Table Of ContentDeterministic and Stochastic Quantum
Annealing Approaches
Demian Battaglia1, Lorenzo Stella1, Osvaldo Zagordi1,
Giuseppe E. Santoro1,2 and Erio Tosatti1,2
1 SISSA † INFM Democritos, Via Beirut 2-4, Trieste, Italy
2 ICTP, Trieste, Italy.
[email protected]
1 Introduction
Theideaofquantumannealing(QA)isalateoffspringofthecelebratedsim-
ulated thermal annealingbyKirkpatricketal.[1].Insimulatedannealing,the
problem of minimizing a certain cost (or energy) function in a large configu-
ration space is tackled by the introduction of a fictitious temperature, which
is slowly lowered in the course of a Monte Carlo or Molecular Dynamics sim-
ulation[1].Thisdeviceallowsanexplorationoftheconfigurationspaceofthe
problem at hand, effectively avoiding trapping at unfavorable local minima
through thermal hopping above energy barriers. It makes for a very robust
and effective minimization tool, often much more effective than standard,
gradient-based, minimization methods.
An elegant and fascinating alternative to such a classical simulated an-
nealing (CA) consists in helping the system escape the local minima through
quantum mechanics, by tunneling through the barriers rather than thermally
overcoming them [2, 3]. Experimental evidence in disordered Ising ferromag-
nets subject to transverse magnetic fields showed that this strategy is not
only feasible but presumably winning in certain cases [4]. These experimen-
tal results were confirmed by a Path-Integral Monte Carlo (PIMC) study of
an Ising glass model, were the crucial role played by Landau-Zener tunneling
events was also pointed out [5].
In essence, in quantum annealing one supplements the classical energy
function – let us denote it by H – with a suitable time-dependent quantum
cl
kinetic term, H (t), which is initially very large, for t ≤ 0, then gradually
kin
reduced to zero in a time τ. For the Ising glass case, for instance, H =
cl
† This project was sponsored by MIUR through FIRB RBAU017S8R004, FIRB
RBAU01LX5H,COFIN2003andCOFIN2004,andbyINFM(“Iniziativatrasver-
salecalcoloparallelo”).EarlycollaborationwithDr.RomanMartonˇ´akandProf.
Roberto Car are gratefully acknowledged.
D.Battagliaetal.:DeterministicandStochasticQuantumAnnealingApproaches,Lect.Notes
Phys.679,171–206(2005)
www.springerlink.com (cid:1)c Springer-VerlagBerlinHeidelberg2005
172 Demian Battaglia et al.
(cid:4)
− J σzσz representsanEdward-AndersondisorderedIsingmodel,while
(cid:2)ij(cid:3) ij i j
a very natural choice for Hkin, suggested b(cid:4)y the experiment [4], is given by
the transverse field term H (t) = −Γ(t) σx. At zero temperature, the
kin i i
quantum state of the system |Ψ(t)(cid:2), initially prepared in the fully quantum
ground state |Ψ (cid:2) of H(t = 0) = H + H (0), evolves according to the
0 cl kin
Scro¨dinger equation
d
i¯h |Ψ(t)(cid:2)=[H +H (t)]|Ψ(t)(cid:2), (1)
dt cl kin
to reach a final state |Ψ(t = τ)(cid:2). A crucial basic question is then how the
residualenergy(cid:5) (τ)=E (τ)−E ,decreasesforincreasingτ.HereE
res fin opt opt
is the absolute minimum of H , and E (τ) is the average energy attained
cl fin
by the system after evolving for a time τ, so that
(cid:4)Ψ(τ)|(H −E )|Ψ(τ)(cid:2)
(cid:5) (τ)=E (τ)−E = cl opt . (2)
res fin opt (cid:4)Ψ(τ)|Ψ(τ)(cid:2)
Generally speaking, this question has to do with the adiabaticity of the
quantum evolution, i.e., whether the system is able, for sufficiently slow
annealing (sufficiently long τ), to follow the instantaneous ground state of
H(t)=H +H (t), for a judiciously chosen H (t). (The fictitious kinetic
cl kin kin
energy H (t) can be chosen quite freely, with the only requirement of being
kin
reasonably easy to implement.) For this reason, this approach has also been
called Quantum Adiabatic Evolution [6].
At the level of practical implementations on an ordinary (classical) com-
puter, the task of following the time-dependent Schro¨dinger evolution in (1)
is clearly feasible only for toy models with a sufficiently manageable Hilbert
space [3, 6, 7]. Actual optimization problems of practical interest usually
involve astronomically large Hilbert spaces, a fact that calls for alternative
Quantum Monte Carlo (QMC) approaches. These QMC techniques, in turn,
are usually suitable to using imaginary time quantum evolution, where the
i¯h∂ in (1) is replaced by −¯h∂ . One of the questions we have recently ad-
t t
dressed,inthecontextofsimplifiedproblems[7],iswhetheranimaginary-time
Schro¨dinger evolution changes the quantum adiabatic evolution approach in
any essential way. The answer to this question appears to be that, as far as
annealing is concerned, imaginary-time is essentially equivalent to real-time,
and, as a matter of fact, can be quantitatively better [7].
A number of recent studies have applied Path-Integral Monte Carlo
(PIMC)strategiestoQA.Acertainsuccesshasbeenobtainedinanumberof
optimizationproblems,suchasthefoldingofoff-latticepolymermodels[8,9],
the random Ising model ground state problem [5, 10] (see Sect. 4.2), and the
Traveling Salesman Problem [11] (see Sect. 4.2). On the other hand, for the
interesting case of Boolean Satisfiability – more precisely, a prototypical NP-
complete problem such as 3-SAT – a recent study of our group shows that
PIMCannealingperformsdefinitelyworsethansimpleCA[12](seeSect.4.2).
Deterministic and Stochastic Quantum Annealing Approaches 173
In view of these results, it is fair to stress that it is a priori not obvious
or guaranteed that a QA approach should do better than, for instance, CA,
on a given problem. Evidently, the comparative performance of QA and CA
dependsindetailontheenergylandscapeoftheproblemathand,inparticular
on the nature and type of barriers separating the different local minima, a
problem about which very little is known in many practical interesting cases
[13].Thatinturndependscruciallyonthetypeandeffectivenessofthekinetic
energy chosen. Unfortunately, there is still no reliable theory predicting the
performance of a QA algorithm, in particular correlating it with the energy
landscape of the given optimization problem. Nevertheless, it is important to
stress that QA is not a universal key to hard NP problems: indeed, one can
think of trivial optimization problems, like the random Ising ferromagnet in
one-dimension [7], where QA (as well as CA) will be by necessity slow.
In order to gain understanding on these problems, we have moved, more
recently, one step back and concentrated attention on the simplest textbook
problems where the energy landscape is well under control: essentially, one-
dimensional potentials, starting from a double-well potential, the simplest
form of barrier. On these well controlled landscapes we have carried out a
detailed and exhaustive comparison between quantum adiabatic Schro¨dinger
evolution, both in real and in imaginary time, and its classical deterministic
counterpart, i.e., Fokker-Planck evolution [7]. This work will be illustrated in
Sect. 2.1. On the same double well-potential, we have also studied [14] the
performance of different stochastic approaches, both classical Monte Carlo
and Path Integral Monte Carlo. Some of this work, which turns out to be
quite instructive, is briefly presented in Sect. 4.4.
The rest of the Chapter is organized as follows: Sect. 2 illustrates the
deterministic annealing approaches applied to toy problems, essentially the
minimizationofafunctionofacontinuouscoordinate.Section3discussesthe
crucial role played by disorder and the issue of Landau-Zener tunneling in
QA. Section 4 introduces the Path-Integral Monte Carlo techniques, and il-
lustratessomeoftherecentapplications,notablyontherandomIsingmodel,
on the Traveling Salesman Problem, and on Boolean Satisfiability problems.
Section 5 discusses alternative approaches to optimization, including a dis-
cussion of Green’s Function Monte Carlo QA, which seems to be a promising
tool for future QA studies. Section 6, finally, contains a brief summary of the
main points, and some concluding remarks.
2 Deterministic Approaches on the Continuum
Conceptually, one of the simplest problems to illustrate is that of finding
the global minimum of an ordinary function of several continuum variables
with many minima. Suppose the classical Hamiltonian H mentioned in the
cl
introduction is just a potential energy V(x), (with x a Cartesian vector of
arbitrary dimension), of which we need to determine the absolute minimum
174 Demian Battaglia et al.
(x , E = V(x )). Assume, generally, a situation in which a steepest-
opt opt opt
descent approach, i.e., the strategy of following the gradient of V, would lead
to trapping into one of the many local minima of V, and would thus not
work.Classically,asanobviousgeneralizationofasteepest-descentapproach,
one could imagine of performing a stochastic (Markov) dynamics in x-space
according to a Langevin’s equation:
1
x˙ =− ∇V(x)+ξ(t), (3)
η(T)
where the strength of the noise term ξ is controlled by the squared correla-
tions ξ (t)ξ (t(cid:6))=2D(T)δ δ(t−t(cid:6)), with ξ¯=0. Both D(T) and η(T) – with
i j ij
dimensions of a diffusion constant and of a friction coefficient and related,
respectively, to fluctuations and dissipation in the system – are temperature
dependent quantities which can be chosen, for the present optimization pur-
pose, with a certain freedom. The only obvious constraint is in fact that the
correct thermodynamical averages would be recovered from the Langevin dy-
namics only if η(T)D(T) = k T, an equality known as Einstein’s relation
B
[15]. Physically, D(T) should be an increasing function of T, so as to lead
to increasing random forces as T increases, with D(T = 0) = 0, since noise
is turned off at T = 0. Classical annealing can in principle be performed
through this Langevin dynamics, by slowly decreasing the temperature T(t)
asafunctionoftime,fromsomeinitially largevalueT downtozero.Instead
0
of working with the Langevin equation – a stochastic differential equation –
one might equivalently address the problem by studying the probability den-
sityP(x,t)offindingaparticleatpositionxattimet.Theprobabilitydensity
is well known to obey a deterministic time-evolution equation given by the
Fokker-Planck (FP) equation [15]:
∂ 1
P(x,t) = div(P∇V) + D(T)∇2P . (4)
∂t η(T)
Here, the second term in the right-hand side represents the well known dif-
fusion term, proportional to the diffusion coefficient D(T), whereas the first
termrepresentstheeffectofthedriftforce−∇V,inverselyproportionaltothe
friction coefficient η(T) = k T/D(T) [15]. Annealing can now be performed
B
bykeepingthesystemforalongenoughequilibrationtimeatalargetemper-
ature T , and then gradually decreasing T to zero as a function of time, T(t),
0
inagivenannealingtimeτ.WecanmodelthisbyassumingT(t)=T f(t/τ),
0
where f(y) is some assigned monotonically decreasing function for y ∈ [0,1],
withf(y ≤0)=1andf(1)=0.InthismannerthediffusionconstantDin(4)
becomes a time-dependent quantity, D = D(T(t)). The FP equation should
t
then be solved with an initial condition given by the equilibrium Boltzmann
distribution at temperature T(t = 0) = T0, i.e., P(x,t = 0) = e−V(x)/kBT0.
The final average potential energy after annealing, in excess of the true mini-
mum value, will then be simply given by:
Deterministic and Stochastic Quantum Annealing Approaches 175
(cid:29)
(cid:5) (τ)= dxV(x)P(x,t=τ) − E ≥0, (5)
res opt
where E is the actual absolute minimum of the potential V.
opt
In a completely analogous manner, we can conceive using Schro¨dinger’s
equationtoperformadeterministicquantumannealing(QA)evolutionofthe
system,byintroducingquantumfluctuationsthroughastandardkineticterm
H (t) = −(h¯2/2m )∇2, with a fictitious time-dependent mass m . We are
kin t t
therefore led to studying the time-dependent Schro¨dinger problem:
" #
∂
ξ¯h ψ(x,t)= −Γ(t)∇2+V(x) ψ(x,t), (6)
∂t
whereξ =iforareal-time(RT)evolution,whileξ =−1foranimaginary-time
(IT) evolution. Here Γ(t)=h¯2/2m will be our annealing parameter, playing
t
the role that the temperature T(t) had in classical annealing. Once again we
may take Γ(t) varying from some large value Γ at t ≤ 0 – corresponding
0
to a small mass of the particle, hence to large quantum fluctuations – down
to Γ(t = τ) = 0, corresponding to a particle of infinite mass, hence without
quantumfluctuations.Again,wecanmodelthiswithΓ(t)=Γ f(t/τ),where
0
f is a preassigned monotonically decreasing function. A convenient initial
condition here will be ψ(x,t = 0) = ψ (x), where ψ (x) is the ground state
0 0
of the system at t ≤ 0, corresponding to the large value Γ(t) = Γ and
0
hence to large quantum fluctuations. For such a large Γ, the ground state
will be separated by a large energy gap from all excited states. The residual
energyafterannealingwillbesimilarlygivenby(5),wherenow,however,the
probability P(x,t=τ) should be interpreted, quantum mechanically, as:
|ψ(x,t)|2
1
P(x,t)= .
dx(cid:6)|ψ(x(cid:6),t)|2
Ingeneral, theresidualenergywillbedifferentforaRToranITSchro¨dinger
evolution. We will comment further on RT versus IT Schro¨dinger evolution
later on.
In the remaining part of this section, we will present some of the results
obtained along the previous lines on simple one-dimensional potential [7],
starting with the simplest example of a problem with two minima separated
by a barrier.
2.1 The Simplest Barrier: A Double-Well Potential
Consider, as a potential V(x) to be optimized, a slightly generalized double-
well potential in one-dimension
V (x2−a2+)2 +δx forx≥0
0 a4
V (x)= + , (7)
asym V (x2−a2−)2 +δx forx<0
0 a4
−
176 Demian Battaglia et al.
with, in general, a+ (cid:6)= a−, both positive, V0, and δ real constants. (The
discontinuity in the second derivative at the origin is of no consequence in
our discussion.) In absence of the linear term (δ = 0), the potential has two
degenerateminimalocatedatx− =−a− andx+ =a+,separatedbyabarrier
of height V0. When a small linear term δ >0 is introduced , with δa± (cid:13)V0,
the two degenerate minima are split by a quantity ∆V ≈ δ(a+ +a−), the
minimum at x ≈ −a− becoming slightly favored. For reasons that will be
clearinamoment,itisusefultoconsiderthesituation, whichwewillreferto
as“asymmetricdouble-well”,inwhichthetwowellspossessdefinitelydistinct
curvatures at the minimum (i.e, their widths differ), realized by taking a (cid:6)=
+
a−. (To lowest order in δ, we have: V(cid:6)(cid:6)(x=x±)=8V0/a2±.) In particular, we
shallexaminethecaseinwhichthemetastable“valley”atx is“wider”than
+
the absolute minimum at x−, which is realized by chosing a+ > a−. This
will have a rather important effect on the quantum evolution, since, as we
shallsee,forintermediatevaluesofthemassoftheparticle,thewavefunction
of the system will be predominantly located on the metastable minimum.
Obviously, if we set a+ =a− =a, and δ =0 we recover the standard double-
well potential.
We now present the results obtained by the annealing schemes introduced
inSect.2above.TheFokker-PlanckandtheSchro¨dingerequation(bothinRT
and in IT) were integrated numerically using a fourth-order adaptive Runge-
Kuttamethod,afterdiscretizingthexvariableinasufficientlyfinerealspace
grid [7]. For the FP classical annealing, the results shown are obtained with
a linear temperature schedule, T(t)=T (1−t/τ), and a diffusion coefficient
0
simply proportional to T(t), D = D (1−t/τ). (Consequently, the friction
t 0
coefficient is kept constant in t, η = k T(t)/D = k T /D .) Similarly, for
t B t B 0 0
theSchro¨dingerquantumannealingweshowresultsobtainedwithacoefficient
of the Laplacian Γ(t) vanishing linearly in a time τ, Γ(t)=Γ (1−t/τ).
0
Figure 1 shows the results obtained for the final annealed probability dis-
tribution P(x,t = τ) at different values of τ, for both the Fokker-Planck
(CA, panel (a)) and the Scro¨dinger imaginary-time case (IT, panel (b)), for
an “asymmetric” double-well potential V (x), with V = 1 (our unit of
asym 0
energy), a+ = 1.25,a− = 0.75, δ = 0.1. Figure 1(c) summarizes the results
obtained for the residual energy (cid:5) (τ) in 5.
res
We notice immediately that QA wins over CA for large enough value of
τ. The RT-QA, which behaves as its IT counterpart for a symmetric double-
well (a+ =a−, see [7]), shows a slightly different behavior from IT-QA in the
present asymmetric case (see below for comments). We discuss first the CA
data(panel(a)and(c)ofFigs.1).StartingfromaninitiallybroadBoltzmann
distributionatahighT =T =V ,P(x,t=0)(solidline),thesystemquickly
0 0
sharpensthedistributionP(x,t)intotwowell-definedandquitenarrowpeaks
located around the two minima x± of the potential. This agrees very well
with what a CA for an harmonic potential would do [7]. If we denote by p±
the integral of each of the two narrow peaks, with p− +p+ = 1, it is clear
that the problem has effectively been reduced to a discrete two-level system
Deterministic and Stochastic Quantum Annealing Approaches 177
10
(a) CA ASYM
8 t=0
τ=1
τ) 6 τ=τ=11000
x,
P( 4
2
0
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x
5
(b) IT ASYM
4 t=0
τ=100
τx,) 3 τ=τ=1406000
P( 2
1
0
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x
10
(c) ASYM 4
x) 2
1 V(
0
-10 1 2
es 0.1 x
εr
CA
0.01
IT
RT L-Z
0.001
1 10 102 103 104 105 106
τ
Fig. 1. (a,b): The annealed final probability distribution P(x,t = τ) at different
valuesoftheannealingtime τ,forboththeFokker-Planckclassicalannealing(CA,
panel(a)),andtheImaginaryTimeSchr¨odingerquantumannealing(IT-QA,panel
(b)). (c) Final residual energy (cid:11) (τ) versus annealing time τ for quantum anneal-
res
ing in Real Time (RT) and Imaginary Time (IT) compared to the Fokker-Planck
classicalannealing(CA).Thesolidlinein(c)isafitoftheCAdata(seetext).The
double well potential (dashed line in (a,b), inset of (c)) is here given by (7) with
a+ =1.25,a− =0.75
178 Demian Battaglia et al.
problem.Thetimeevolutionofp±,therefore,obeysadiscreteMasterequation
which involves the thermal promotion of particles over the barrier V , of the
0
form presented and discussed by Huse and Fisher in [16], where they show
that, apart from logarithmic corrections, the leading behavior of the residual
energy is of the form (cid:5)res ∼τ−∆V/B, with the power-law exponent controlled
by the ratio ∆ /B between the energy splitting of the two minima ∆ and
V V
the barrier B =V −V(x ). As shown in Fig. 1(c) (solid lines through solid
0 +
circles),theasymptoticbehavioranticipatedbyHuseandFisherfitsnicelyour
CA residual energy data (solid circles), as long as the logarithmic corrections
are accounted for in the fitting procedure [7]. Obviously, we can make the
exponent as small as we wish by reducing the linear term coefficient δ, and
hence the ratio ∆ /B, leading to an exceedingly slow classical annealing.
V
The behavior of the QA evolution is remarkably different. Observe, as a
first point, that the final annealed wavefunctions only slowly narrows around
the minimum of the potential, although the residual energy asymptotics of
QA is clearly winning. The asymptotic behavior of the QA residual energy is
(cid:5) (τ)∝τ−1/3, indicated by the dashed line in Fig. 1(c): this rather strange
res
exponent turns out to be the appropriate one for the Schro¨dinger annealing
withalinearscheduleΓ(t)withinanharmonicpotential(thelowerminimum
valley, see [7] for details). Going back to Fig. 1(b), the initial wavefunction
squared |ψ(x,t = 0)|2 corresponds to a quite small mass (a large Γ = 0.5),
0
and is broad and delocalized over both minima (solid line). As we start an-
nealing, and if the annealing time τ is relatively short – that is, if τ < τ ,
c
with a characteristic time τ which depends on which kind of annealing, RT
c
or IT, we perform – the final wavefunction becomes mostly concentrated on
the wrong minimum, roughly corresponding to the ground state with a still
relatively large Γ < Γ (see also Fig. 2 and accompaning discussion). The
1 0
larger width of the wrong valley is crucial, giving a smaller quantum kinetic
energy contribution, so that tunneling to the other (deeper) minimum does
not yet occur. By increasing τ, there is a crossover: the system finally recog-
nizes the presence of the other minimum, and effectively tunnels into it, with
a residual energy that, as previously mentioned, decays asymptotically as
(cid:5) (τ)∝τ−1/3 (dashed line in Fig. 1(c)). There is a characteristic annealing
res
timeτ –differentinthetwoScro¨dingercases,RTandIT–abovewhichtun-
c
neling occurs, and this shows up as the clear crossover in the residual energy
behavior of both IT and RT, shown in Fig. 1(c).
These findings can be quite easily rationalized by looking at the in-
stantaneous (adiabatic) eigenvalues and eigenstates of the associated time-
independent Schro¨dinger problem, which we show in Fig. 2(a,b). Looking at
the instantaneous eigenvalues shown in Fig. 2(a) we note a clear avoided-
crossing occurring at Γ = Γ ≈ 0.038, corresponding to a resonance con-
LZ
dition between the states in the two different valleys of the potential. For
Γ >Γ thegroundstatewavefunctionispredominantlyconcentratedinthe
LZ
wider but metastable valley, while for Γ < Γ it is mostly concentrated on
LZ
thedeeperandnarrowerglobalminimumvalley.Inthefulltime-dependentRT
Deterministic and Stochastic Quantum Annealing Approaches 179
(a)ASYM
10
Γ) 1
(n
E
0.1
L-Z
0.01
0.001 0.01 0.1 1
Γ
4
(b)ASYM
Γ= 0.5
3 Γ= 0.04
2 Γ= 0.035
(x)|0 2 Γ= 0.005
Ψ
|
1
0
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x
Fig. 2. Instantaneous eigenvalues (a) and ground state wavefunctions (b) of the
Schro¨dinger problem Hψ = Eψ for different values of Γ, for the potential in (7)
with a+ = 1.25,a− = 0.75. Notice the clear Landau-Zener avoided crossing in (a),
indicated by the arrow and magnified in the inset
evolution, transfertothelowervalleyisaLandau-Zenerproblem[17,18]:the
characteristic time τ for the tunneling event is given by τ = h¯αΓ /2π∆2,
c LZ 0
where α is the relative slope of of the two crossing branches as a function of
Γ, 2∆ is the gap at the avoided-crossing point, and Γ is the initial value of
0
theannealingparameter.(ForthecaseshowninFig.2,wehave2∆=0.0062,
α = 2.3, hence τ ≈ 18980, see rightmost arrow in Fig. 1(c).) The Landau-
LZ
Zener probability of jumping, during the evolution, from the ground state
onto the “wrong” (excited) state upon fast approaching of the avoided level
crossing is Pex = e−τ/τLZ, so that adiabaticity applies only if the annealing
is slow enough, τ > τ . Notice that the gap 2∆, and hence the probability
LZ
of following adiabatically the ground state, can be made arbitrarily small by
increasing the asymmetry of the two well, i.e., by making a+ (cid:18) a−. The IT
characteristictimeissmaller,inthepresentcase,thantheRTone.Thispoint
isdiscussedinsomedetailin[7].Inanutshell,thereasonforthisisthefollow-
ing. After the system has jumped into the excited state, which occurs with a
probabilityPex =e−τ/τLZ,theresidualITevolutionwillfilterouttheexcited
state; this relaxation towards the ground state is controlled by the annealing
rate as well as by the average gap seen during the residual evolution. Numer-
ically, the characteristic time τ seen during the IT evolution is of the order
c
180 Demian Battaglia et al.
of h¯/(2∆), see leftmost arrow in Fig. 1(c), rather than being proportional to
1/∆2 as τ would imply.
LZ
Obviously, instantaneous eigenvalues/eigenvectors can be studied for the
Fokker-Planckequationaswell;theirproperties,however,areremarkablydif-
ferentfromtheLandau-ZenerscenariojustdescribedfortheSchro¨dingercase.
Figure 3(c) shows the first four low-lying eigenvalues of the FP equation as
a function of T, while Fig. 3(a,b) show the corresponding eigenstates for two
values of the temperature, T/V = 1 and T/V = 0.1 (the data refer to a
0 0
4
T = 0.1
3
x) 2
(n
P 1
0
-1
-2 -1 0 1 2
x
1
T = 1
0.5
(x)n 0
P
-0.5
-1
-2 -1 0 1 2
x
11
1
9 2
T) 7 3
E(n 5
3
1
0.01
10-4
10-6
0 1 2 3 4 5 6 7 8 9 10
1/T
Fig. 3. Instantaneous eigenvalues of the Fokker-Planck equation (panel (c), the
lowest eigenvalue E = 0 is not shown) as a function of temperature T, and the
0
corresponding eigenstates for two values of T (panels (a) and (b)). The potential
hereissymmetric,i.e.,Vasym in7withV0 =1,a+ =a− =1,δ=0.1.Similarresults
(not shown) are obtained for asymmetric choices of the double well potential
Description:only feasible but presumably winning in certain cases [4] The final average potential energy after annealing, in excess of the true mini- mum value
