanical computers.Found. Phys., 16(6):507–31, 1986.
D. de Falco and D. T.
Speed and entropy of an interacting continuous time quantum walk.J. Phys. A: Math. Gen., 39:5873–5895, 2006.
F. Dominiguez-Adame and V. Malyshev.
A simple approach to Anderson localization in one-dimensional disordered lattices.Am. J. Phys., 72:226–230, 2004.
H. Perets et al..
Realization of quantum walks with negligible decoherence in waveguide lattices.Phys. Rev. Lett., 100:170506, 2008.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Optimization problems and heuristic approach
Combinatorial optimization: find good approximations to solution(s) ofhard minimization problems.
Thermal simulated annealing1: temperature dependent random walk onthe space of all admissible solutions, endowed with a potential profiledependent on the cost function.
1Kirkpatrick et al., Science, 220, 671-680, 1983de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum annealing
SA
QA
x
V(x)
Thermal jumps vs. Tunnelling
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum annealing
Use a Quantum Walk to explore the solution space: suggested by thebehaviour of the stochastic process qν(t) associated (G. Jona-Lasinio etal. 1981) to the ground state φν of the Hamiltonian
Hν = −ν2
2∂2
∂x2 + V (x) (1)
where V (x) encodes the cost function to be minimized.
100 200 300 400 500t
-4
-2
2
4
x
-4 -2 2 4x Markov chain with state space
determined by stableconfigurations.
Long sojourns around minima ofV (x)
Rare large fluctuations from oneminimum to another.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
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 φtrial (x) . . .2 . . . let it evolve under the Hamiltonian semigroup e−tHν ,
Hν = −ν2
d2
dx2 + V (x)
for “some” time . . .3 . . . get an (unnormalized) approximation of φν .4 . . . decrease ν . . .5 . . . go back to step 2 until ν → 0 (slowly enough).
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Adiabatic computation vs. Dissipative dynamics
Can we use viscous friction instead of an adiabatic change of theHamiltonian to reach the ground state?
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
The SLK EquationToy Models
Outline
1 Introduction
2 The Schrödinger-Langevin-Kostin equation: Continuous caseThe SLK EquationToy Models
3 SLK Equation: Discrete CaseQuantum optimizationBloch oscillationsAnderson localization
4 Conclusions and Outlook
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
The SLK EquationToy Models
The SLK Equation
SLK (Kostin, 1972) derives from Heisenberg-Langevin equation (Ford,Kac, Mazur, 1965).
Represents the quantum analogue of classical motion with frictionalforce proportional to velocity (rough Drude-Lorentz model of Ohmicfriction).
If ψ(t , x) =pρ(t , x)etS(t,x) is a solution of the nonlinear, norm preserving,
SLK equation
i∂ψ(t , x)
∂t= −ν
2
2∂2ψ(t , x)
∂x2 + V (x)ψ(t , x) + βS(t , x) (2)
then it satisfiesddt〈 ψ(t) |Hν | ψ(t) 〉 ≤ 0, β ≥ 0.
In the following we will show how this dissipative evolution can drive asuitable initial condition ψ(0, x) to the ground state φν(t).
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
The SLK EquationToy Models
Outline
1 Introduction
2 The Schrödinger-Langevin-Kostin equation: Continuous caseThe SLK EquationToy Models
3 SLK Equation: Discrete CaseQuantum optimizationBloch oscillationsAnderson localization
4 Conclusions and Outlook
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
The SLK EquationToy Models
Toy model 1: warm up and numerical check
Require
φν(x) = c+ exp„− (x − a)2
4σ2+
«+ c− exp
„− (x + a)2
4σ2−
«+ c0 exp
„− x2
4σ20
«to be
1 the ground state of an Hamiltonian Hν2 belonging to the eigenvalue 0.
Those requirements determine the potential
V (x) =ν2
2φν(x)
d2
dx2 φν(x).
Set the initial condition ψ(0, x) =exp
− (x+a)2
4σ2−
!(2πσ2
−)1/4
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
The SLK EquationToy Models
Toy model 1
VHxLt=0t=tmax
-4 -2 2 4
0 10 20 30 40 50t
0.02
0.04
0.06
0.08
0.10
0.12
0.14
EΨt
0 10 20 30 40 50t
0.8
0.85
0.9
0.95
1.
ÈXΨt ÈΦΝ\È2
〈 ψ(t) |Hν | ψ(t) 〉 decreaseswith time
the “vacuum overlap”|〈 ψ(t) | φν 〉|2 approaches thevalue 1
some probability mass“tunnels” from the leftmost(local) to the rightmost (global)minimum.
Since we know the ground state wecan use this class of examples tocheck properties of the dynamics(convergence, convergence speed)and of the numerical methods used.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
The SLK EquationToy Models
Toy model 2
Double-well potential:
V (x) =
8><>:V0
(x2−a2+)2
a4+
+ δx , for x ≥ 0
V0(x2−a2
−)2
a4−
+ δx , for x < 0.(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 tofind it.
2G.E. Santoro and E. Tosatti. Optimization using quantum mechanics: quantum annealingthrough adiabatic evolution. J. Phys. A: Math. Gen., 39:R393-R431, 2006.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
The SLK EquationToy Models
Toy model 2
VHxLt=0t=tmax
-4 -2 2 4 6
0 10 20 30 40 50t
0.89
0.55
EΨt
-4 -2 2 4x
0.5
1.0
1.5
2.0
WtHxL
〈 ψ(t) |Hν | ψ(t) 〉 decreaseswith time
ψ(tmax , x) is a goodapproximation of the unknownground state: compare V (x)with the potentialW (t , x) = 1
2ψ(t,x)∂2
∂x2ψ(t , x) +
〈 ψ(t) |Hν | ψ(t) 〉 t → tmax , ofwhich ψ(tmax , x) is the groundstate belonging to theeigenvalue 0.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Outline
1 Introduction
2 The Schrödinger-Langevin-Kostin equation: Continuous caseThe SLK EquationToy Models
3 SLK Equation: Discrete CaseQuantum optimizationBloch oscillationsAnderson localization
4 Conclusions and Outlook
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Quantum Optimization: General framework
To make the intuition developed so far available on the context ofquantum optimization (quantum annealing or adiabatic computation) wasthe 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 = Qn = {−1, 1}n: search on the hypercube with edges betweenvertices at 1 Hamming distance.
3B. Apolloni, C. Carvalho, and D. de Falco. Quantum stochastic optimization. Stoc. Proc. andAppl., 33:223-244, 1989.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
A simple case
Here, we take the much simpler following example:
Q = Λs = {1, 2, . . . , s};E = {{i, j} : (i, j) ∈ Λs × Λs ∧ |i − j| = 1} .Search by an interacting continuous time quantum walk governed byh = − 1
2
Ps−1j=1 | j + 1 〉〈 j |+ | j 〉〈 j + 1 |+
Psj=1 V (j)| j 〉〈 j |.
Quantum search of this kind can suffer from two quantum effects:
Bloch oscillations
Anderson localization
but . . .
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Viscosity as a resource
. . . both problems can benefit of the introduction of a certain amount ofviscous friction.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Outline
1 Introduction
2 The Schrödinger-Langevin-Kostin equation: Continuous caseThe SLK EquationToy Models
3 SLK Equation: Discrete CaseQuantum optimizationBloch oscillationsAnderson localization
4 Conclusions and Outlook
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Bloch oscillations
x
t
20 40 60 80 100
100
200
300
400
x
t
20 40 60 80 100
100
200
300
400
The effect on ballistic (in the sense of deFalco, D.T 2006) evolution of a linearpotential V (x) = −gx :
energy-momentum relation E(p) = 1− cos p determines Blochoscillations;this prevents the wave packet from approaching the point x = s wherethe minimum of V (x) is located;greedy (gradient) optimization hindered by the fact that, on a lattice,v(p) = sin p.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
IDEA
Idea: add some friction to prevent first Brillouin zone crossing.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Discrete Kostin equation
Given the discrete equation
i∂ψ(t , x)
∂t= (h ψ)(t , x) + K (t , x)ψ(t , x) = (4)
= −12
(ψ(t , x + 1) + ψ(t , x − 1)) + V (x)ψ(t , x)| {z }h
+ K (t , x)ψ(t , x)| {z }dissipation
. (5)
the requirement
ddt〈 ψ(t) |h| ψ(t) 〉 = −
s−1Xx=1
(K (t , x + 1)− K (t , x)) sin (S(t , x + 1)− S(t , x)) ,
(6)
is saftisfied, for example, if
K (t , x) = β
xXy=2
sin (S(t , y)− S(t , y − 1)) (7)
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Bloch oscillations
Linear potential:
x
t
20 40 60 80 100
100
200
300
400
g = 0, β = 0;s = 100, ε = 17g0 = 2/s,0 ≤ t ≤ 4s.
Linear Potential + Viscous force:
x
t
20 40 60 80 100
100
200
300
400
x
t
20 40 60 80 100
100
200
300
400
100 200 300 400t
0.2
0.4
0.6
0.8
1.0
100 200 300 400t
0.2
0.4
0.6
0.8
1.0
g = 3g0, β = 2g0 g = 3g0, β = 4g0:
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Outline
1 Introduction
2 The Schrödinger-Langevin-Kostin equation: Continuous caseThe SLK EquationToy Models
3 SLK Equation: Discrete CaseQuantum optimizationBloch oscillationsAnderson localization
4 Conclusions and Outlook
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Quantum optimizationBloch oscillationsAnderson localization
Anderson localization
x
t
20 40 60 80 100
500
1000
1500
2000
x
t
20 40 60 80 100
100
200
300
400
x
t
20 40 60 80 100
500
1000
1500
2000
500 1000 1500 2000t
0.2
0.4
0.6
0.8
1.0
Random Gaussiannoise:2σ0 = 2 (10/s)3/2.
Linear potential:g = 3g0.
Kostin friction: β = 4g0.
pseudo-ballistic motionis much more stablethan the truly inertialone with respect to theonset of Andersonlocalization.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Conclusions
Bloch oscillations and Anderson localization can hinder a completesearch of the solution space of the optimization problem.
“Viscous” friction of Kostin suppresses both these effects.
Bloch oscillations are suppressed by preventing the wavepacketmomentum from crossing the first Brillouin zone.
Anderson localization is avoided by probability percolation through theirregular potential profile.
de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Outlook
The same framework developed in this note for the combinatorialoptimization metaphor can be used, with minor changes, to describe anexcitation travelling along a spin chain or a light pulse propagatingthrough a waveguide lattice.4
SLK dynamics can be exploited also in these fields.Increase the fidelity of state transmission, in presence of imperfectionsalong a spin chain, by applying a “tension” at both ends of it.5
The sole convergence toward the ground state could, instead, findapplications in all-optical switching of light in waveguide arrays 6: theinjected light pulse can be steered toward a given position by a suitabletuning of the thermal gradient which determines the potential profile ofthe lattice.So far we considered the dissipative part. Inclusion of fluctuations isrequired.Extension to general graphs. First step: Hypercube.
4H. Perets et al.. Phys. Rev. Lett., 100:170506, 2008.5R.P. Feynman. Quantum mechanical computers. Found.Phys., 16(6):507-31, 1986.6D.N. Christodoulides, F. Lederer, and Y. Silberberg. Discretizing light behaviour in linear and
nonlinear waveguide lattices. Nature, 424:817-823, 2003de Falco D., Tamascelli D. QA and SLK Equation
IntroductionThe Schrödinger-Langevin-Kostin equation: Continuous case
SLK Equation: Discrete CaseConclusions and Outlook
Thank You!
de Falco D., Tamascelli D. QA and SLK Equation