Quantum Annealing and Analog Quantum Computation

22 Pages·2008·0.43 MB·English

Reviews of Modern Physics (in press)
Quantum Annealing and Analog Quantum Computation
Arnab Das and Bikas K. Chakrabarti
∗ †
Theoretical Condensed Matter Physics Division and
Centre for Applied Mathematics and Computational Science, Saha Institute of Nuclear Physics, 1/AF,
Bidhannagar,Kolkata-700064, India.
We review herethe recent success inquantum annealing, i.e.,optimization of the cost or energy
8 functionsofcomplexsystemsutilizingquantumfluctuations. Theconceptisintroducedinsucces-
sivestepsthroughthestudiesofmappingofsuchcomputationallyhardproblemstotheclassical
0
spin glass problems. The quantum spin glass problems arise with the introduction of quantum
0
fluctuations, and the annealing behavior of the systems as these fluctuations are reduced slowly
2
tozero. Thisprovidesageneral frameworkforrealizinganalogquantum computation.

r
a
M
PACSnumbers: 0.2.10.Ox,0.2.60.Pn,0.2.70Ss,0.3.65Xp,03.67.Lx,64.60.Cn,64.70.Pf

4
2
Contents I. INTRODUCTION

]
h I. Introduction 1
p
- II. Optimizationand Annealing 3 Utilization of quantum mechanical tunneling through
t
n A. CombinatorialOptimizationProblems 3 classically localized states in annealing of glasses has
a B. Statistical MechanicsoftheOptimizationProblems opened up a new paradigm for solving hard optimiza-
u andThermalAnnealing 4 tion problems through adiabatic reduction of quantum
q C. SpinGlassandOptimization 5
fluctuations. This will be introduced and reviewed here.
[ 1. FindingTheGroundStates oftheClassicalSpin
Consider the example of a ferromagnet consisting of
Glasses 5
3 N tiny interacting magnetic elements; the spins. For a
2. TheTravelingSalesmanProblem 6
v
D. QuantumSpinGlassesandAnnealing 7 macroscopicsample N is very large;ofthe order ofAvo-
3
gadro number. Assume that each spin can be in any of
9
III. Quantum Annealing 9 the two simple states: up or down. Also, the pairwise
1
2 A. QuantumMonteCarloAnnealing 10 interactions between the spins are such that the energy
. 1. AShort-RangeSpinGlass 11 of interaction(potential energy or PE)between any pair
1 2. TheTravelingSalesmanProblem 12 ofspinsis negative(smaller)ifboththe spinsin the pair
0
3. Random FieldIsingModel: HowaChoiceof
8 KineticTermImprovesAnnealingResults 13 are in the same state and is positive (higher) if their
0 states differ. As such, the collective energy of the N-
B. QuantumAnnealingUsingReal-timeAdiabatic
v: Evolution 13 spin system (given by the Hamiltonian H) is minimum
i C. AnnealingofaKineticallyConstrainedSystem 15 when all the spins are aligned in the same direction; all
X D. ExperimentalRealizationofQuantumAnnealing 17 up or all down, giving the full order. We call these two
r minimum energy configurations the ground states. The
a IV. Convergence of Quantum Annealing Algorithms 18 rest of the 2N configurations are called excited states.
The plot of the interaction energy of the whole system
V. Quantum Quenching 18
with respect to the configurations is called the potential
energy-configuration landscape, or simply, the potential
VI. Summary and Discussions 18
energy landscape (PEL). For a ferromagnet, this land-
Acknowledgments 19 scape has a smooth double-valley structure (two mirror-
symmetricvalleyswiththetwodegenerategroundstates,
VII. Appendix 19 allupandalldown,attheirrespectivebottoms). Atzero-
1. Suzuki-Trotter Formalism 19
temperature the equilibrium state is the state of mini-
2. Quantum QuenchingofaLongRangeTIM 20
mum potential energy, and the system stably resides at
the bottom of any of the two valleys. At finite temper-
References 20
ature, the thermal fluctuations allow the system to visit
higher energy configurations with some finite probabil-
ity (givenby the Boltzmannfactor)and thus the system
∗Electronicaddress: [email protected] spends time in other part of the PEL also. The proba-
†Electronicaddress: [email protected] bility that a system is found in a particular macroscopic
2
state depends not only on the energy of the state (as at the minima diverges(or become very high), as in case of
zero temperature), but also on its entropy. The thermo- a spin glass, due to frequent trapping of the system in
dynamic equilibrium state corresponds to the minimum such local minima (glassy behavior). In case such bar-
of a thermodynamic potential called free energy F given riers are very narrow, quantum-mechanical fluctuations
by the difference between the energy of the state and (fluctuations in a quantum observable due to its non-
theproductofitsentropyandthetemperature. Atzero- commutativity with the Hamiltonian of the system) can
temperature,theminimaofthefreeenergycoincideswith helptunnelingthroughthem,therebyleadingtosuccess-
the minimum for energy and one gets the highest order ful quantum annealing (see. Fig. 1).
(magnetization). As the temperature is increased, the We introduce here these ideas in details through suc-
contributionofentropygetsmagnifiedandthe minimum cessive steps.
of the free energy is shifted more and more towards the a) The physics of classical spin glasses has already con-
states with lower and lower order or magnetization, un- tributed enormously to our knowledge of the landscape
til at (and beyond) some transition temperature Tc, the structure of the energy or the thermodynamic poten-
order disappears completely. For antiferromagnetic sys- tials and that of the unusually slow (glassy) dynamics
tems, the spin-spin pair interactions are such that it is of many-body systems in the presence of frustrationand
lower if the spins in the pair are in opposite states, and disorder. Mapping of computationally hard problems,
higher if their states are same. For antiferromagnetsone like the traveling salesman problem etc, to classical spin
can still define a sub-lattice order or magnetization and glassmodelsalsohelpedunderstandingtheircomplexity.
the PEL still has the double-well structure like in a fer-
b) The ground (and some low-lying) state structures of
romagnet for short range interactions. The free energy
frustrated random systems in the presence of quantum
andtheorder-disordertransitionalsoshowsidenticalbe-
fluctuations have also been studied in the context of
havior as observed for a ferromagnet.
quantum spin glasses. It has been shown that because
In spin glasses, where different spin-spin interactions of the possibility of tunneling through the barriers in
are randomly ferromagnetic or antiferromagnetic and thepotentialenergylandscape,quantumfluctuationscan
frozenintime(quencheddisorder),thePELbecomesex- help the dynamics to be ‘more ergodic’than the dynam-
tremelyrugged;variouslocalandglobalminimatrapped ics induced by the classical fluctuations and thus help
between potential energy barriers appears in the PEL. exploring the landscape much better. Ergodicity here
The ruggedness and the degeneracies in the minima meansthelossofthememoryoftheinitialstateincourse
comes from the effect of frustration or competing inter- of evolution (weak ergodicity) and the convergence to a
actions between the spins; none of the spin states on a stationarydistributionirrespectiveofthetheinitialstate
cluster or a placket is able to satisfy all the interactions (strong ergodicity). The nature of these quantum phase
in the cluster. The locally optimal state for the spins in transitions in such systems have also been extensively
the cluster is therefore degenerate and frustrated. studied. These studies (Sec. IID) endow one with the
Similar situations occur for multivariate optimization knowledge of the phase diagram and the location of the
problems like the traveling salesman problem (TSP). quantum critical point or the phase boundary, which is
Here a salesman has to visit N cities placed randomly crucial for choosing the proper quantum kinetic terms
on a plane (country). Of the N!/N distinct tours pass- and the annealing path (Sec. IIIA and Sec. IIIB).
ing through each city once, only few corresponds to the c) The most natural connection between the paradigm
minimum (ground state) travel distance or travel cost. of classical spin glasses and hard optimization problems
Therestscorrespondstothehighercosts(excitedstates). comes through a widely used and well established op-
The cost function, when plotted against different tour timization technique, namely, simulated annealing algo-
configurations, gives a similar rugged landscape, equiva- rithm as discussed earlier. The possibility of quantum
lent to the PEL of a spin glass (henceforth we will use tunneling through classically impenetrable barriers, as
PEL to mean cost-configuration landscapes also). indicated from the studies of quantum spin glasses, nat-
Obviously, an exhaustive search for the global mini- urally suggests an elegantand often more effective alter-
mum of a rugged PEL requires an exponentially large native to simulated annealing.
(or higher) number (in N) of searches (2N or N! or- Inquantumannealing,onehasaclassicalHamiltonian
der of searches for an N-spin spin glass and an N-city (oramultivariatecostfunctionviewedasthesame)tobe
TSP respectively). The computational effort or time for optimized, to which, one adds a (non-commuting) quan-
suchsearchesarethereforegenerallynotboundedbyany tum kinetic term and reduce it from a very high initial
polynomialinthe problemsizeN. Alternatively,agrad- valuetozeroeventually. Thisreduction,whendonecom-
ualenergyorcostdissipativedynamics(annealing),with pletelyadiabatically,assuresreachingofthegroundstate
Boltzmann like thermal fluctuations or some noise fac- of the classical glass at the end, assuming that there is
tor (in order to get out of the the local minima) in the no crossing of energy levels with the ground state in the
PEL may help reaching deep enough minima more eas- course of evolution, and provided that the starting state
ily. This simulated thermal annealing scheme is consid- wasthe groundstateofthe initialHamiltonian. Tostart
ered to be a very successful technique now. However, with, the tunneling field is much higher than the inter-
suchatechniqueoftenfailsifthebarrierheightsbetween action term, so the ground state (a uniform superposi-
3
binatorial in nature. If the variables S are discrete and
i
eachtakesupafinitenumberofvalues,thentheproblem
Thermal is clearly a combinatorial one. Moreover, certain prob-
Jump
lems with continuous variables (like linear programming
nergy problem) can also be reduced to combinatorialproblems
E
Cost/ C TQuunanneltiunmg (oPfaoppatdimimizitartiioouneptroabll1e9m9,8)a.ndHearsesuwmeefotchuastownethhiasvteyptoe
minimize ( S ) with respect to the discrete set of the
i
H { }
variables S .
C’ i
Configuration
FIG. 1 While optimizing the cost function of a computationally An optimizationproblem is said to belong to the class
hard problem (like the ground state energy of a spin glass or the P (P for Polynomial), if it can be solved in polynomial
minimumtraveldistanceforatravelingsalesmanproblem),onehas
time (i.e., the evaluation time goes like some polynomial
to get out of a shallower local minimum like the configuration C
(spinconfigurationortravelroute),toreachadeeperminimumC′. in N) using polynomially (in N, again) bound resources
Thisrequiresjumpsortunnelinglikefluctuationsinthedynamics. (computer space, processors etc). Existence of such a
Classicallyonehastojumpovertheenergyorthecostbarrierssep- polynomialboundonthe evaluationtime is somehowin-
aratingthem,whilequantummechanicallyonecantunnelthrough terpreted as the “easiness” of the problem. However,
the same. If the barrier is high enough, thermal jump becomes
many important optimization problems seem to fall out-
very difficult. However, if the barrier is narrow enough, quantum
tunnelingoftenbecomes quiteeasy. side this class, like, the famous traveling salesman prob-
lem (see Sec. IIC.2).
tion of all classical configurations) is trivially realizable.
Thereisanotherimportantclassofproblemswhichcan
Simulationsclearlydemonstratethatquantumannealing
be solved in polynomial time by non-deterministic ma-
canoccasionallyhelpreachingthegroundstateofacom-
chines. This class is the famous NP (Non-deterministic
plex glassysystemmuchfaster than couldbe done using
Polynomial) class (Garey and Johnson 1979). P is in-
thermalannealing(discussedlaterinSec. III).Anexper-
cluded completely in the NP class, since a deterministic
imentcomparingtheclassicalandquantumannealingfor
Turingmachineisjustaspecialcaseofnon-deterministic
a spin glass also shows that the relaxations in course of
Turing machines. Unlike a deterministic machine, which
quantumannealingareoftenmuchfasterthanthosedur-
takes a specific step deterministically at each instant
ingthe correspondingclassicalannealing,asdiscussedin
(and hence follows a single computational path), a non-
Sec. IIID. What makes quantum annealing fundamen-
deterministic machine has a host of different ‘allowed’
tally different from the classical annealing, is the non-
steps at its disposal at every instant. At each instant
local nature (Sec. III) and its higher tunneling ability
it explores all the ‘allowed’ steps and if any of them
(Secs. IID & IIIC).
leads to the goal, the job is considered to be done. Thus
Quantum annealing thus permits a realization of ana-
it explores in parallel many paths (whose number goes
log quantum computation, which is an independent and
roughly exponentially with time) and checks if any one
powerful complement to digital quantum computation,
of them reaches the goal.
where discrete unitary transformations are implemented
through quantum logic gates.
Among the NP problems, there are certain problems
(known as NP-complete problems) which are such that
II. OPTIMIZATION AND ANNEALING anyNP problemcanbe “reduced”to themusing a poly-
nomial algorithm. The famous 3-SAT problem (see Sec.
A. Combinatorial Optimization Problems IIIA.3) is a representative of the class. This roughly
means that if one has a routine to solve an NP-complete
The occurrence of multivariate optimization problems is problem of size N then using that routine one can solve
ubiquitous in our life, wherever one has to choose the anyNPproblematthe costofanextraoverheadintime
best bargain from a host of available options that de- that goes only polynomially with N. The problems in
pend on many independent factors. In many cases, such this class are considered to be hard, since so far no one
a task can be cast as a problem of minimizing a given cansimulateageneralnondeterministicmachinebyade-
cost or energy function (S ,S ,...S ) with respect to terministic Turing machine (or any sequential computer
1 2 N
H
N variablesS ,S ,...S (sometimessubjecttosomecon- withpolynomiallyboundresources)withoutanexponen-
1 2 N
straints). Thetaskistofindasetofvaluesforthesevari- tialgrowthofexecutiontime. Infactitislargelybelieved
ables (a configuration) for which the function ( S ) (thoughnotprovedyet)thatitisimpossibletodoso(i.e.,
i
H { }
has the minimum value (cf. Fig. 1). In many important P=NP) in principle. However, assuming this to be true,
6
optimization problems, the set of feasible configurations onecanshowthatthereareindeedproblemsinNP class
fromwhichanoptimumistobechosenisafiniteset(for thatareneither NP-completenorP(GareyandJohnson
finiteN). Insuchacase,wesaythattheproblemiscom- 1979).
4
B. Statistical Mechanics of the Optimization Problems and figurations is relatively high) and slowly fluctuations are
Thermal Annealing reduced until finally they are tuned off completely. In
the mean time the system gets a fair opportunity to ex-
There are some excellent deterministic algorithms for plore the landscape more exhaustively and settle into a
solvingcertainoptimizationproblemsexactly(Papadim- reasonably deep cost or energy minimum. Kirkpartick
itriou and Steiglitz 1998, Hartmann and Rieger, 2002). et al (1983) suggested an elegant way: A fluctuation is
These algorithms are, however, quite small in number implemented by introducing an “artificial” temperature
andarestrictlyproblemspecific. ForNPorharderprob- T into the problem such that the transition probability
lems, only approximate results can be found using these from a configuration Ci to a configuration Cf is given
algorithmsin polynomialtime. These approximatealgo- by min 1,exp [∆if/T] , where ∆if = Ef Ei, with
{ − } −
rithms too are also strictly problem specific in the sense Ek denoting the cost or energy of the configuration Ck.
that if one can solve a certain NP-complete problem up A corresponding Monte Carlo dynamics is defined, say,
to a certain approximation using some polynomial algo- basedon detailed balance,and the thermal relaxationof
rithm, then that does not ensure that one can solve all thesystemissimulated. Incourseofsimulation,thenoise
otherNPproblemsusingituptothesaidapproximation factor T is reduced slowly from a very high initial value
in polynomial time. to zero following some annealingschedule. At the end of
the simulation one is expected to end up with a config-
Exactalgorithmsbeingscarce,onehastogoforheuris-
uration whose cost is a reasonable approximation of the
tics algorithms, which are algorithms based on certain
globally minimum one. If the temperature is decreased
intuitive moves, without any guarantee on either the
slow enough, say,
accuracy or the run time for the worst case instance.
However, these algorithms are generally easy to formu-
late and are quite effective in solving most instances of
a the intended problems. A general approach towards T(t) N/logt, (1)
≥
formulating such approximate heuristics may be based
on stochastic (randomized) iterative improvements. The
most preliminary one in this family is the local mini- where t denotes the cooling time and N the system size,
mization algorithm. In this algorithm one starts with a thentheglobalminimumisattainedwithcertaintyinthe
random configuration C0 and makes some local changes limit t (Geman and Geman 1984). Even within a
intheconfigurationfollowingsomeprescription(stochas- finiteti→me∞andwithafastercoolingrate,onecanachieve
tic or deterministic) to generate a new configuration C1 areasonablygoodapproximation(acrystalwithonlyfew
and calculates the corresponding change in the cost. If defects) in practice. This simulated annealing method is
the cost is lowered by the change, then the new config- nowbeingusedextensivelybyengineersfordevisingreal-
uration C1 is adopted. Otherwise the old configuration life optimization algorithms. We will refer to this as the
is retained. Then in the next step a new local change classicalannealing (CA), to distinguishit fromquantum
is attempted again, and so on. This reduces the cost annealing (QA) which employs quantum fluctuations. It
steadily untila configurationis reachedwhichminimizes isimportanttonotethatthoughinthistypeofstochastic
the cost locally. This means that no further lowering of algorithmsthesystemhasmanydifferentstepswiththeir
cost is possible by changing this configuration using any respective probabilitiesat its disposal,it finally takes up
of the prescribed local moves. The algorithm essentially a single one, say by tossing coins, and thus finally follow
stopsthere. Butgenerallyinmostoptimizationproblems a single (stochastically selected) path. Hence it is not
(such as in spin glasses), there occur many local minima equivalent to a non-deterministic machine, where all the
in the cost-configuration landscape and they are mostly allowed paths are checked in parallel at every time-step.
far above the global minimum (see Fig. 1). It is likely
thatthealgorithmthereforegetsstuckinoneofthemand As has been mentioned already, many combinatorial
ends up with a poor approximation. One can then start optimization problems can be cast into the problem of
afresh with some new initial configuration and end up findingthegroundstateofsomeclassical(spinglasslike)
withanotherlocalminimum. Afterrepeatingthisforsev- Hamiltonian H( S ). One can therefore analyze the
i
{ }
eraltimes,eachtimewithanewinitialconfiguration,one problem by using statistical mechanics so as to to ap-
may choose the best result from them. But much better ply physical techniques like simulated annealing. If one
idea would be to get somehow out of shallow local min- naivelytakesthe number ofvariablesN asthe size,then
ima. Onecanintroducesomefluctuationsornoiseinthe the entropy and the energy are often found to scale dif-
processsothatthemovementisnotalwaystowardslower ferently with N andthe applicationof standardthermo-
energy configurations,but there is also a finite probabil- dynamic arguments become difficult. One needs to scale
ity to go to higher energy configurations (the higher the temperature and some other quantities properly with N
final energy, the lower the probability to move to that), sothatonecantalkintermsofthe conceptslikefreeen-
and consequently there appear chances to get out of the ergyminimizationetc. Moreover,theconstraintspresent
shallow local minima. Initially, strong fluctuations are in the problems are often very difficult to take into ac-
adopted(i.e., the probabilityto goto higher energycon- count.
5
C. Spin Glass and Optimization boththe spatialandthe temporalaveragesofthe single-
site magnetization of course vanishes. A relevant order
1. Finding The Ground Statesof theClassical Spin Glasses parameter for this freezing is therefore
As has been mentioned already, the difficulty faced by 1 N
a physically motivated optimization heuristic (one that q = N hSii2T, (5)
follows physical relaxation dynamics, classical or quan- i
X
tum, to search for the solution), in finding the solution
(overhead bar denoting the average over disorders (the
of a hard optimization problem, is similar to that faced
distribution of J ) and ... denotes the thermal aver-
ij T
by a glassy system in reaching its ground state. In fact, h i
age). Here q = 0 for T < T , while q = 0 for T T .
c c
finding the groundstates of the spin glassesis an impor- 6 ≥
Aswillbeseeninthefollowing,theexistenceofaunique
tant class of combinatorial optimization problem, which
order parameter q will indicate ergodicity.
includes an NP-complete problem (Barahona 1982), and
Inthespinglassphase(T <T ),thewholefreeenergy
c
many other apparently different ones (like the traveling
landscape gets divided (cf. Fig. 1) into many valleys
salesmanproblem)canberecastinthisform. Hencehere
(localminima offreeenergy)separatedbyveryhighfree
we discuss verybriefly the nature of the spin glassphase
energy barriers. Thus the system, once trapped in a val-
and the difficulty in reaching its ground state.
ley,remainsthereforaverylongtime. Thespinsofsuch
Theinteractionenergyofatypicallyrandomandfrus-
a confined system are allowed to explore only a very re-
trated Ising spin glass (Binder and Young 1986, Nishi-
stricted (and correlated) part of the configuration space
mori 2001, Dotsenko 2001) may be represented by a
and thus makes them “freeze” to have a magnetization
Hamiltonian of the form
that characterizes the state (valley) locally.
So far, two competing pictures continue to represent
the physics of the spin glasses. The mean-field picture
N
of replica symmetry-breaking is valid for infinite-ranged
= J S S , (2)
ij i j
H − spin glass systems like the SK spin glass. In this pic-
i>j
X ture, below the glass transition temperature T , the bar-
c
where S denote the Ising spins and J the interactions riers separating the valleys in the free energy landscape
i ij
between them. These J s here are quenched variables actually diverge (in the limit N ), giving rise to
ij → ∞
whichvaryrandomlybothinsignandmagnitudefollow- a diverging timescale for the confinement of the system
ing some distribution ρ(J ). The typical distributions in any such valley once the system gets there somehow.
ij
are (zero mean) Gaussian distribution of positive and This means there is a loss of ergodicity in the thermal
negative Jij values dynamics of the system at T < Tc. Thus one needs a
distribution P(q) of order parameters, instead of a sin-
J2 gle orderparameterto characterizethe whole landscape,
ρ(Jij)=Aexp −2Ji2j , (3) asemergesnaturallyfromthereplicasymmetry-breaking
!
ansatzofParisi(1980). Tobeabitmorequantitative,let
usimaginethattwoidenticalreplicas(havingexactlythe
and binary distribution
same setJ s) ofa spinglass sample are allowedto relax
ij
thermally below T , starting from two different random
ρ(J )=pδ(J J)+(1 p)δ(J +J), (4) c
ij ij ij
− − (paramagnetic) initial states. Then these two replicas
with probability p of having a +J bond, and (1 p) of (labeled by µ and ν, say) will settle in two different val-
having a J bond. Two well studied models a−re the leys,eachbeingcharacterizedbyalocalvalueoftheorder
SK model−(introduced by Sherrington and Kirkpatrick parameterandthecorrespondingoverlapparametersqµν
1975) and the EA model (introduced by Edward and having a sample-specific distribution
Anderson 1975). In the SK model the interactions are
Fµ+Fν
infinite-ranged and for the sake of extensively (for rules PJ(q)= e− T δ(q qµν);
−
of equilibrium thermodynamics to be applicable, energy Xµ,ν
should be proportional to the volume or its equivalent
thatdefinesthesystem-size)onehastoscaleJ 1/√N,
while in the EA model, the interactions are bet∼ween the P(q)= dJijρ(Jij)PJ(q). (6)
nearestneighborsonly. Forbothofthem,however,ρ(J ) Z Yi>j
ij
is Gaussian; A = (N/2πJ2)1/2 for normalization in the Here the subscript J denotes a particular sample with a
SK model. given realization of quenched random interactions (J s)
ij
The freezing (below T ) is characterizedby some non- between the spins, and finally averaging P (q) over dis-
c J
zero value of the thermal average of the magnetization orderdistributionρ(J)in(2)or(3)onegetsP(q). Phys-
at each site (local ordering). However, since the interac- ically, P(q) gives the probability distribution for the two
tions are random and competing, the spatial average of purestatestohaveanoverlapq,assumingthattheproba-
single-site magnetization (below T ) is zero. Above T , bilityofreachinganypurestateµstartingfromarandom
c c
6
(high temperature) state is proportional to the thermo- rows being identical, and so is not any two columns. In
dynamicweightexp F ofthestateµ. Theotherpic- termsofU s,the lengthofatourcanbe representedby
µ ij
{− }
tureis dueto thedropletmodelofshort-rangespinglass
N
(Bray and Moore 1984, Fisher and Huse 1986), where 1
= d U . (7)
there is no divergence in the typical free energy barrier H 2 ij ij
height and the relevant timescale is taken to be that of iX,j=1
crossing the free energy barrier of formation of a typi- One canrewrite the aboveHamiltonianin terms of Ising
cal droplet of same (all up or all down) spins. Based on spins S s as
ij
certain scaling ansatz, this picture leads to a logarith-
mically decaying(with time) self-correlationfunction for 1 N (1+S )
ij
the spins below the freezing temperature Tc. HTSP = 2 dij 2 (8)
The issue of the validity of the mean-field picture (of i,j=1
X
thereplicasymmetry-breaking)inthecontextofthereal- where S = 2U 1 are the Ising spins. The Hamilto-
ij ij
life spin glasses, where the interactions are essentially −
nianis similar to that of a non-interactingIsing spins on
short-range,isfarfromsettled(seee.g.,Mooreetal1998,
an N N lattice, with random fields d on the lattice
ij
Marinari et al 1998, Krzakala et al 2001 and Gaviro et ×
points i,j . The frustrationis introduced by the global
al 2006). However, the effective Hamiltonian (cost func- { }
constraintsonthespinconfigurationsinordertoconform
tion) formany other optimizationproblems maycontain withthe structureofthe matrixUdiscussedabove. The
long-range interactions and may even show the replica
problem is to find the ground state of the Hamiltonian
symmetry-breaking behavior shown in the Graph Parti- subject to these constraints. There are N2 Ising spins,
tioning Problem (Fu and Anderson 1986). Of course, no which can assume 2N2 configurations in absence of any
resultofQAforsuchsystem(forwhichreplicasymmetry-
constraint, but the constraint here reduces the number
breakingis shownexplicitly)has been reportedyet. The
of valid configurations to that of the number of distinct
successesofQAreportedsofararemostlyforshort-range
tours, which is (N!)/2N.
systems. Thus, the scope of quantum annealing in those
Mainly two distinct classes of TSP are studied: one
long-rangesystemsstillstandsoutasaninterestingopen
with an Euclidean d in finite dimension (where d are
question. ij ij
strongly correlated through triangle inequalities, which
means, for any three cities A, B and C, the sum of any
two of the side AB, BC and CA must be greater than
2. The Traveling Salesman Problem the remaining one), and the other with random d in
ij
infinite dimension.
In the traveling salesman problem (TSP), there are N In the first case, N cities are uniformly distributed
citiesplacedrandomlyinacountryhavingadefinitemet- within a hypercube in a d-dimensional Euclidean space.
ric to calculate the inter-city distances. A salesman has Finding a good approximation for large N is easier for
to make a tour to cover every city and finally come back this case, since the problem is finite-ranged. Here a d-
to the starting point. The problem is to find a tour of dimensional neighborhood is defined for each city, and
minimum length. Aninstance ofthe problemis givenby the problem can be solved by dividing the whole hyper-
a set d ; i,j = 1,N , where d indicates the distance cube into a number of smaller pieces and then searching
ij ij
{ }
between the i-th and the j-th city, or equivalently, the for the least path within each smaller part and joining
cost for going from the former to the later. We mainly them back. The correctionto the true least path will be
focus on the results of symmetric case, where d = d . due to the unoptimized connections across the bound-
ij ji
The problem can be cast into the form where one min- aries of the subdivisions. For a suitably made division
imizes an Ising Hamiltonian under some constraints, as (not too small), this correction will be of the order of
shown below. A tour can be represented by an N N the surface-to-volume ratio of each division, and thus
matrix T with elements either 0 or 1. In a given t×our, will tend to zero in the N limit. This method,
→ ∞
if the city j is visited immediately after visiting city i, knownas“divideandconquer”,formsareasonablestrat-
then T = 1, or else T = 0. Generally an additional egyforsolvingapproximatelysuchfinite-rangeoptimiza-
ij ij
constraintisimposedthatonecityhastobevisitedonce tionproblems(includingfinite-rangespinglasses)ingen-
and only once in a tour. Any valid tour with the above eral. Inthesecondcase,d sareassignedcompletelyran-
ij
restrictionmayberepresentedbyaTmatrixwhoseeach domly,withnogeometric(e.g.,Euclidean)correlationbe-
rowandeachcolumnhasoneandonlyoneelementequal tweenthem. Theprobleminthis casebecomesmorelike
to 1 and rest all are 0s. For symmetric a metric, a a long-range spin glass. A self avoiding walk represen-
tour and its reverse tour have the same length, and it is tation of the problem was made using an m-component
more convenientto workwith anundirectedtour matrix vectorfield,andthereplicaanalysiswasdone(Mezardet
U= 1(T+T˜), where T˜, the transpose of T, represents al1987)for finite temperature,assumingreplicasymme-
2
the reverseof the tour givenby T. Clearly,U must be a tryansatztohold. Moreover,truebreakingofergodicity
symmetric matrix having two and only two distinct en- may occur only in infinite systems, not in any finite in-
tries equal to 1 in every row and every column, no two stanceoftheproblem. Theresults,whenextrapolatedto
7
zero-temperature,do notdisagreemuchwiththe numer- is crucially important for its annealing, as it gives an
icalresults(Mezardetal1987). Thestabilityofareplica ideaofthe locationofthe quantumcriticalpoints onthe
symmetric solutionhas not yetbeen provedfor lowtem- phase diagram, and thus offers a guideline to choose the
perature region. However, numerical results of thermal proper kinetic terms (that maintain a sizable gap) and
annealingforinstancesofsizeN =60toN =160yielded the suitable annealing paths (see Sec. IIIA and IIIB).
many near optimal tours, and the corresponding over- In quantum spin glasses (Chakrabarti 1981, Ishii and
lap analysis shows a sharply peaked distribution, whose Yamamoto1985,Bhatt1998,Rieger2005,Sachdev1999,
width decreases steadily with increase in N. This indi- Ye 1993), the order-disorder transition (i.e., from the
cates the existence of a replica symmetric phase for the frozen phase to the high kinetic energy phase, termed
system (Mezard et al 1987). the para phase) can be driven both by thermal fluctua-
An analytical bound on the average (Normalized by tions as well as by quantum fluctuations. Quantum spin
N1/2) value of optimal path-length per city (Ω) calcu- glasses can be of two types: vector spin glasses, where
latedforTSPon2-dimensionalEuclideanplanehasbeen the quantum fluctuations cannot be adjusted by chang-
foundtobe5/8<Ω<0.92(Bearwoodetal1959). Care- ing some laboratory field, and the other, a classical spin
ful scaling analysis of the numerical results obtained so glassperturbedbysomequantumtunneling term,where
far indicates the lower bound to be close to 0.72 (Percus the quantum fluctuations are controlled through, say, a
and Martin 1996,Chakraborti and Chakrabarti2000). transverse laboratory field.
Simulated (thermal) annealing of a Euclidean TSP on The amount of the quantum fluctuations being ad-
a squarehavinglengthN1/2 (so asto renderthe average justable, this transverse Ising spin glass (TISG) model
nearestneighbordistanceindependentofN)hasbeenre- is perhaps the simplest model in which the quantum ef-
ported (Kirkpatrick et al 1983). In this choice of length fects in a random system can be and have been studied
unit, the optimaltour lengthper step (Ω) becomesinde- extensively and systematically (Chakrabarti et al 1996).
pendent of N for large N. Thermal annealing rendered Here we will focus only on TISG, since the reduction of
Ω 0.95 for N up to 6000 cities. This is much better the quantum fluctuations is the key feature required for
tha≤n what is obtained by the so called “greedy heuris- quantum annealing.
tics”, (where being at some city in a step, one moves to The interest in the zero-temperature quantum spin
thenearestcitynotinthetourinthenextstep)forwhich glass phases in the TISG models have been have been
Ω 1.12 on average. Later we will see that (Sec. IIIA) complimented all along by the experimental studies
∼
thatquantumannealingcandoevenbetterthanthermal in several systems which have been established to be
annealing in context of random TSP. represented accurately by transverse field Ising model
To summarize, when cast in some energy minimiza- (TIM). Recent discovery of the compound material
tion problem, the combinatorial optimization problems LiHoxY1 xF4 with the magnetic Ho ion concentration
may exhibit glassy behavior during thermal annealing. x= 0.167−(Aeppli and Rosenbaum 2005, Wu et al 1991,
Even replica symmetry-breaking behavior may be ob- Wuetal1993a;seealsoBrookeetal2001,Silevitchetal
served(likeincaseofGraphPartitioningProblem,seeFu 2007)representingaccuratelyarandomlong-rangetrans-
and Anderson 1986), since the underlying Hamiltonian verseIsingsystem,hasledtorenewedinterest. Here,the
need not be short-range, and the constraints can bring strongspin-orbitcouplingbetweenthespinsandthehost
frustrations into the problem. One can intuitively con- crystalsrestrictstheeffective“Ising”spinstoaligneither
clude that thermal annealing or other heuristics would paralleloranti-paralleltothespecificcrystalaxis. Anap-
not be able to solve such problems easily to a good ap- pliedmagneticfield,transversetothepreferredaxis,flips
proximation within reasonable time. Moreover, practi- the “Ising” spins. This feature, together with the ran-
cally nothing can be said about the time required to domness in the spin-spin interaction, makes it a unique
solve the worst case instance exactly. Specifically, in TISG-like system. Most interestingly, it has been shown
somecases,wheregoodsolutionsarethermodynamically that in spite of the presence of all the three ingredients
very insignificant in number and there is no monotonic - frustrations, randomness and the long-range (dipolar)
gradient towards them, the entropy might make a clas- interactions, that are necessary for the formation of a
sicalsearchexponentially difficult, thoughthe landscape spin glass, the spin glass phase of LiHoxY1 xF4 is ac-
might still remain completely ergodic. Later we will see tually destroyed by any finite transverse field−(Schechter
quantum searches can bring about spectacular improve- andLaflorencie2006). This indicates the effectiveness of
ments in some such cases (see Sec. IIIB and Fig. 4). quantum tunneling in the exploration of a rugged PEL
with formidable potential energy barriers. The TISG
model described here, is given by the Hamiltonian
D. Quantum Spin Glasses and Annealing
N N
= J SzSz Γ Sx, (9)
In QA one adds a kinetic (tunneling) term to the inter- H − ij i j − i
i>j i
X X
action part of the classical glass Hamiltonian. The ob-
ject that results in, is called a quantum spin glass. The where Γ denotes the tunneling strength at each site and
knowledge of the phase-diagramof a quantum spin glass J s are distributed randomly following the distribution
ij
8
ρ(J ) given by (3) or (4). Generally, we denote the d-dimensionalquantumHamiltoniantoaneffectived+1
ij
strength of the quantum kinetic term by Γ. dimensional anisotropic classical Hamiltonian (see also
The unique interest in such quantum spin glass sys- Hatano and Suzuki 2005). The partition function gives
temcomesfromthe possibilityofmuchfastercrossingof the effective classical Hamiltonian in the Mth Trotter
the high barriers occurring in the potential energy land- approximationas
scapes of the classicalspin glasses by means of quantum
tunneling induced by the transverse field, compared to
that done thermally by scaling such barriers with the N M N M
= K S S KS S , (13)
temperature. ij ik jk ik ik+1
H −
The phase transitions in quantum spin glasses can be Xi>jXk Xi Xk
driven both by thermal and quantum fluctuations as
with
mentioned before and the equilibrium phase diagrams
also indicate how the optimized solution (in SG phase)
J 1 Γ
ij
canbeobtainedeitherbytuningofthetemperatureT or K = ; K = lncoth , (14)
ij
MT 2 MT
the tunneling field Γ, or by both. We will show later (in (cid:18) (cid:19)
context of quantum annealing) that reaching the phase
where S denotes the Ising spin defined on the lattice
by tuning Γ may often be more advantageous (faster) ik
site (i, k), i denoting the position in the in the original
than that by tuning T.
SK model and k denoting the position in the additional
The short-range version of this TISG model was first
Trotter dimension. Although the equivalence between
studied by Chakrabarti (1981), and the long-range ver-
classical and the quantum model holds exactly in the
sion, discussed here, was first studied by Ishii and Ya-
limit M , one can always make an optimum choice
mamoto (1985). Several analytical studies have been → ∞
for M. The equivalent classical Hamiltonian has been
madetoobtainthephasediagramofthetransverseIsing
studied using standard Monte Carlo technique. The nu-
SK model (Miller and Huse 1993). The problem of an
mericalestimatesofthephasediagrametc. arereviewed
SK glass in a transverse field becomes a nontrivial one
in details in (Bhatt 1998,Rieger 2005). Ray et al (1989)
due to the presence of non-commuting spin operators in
took Γ << J and their results indeed indicated a sharp
the Hamiltonian. This leads to a dynamical frequency-
lowering of T (Γ). Such sharp fall of T (Γ) with large
dependent self-interaction for the spins. c c
Γ is obtained in almost all the theoretical studies of the
One can study an effective-spin Hamiltonian for the
phasediagramofthemodel(MillerandHuse1993;Yeet
SK model in a transverse field within the mean field
al1993;seealsoBhatt1998andRieger2005). Quantum
framework very easily. The spin glass order parameter
Monte Carlo (Alverez and Ritort 1996) as well as real-
inaclassicalSKmodelis givenbyarandom‘meanfield’
time Schr¨odinger evolution (the true dynamics given by
h(r)havingGaussiandistribution(seeBinderandYoung
the time-dependent Schr¨odinger equation) studies of SK
1986)
spin glass in transverse field were made (Lancaster and
+ Ritort, 1997).
q = ∞dre−r2/2tanh2(hz(r)/T); hz(r)=J√qr+hz IntheHamiltonianfortheEAspinglassinpresenceof
Z−∞ (10) a transverse field, given by (9), the random interactions
are restricted among the nearest neighbors and satisfy
where hz denotes the external field (in z direction), the
a Gaussian distribution with zero mean and variance J,
mean-field h(r) being also in the same direction. In the
as given by Eq. (3). Here, the variation of correlations
presence of the transverse field, as in (9), h(r) has com-
in the equivalent (d+1) dimensional classical model fit-
ponents both in z and x directions
ted very well (Guo et al 1994) with the scaling fit with
~h(r)= hz(r)zˆ Γxˆ; h(r)= hz(r)2+Γ2, (11) a unique order parameter and a critical interval corre-
− − sponding to a phase diagram whose features are similar
andonereplacestheorderingtermtpanh2(h(r)/T)in(10) tothosediscussedabove(seealsoChakrabartietal1996,
by its component [hz(r)/h(r)]2tanh2(h(r)/T) in the Bhatt 1998 and Rieger 2005).
z-direction. Puttin|ghz =| 0| and|q 0on|eget|sthephase Asdiscussedearlierinthissection,LiHoxY1 xF4 with
→ x = 0.167 provides a spin glass system, for −which the
boundary equation as (see Chakrabarti et al 1996)
external magnetic field transverse to the preferred axis
Γ Γ scales like the square root of the tunneling field Γ in
=tanh . (12)
J T (9). With increasingtransversefield, the glasstransition
(cid:18) (cid:19) temperature decrease monotonically, as shownin Fig. 2.
This gives Γ (T = 0) = J = T (Γ = 0) and a A quantum tunneling term allows for overlap between
c c
phase diagram qualitatively similar to the experimental two classically localized states and the dynamics near
one shown in Fig. 2. the ground states of such glasses show better ergodicity
Several Monte Carlo studies have been performed for properties. In order to investigate this aspect of quan-
the SK spin glass model in transverse field applying the tum spin glasses, one can study the overlap distribution
Suzuki-Trotter formalism (see Appendix 1), mapping a function P(q) given by Eq. (6).
9
minimum form any point in the configuration space, the
search has to involve visiting any substantial fraction of
the configurations. Thus a PEL without a guiding gra-
dient poses a problem which is clearly of an exponen-
tial or higher order in complexity (depending how the
size of the configuration space scales with the system-
size N), and the CA algorithms can do no better than
a random search algorithm. This is the case for golf-
course type potential-energy landscape (where there is a
sharp potential minimum on completely flat PEL) (see
FIG. 2 Phase diagram of LiHo0.167Y0.833F4 according to the Sec. IIIB, Fig. 4). One can imagine that quantum me-
dynamical measurements (filled circles) and of the nonlinear sus- chanics might have some solutions to both these prob-
ceptibilitymeasurements(opencircles). Filledsquaresindicatethe lems, at least, to some extent. This is because quan-
freezing boundary obtained from AC susceptibility measurements tum mechanics can introduce classically unlikely tunnel-
(taken fromWuetal1993b). ingpathseventhroughveryhighbarriersiftheyarenar-
rowenough(Rayetal1989;seealsoApollonietal1989,
1990). Thiscansolvetheergodicityproblemtosomeex-
If the ergodicity is recovered,at least for a part of the
tent,asdiscussedearlier. Eveninplaceswhereergodicity
phasediagram,theabovefunctionshouldtendtoadelta
breaking does not take place in true sense, once the en-
function form, peaking at some finite value of the order-
ergy landscape contains high enough barriers (specially
parameter q in thermodynamic limit. In the para-phase,
forinfinite-rangedquenchedinteractions),quantumtun-
of course, the distribution becomes a delta function at
neling may providemuch faster relaxationto the ground
q = 0 for the infinite system. In spite of several investi-
state (Santoro et al 2002, Marton˘a´k et al 2002; see also
gations ( see e.g., Ray et al 1989, Thirumalai and Kirk-
Santoro and Tosatti 2007, Somma et al 2007). In addi-
patrick 1989,Goldschmidt and Lai 1990,Chakrabartiet
tion,aquantummechanicalwavefunctioncandelocalize
al1996,KimandKim2002)the point ofreplicasymme-
over the whole configuration space (i.e., over the poten-
try restoration in spin glasses by quantum fluctuation is
tial energy landscape) if the kinetic energy term is high
not settled yet. However, slow withdrawal (see Eq. 16
enough. Thus it can actually “see” the whole landscape
for the characteristic slowness) of the tunneling field in
simultaneously at some stage of annealing. These two
these quantum spin glasses can help annealing the sys-
aspects can be expected to improve the search process
tem close to the ground state of the classical spin glass
when employedproperly. In fact such improvements can
eventually, as has been described in the next section.
indeed be achieved in certain situations, though quan-
tum mechanics is not a panacea for all such diseases as
ergodicitybreaking,spinglassbehavioretc,andcertainly
III. QUANTUM ANNEALING
has its own inherent limitations. What intrigues one is
the fact that the limitations due to quantum nature of
Intheprevioussectionswehaveseenhowthermalfluctu-
an algorithm are inherently different from those faced
ations can be utilized to devise fast heuristics to find an
by its classical counterpart, and thus it is not yet clear
approximate ground state of a glassy system, or equiva-
in general which wins when. Here we discuss some re-
lently,anear-optimalsolutiontocombinatorialproblem,
sults regardingthe quantumheuristics andsomeoftheir
whose cost-configuration landscape has glassy behavior
general aspects understood so far. For more detailed re-
due to the occurrence of many local minima. There
views of the subject we refer to the articles in Das and
are two aspects of an optimization problem which might
Chakrabarti(2005) and Santoro and Tosatti (2006).
render thermal annealing to be a very ineffective one.
First, in a glassy landscape, there may exist very high Some basic aspects ofQA canbe understoodfromthe
cost/energybarriersaroundlocalminimawhichdoesnot simple case of QA in the context of a double-well po-
correspond to a reasonably low cost (see Fig. 1). In the tential (Stella 2005, Battaglia et al 2005). Typically a
case of infinite-range problems, these barriers might be particle in a double-well consisting of a shallower but
proportional to the system-size N, and thus diverge in wider well and a deeper but narrower well is annealed
thermodynamic limit. Thus there might occur manyun- (it is likely that the deeper well, i.e., the target state,
satisfactorylocalminima, anyofwhichcantrapthe sys- is narrower,otherwise the searching becomes easier even
tem for very long time (which actually diverges in ther- classically). The kinetic energy (inverse mass) is tuned
modynamic limit for infinite-range systems) in course of from a very high value to zero linearly within a time τ.
annealing. Thesecondproblemistheentropyitself. The For a very high value of initial kinetic energy, the wave
numberofconfigurationsgrowveryfastwiththenumber function, which is the ground state, is delocalized more
ofvariables(roughlyexponentially;nIsingspincanbein or less over the whole double-well. As kinetic energy is
2nconfigurations)andaclassicalsystemcanonlyassume reducedbutstillquitehigh,thegroundstatecorresponds
one configuration at a time and unless there is a gradi- to a more pronounced peak on the shallower minimum,
ent that broadly guides the system towards the global since it is wider. This is because at this stage, to obtain
10
the minimal (groundstate)energy,it ismore effective to where
minimize the kinetic energy by localization over a wider
d
space, rather than minimizing the potential energy by ˙ = max φ (t) Htot φ (t) ,
tot max 0 1
localizing in the deeper well. However, as kinetic en- |hH i| 0≤t≤τ(cid:20)(cid:12)(cid:28) (cid:12) ds (cid:12) (cid:29)(cid:12)(cid:21)
ergy is tuned down further, the potential energy term (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
becomes dominating, and the ground state has a taller ∆2 = min ∆2(t) ;(cid:12) s=t/(cid:12)τ; 0(cid:12) s 1(cid:12), (17)
peak around the deeper minimum. The evolving wave min 0 t τ ≤ ≤
≤ ≤
function can roughly follow this ground state structure (cid:2) (cid:3)
φ (t) and φ (t) being respectively the instantaneous
all the way and finally settle to the deeper minimum if 0 1
| i | i
groundstateandthefirstexcitedstateofthetotalHamil-
the annealing time τ is greater than some τ . When
c
tonian , and ∆(t) the instantaneous gap between
T < T it fails to tunnel from its early state localized in tot
c H
the ground state and the first excited state energies (see
the shallower well to the deeper well as the kinetic en-
Sarandy et al 2004). One may wonder that while en-
ergy is decreased. This result is qualitatively the same
tering the ordered phase (Γ < Γ ) from the para phase
for both real-time and quantum Monte Carlo annealing, c
(Γ > Γ ) in course of annealing, the gap ∆ may vanish
excepting for the fact that the τ ’s are different in the c
c
at the phase boundary (Γ=Γ ) in the N limit. In
two cases. c
→∞
fact, in such a case, QA cannot help anyway in finding
TherealizationofQAconsistsofemployingadjustable
the ground state of an infinite system. However, for any
quantumfluctuations into the probleminsteadofa ther-
finite sample, this gap is very unlikely to vanish for a
mal one (Kadowaki and Nishimori 1998, Amara et al
random system, and QA may still work out nicely.
1993, Finnila et al 1994). In order to do that, one needs
However, it is impossible to follow, even for a finite
tointroduceanartificialquantumkinetictermΓ(t) ,
kin
H N, the evolution of a full wave function in a classical
which does not commute with the classical Hamiltonian
computer using polynomial resources in general, since
representing the cost function. The coefficient Γ is
C
H it requires tracking the amplitudes of all the basis vec-
the parameter which controls the quantum fluctuations.
tors(allpossible classicalconfigurations),whosenumber
The total Hamiltonian is thus given by
grows exponentially with system-size N. Such an adi-
abatic evolution may be realized within polynomial re-
= +Γ(t) . (15)
tot C kin
H H H sources only if one can employ a quantum mechanical
system itself to mimic the dynamics. However, one may
The ground state of is a superposition of the eigen-
tot
H employquantumMonteCarlomethodstosimulatesome
states of H . For a classical Ising Hamiltonian of the
C
dynamics(notthe real-timequantumdynamics)tosam-
form (2), the corresponding total quantum Hamiltonian
might have the form (9), where = iNSzSz ple the ground state (or a mixed state at low enough
HC − i>j i j temperature) for a given set of parameter values of the
and = NSx. Initially Γ is kept very high so
Hkin − i i P Hamiltonian. Annealingisdonebyreducingthestrength
thatthe dominatesandthegroundstateistrivially
kin ΓofthequantumkinetictermintheHamiltonianfroma
H P
auniformsuperpositionofalltheclassicalconfigurations.
veryhighvaluetozerofollowingsomeannealingschedule
One starts with that uniform superpositionas the initial
in course of simulation. In case of such Monte Carlo an-
state, and slowly decreases Γ following some annealing
nealing algorithm, there is no general bound on success
schedule, eventually to zero. If the process of decreasing
time τ like the one provided by the adiabatic theorem
is slow enough, the adiabatic theorem of quantum me-
for true Schr¨odinger evolution annealing. Here we will
chanics(Sarandyetal2004)assuresthatthe systemwill
separatelydiscusstheresultsofreal-timeQAandMonte
always remain at the instantaneous ground state of the
Carlo QA. Apart from these quasi-stationary quantum
evolving Hamiltonian . When Γ is finally brought to
tot annealingstrategies,wherethesystemalwaysstaysclose
H
zero, will coincide with the originalclassicalHamil-
tot tosomestationarystate(orlow-temperatureequilibrium
H
tonian and the system will be found in the ground
C state), there may be cases where quantum scatterings
H
state of it, as desired. The special class of QA algo-
(with tunable amplitudes) are employed to anneal the
rithms, where strictly quasi-stationary or adiabatic evo-
system (Das et al 2005).
lutions are employed are also known as Quantum Adia-
batic Evolution algorithms (Farhi et al 2001).
Two important questions are how to choose an appro- A. Quantum Monte Carlo Annealing
priate and how slow the evolution needs to be in
kin
H
order to assure adiabaticity. According to the adiabatic In quantum Monte Carlo annealing, one may employ ei-
theorem of quantum mechanics, for a non-degenerate ther a finite (but low) temperature algorithm, or a zero-
spectrum with a gap between the ground state and first temperature algorithm. Most of the Monte Carlo QA
excited state, the adiabatic evolution is assured if the (Das and Chakrabarti 2005, Santoro and Tosatti 2006)
evolution time τ satisfies the following condition- aredoneusingafinitetemperatureMonteCarlo,namely,
the Path Integral Monte Carlo (PIMC), since its imple-
˙
tot max mentation is somewhat simpler than that of other zero-
τ >> |hH i| , (16)
∆2 temperature Monte Carlo methods .
min