Adiabatic Quantum Computing for Random Satisfiability Problems

0.2 MB·English

by  Tad Hogg

Adiabatic Quantum Computing for Random Satisfiability Problems∗
Tad Hogg
HP Labs
Palo Alto, CA 94304
Thediscreteformulationofadiabaticquantumcomputingiscomparedwithothersearchmethods,
classicalandquantum,forrandomsatisfiability(SAT)problems. Withthenumberofstepsgrowing
only as thecubeof thenumberof variables, theadiabatic method gives solution probabilities close
to 1 for problem sizes feasible to evaluate via simulation on current computers. However, for these
sizes the minimum energy gaps of most instances are fairly large, so the good performance scaling
seen for small problems may not reflect asymptotic behavior where costs are dominated by tiny
gaps. Moreover, theresultingsearch costsaremuchhigherthanforothermethods. Variantsofthe
quantum algorithm that do not match the adiabatic limit give lower costs, on average, and slower
4 growth than theconventional GSATheuristic method.
0
0 PACSnumbers: 03.67.Lx
2

n
I. INTRODUCTION 1/2. Thus clauses are selected uniformly from among
a
J the M = n 2k possible clauses. We focus on the deci-
k
sion problem, i.e., finding a solution, rather than the re-
3 Quantumcomputers[1,2,3,4]canrapidlyevaluateall
(cid:0) (cid:1)
2 searchstates of nondeterministic polynomial (NP) prob- latedoptimization problem,i.e., finding a minimum cost
lems[5],butappearunlikelytogiveshortworst-casesolu- state. Thisallowsdirectcomparisonwithpriorempirical
2 studies of heuristic methods for SAT. The algorithms
tiontimes[6]. Ofmorepracticalinterestiswhethertheir
v we consider are probabilistic, so cannot definitively de-
averageperformanceimprovesonconventionalheuristics.
9
terminenosolutionexists. Thusweusesolubleinstances:
5 Adiabatic quantum computing, using a slowly chang-
after random generation, we solve the instances with an
0 ing time-dependent Hamiltonian, appears to give poly-
exhaustive conventional method and only retain those
6 nomial average cost growth for some NP combinatorial
0 searchproblems[7]. Theseobservations,while encourag- with a solution. This ensemble has a high concentra-
2 tion of hard instances near a phase transition in search
ing, are limited to small problems for which other meth-
0 difficulty [10, 11, 12, 13]. For 3-SAT, we generate in-
ods,bothconventionalandquantum,canhaveevenlower
/ stances near this transition by using µ m/n = 4.25,
h costs. Furthermore, although adiabatic methods appar- ≡
p entlyshowexponentialcostscalingforsetpartitioning[8] though for those n not divisible by 4, half the samples
- had m= 4.25n and half had m larger by 1.
t andfindingthegroundstateofspinglasses[9],thetypical ⌊ ⌋
n Theremainderofthispaperdescribesseveralquantum
performance of adiabatic quantum computing for large
a search algorithms in the context of satisfiability prob-
NP search problems remains an open question. Thus it
u
lems, and then compares their behavior.
q isofinteresttocomparetheadiabaticmethodwithother
: techniques for NP problems having a well-studied class
v
of hard instances.
i II. ALGORITHMS
X This paper provides such a comparison for k-
r satisfiability (k-SAT), consisting of n Boolean variables
a and m clauses. A clause is a logical OR of k variables, The adiabatic technique [7] is based on two Hamil-
each of which may be negated. A solution is an assign- tonians H(0) and H(c). The first is selected to have a
ment, i.e., a value, true or false, for each variable, satis- knowngroundstate,whilethe groundstatesofH(c) cor-
fying all the clauses. An example 2-SAT instance with 3 respond to the solutions of the problem instance to be
variablesand2clausesisv OR(NOTv )andv ORv , solved. The algorithm continuously evolves the state of
1 2 2 3
whichhas4solutions,e.g.,v =v =falseandv =true. thequantumcomputerusingH(f)=(1 f)H(0)+fH(c)
1 2 3 −
For a giveninstance, let the cost c(s) of anassignments with f ranging from 0 to 1. Under suitable conditions,
be the number of clauses it does not satisfy. i.e., with a nonzero gap between relevant eigenvalues of
H(f), the adiabatic theorem guarantees that, with suffi-
For k 3, k-SAT is NP-complete [5], i.e., among the
≥ cientlyslowchangesinf,the evolutionmapstheground
mostdifficultNPproblemsintheworstcase. Foraverage
stateofH(0) intoagroundstateofH(c),soasubsequent
behavior we use the random k-SAT ensemble, in which
measurement gives a solution. The choices of H(0), H(c)
the m clauses are selected uniformly at random. I.e.,
and how f varies as a function of time are somewhat
for each clause, a set of k variables is selected randomly,
arbitrary.
and each selected variable is negated with probability
Inmatrixform,oneHamiltonianwithminimal-costas-
(c)
signments as ground states is H =c(s)δ , for assign-
r,s r,s
ments r and s, where δ is 1 if r = s and 0 otherwise.
r,s
∗appearedinPhys Rev A67,022314(2003) This Hamiltonian introduces a phase factor in the am-
2
plitude of assignment s depending on its associated cost algorithm parameters phase functions
c(s). T ∆
For H(0), we introduce a nonnegative weight wi for adiabatic T ∆ 0 ρ(0)=0=τ(1)
variable i, let ω ≡ ni=1wi and take discrete adiabatic T →→∞∞ cons→tant ρ(0)=0=τ(1)
heuristic constant ∆ 0 suitable ρ,τ
ω/2 Pif r=s →
Hr(,0s) = wi/2 if r and s differ only for variable i (1)
(0− otherwise TABLE I: Summary of quantum search algorithms using
problemstructure. Theheuristicmethodrequiresfindingap-
This Hamiltonian can be implemented with elementary
propriatechoicesfor thephasefunctionstogivegood perfor-
quantum gates by use of the Walsh-Hadamard trans- manceandforthenumberofstepsj toincreasewithproblem
form W, with elements Wr,s = 2−n/2( 1)r·s (treating sizen. Theadiabaticmethodsrequiresufficientlylargevalues
−
the states r and s as vectors of bits so their dot product of T = j∆. A constant value for a parameter in this table
counts the number of variables assigned the value 1 in means it is taken to beindependent of n and j.
both states). Specifically, H(0) = WDW where D is a
diagonal matrix with the value for state r given by the
weighted sum of the bits: n w r with r represent- spondence with the continuous evolution for sufficiently
i=1 i i i
ing the value of the ith bit of r. In particular, if all the large j.
weightsequal1,D justcoPuntsthenumberofbitsequal TheunweightedH(0)usesequalweights: w =1soω =
r,r i
to 1. n. Alternatively,w canbethenumberoftimesvariablei
i
Theadiabaticmethodisacontinuousprocess. Tocom- appearsinaclause[7],asalsousedbysomeconventional
pare with other algorithms, we use the algorithmically heuristics to adjust the importance of changes in each
equivalentdiscreteformulation[14,15]actingontheam- variable. This choice gives ω = mk. By matching H(0)
plitude vector initially in the ground state of H(0), i.e., totheprobleminstance,onemightexpectsuchweightsto
ψ(0) =2−n/2. This formulation consists of j steps and a improve performance. Instead, for random 3-SAT these
s
parameter ∆. Step h is a matrix multiplication: weights give higher costs C, requiring about n times as
many steps to achieve the same P as the unweighted
soln
ψ(h) =e−iτ(f)H(0)∆ e−iρ(f)H(c)∆ ψ(h−1) (2) choice. If instead these weights are normalized so their
averagevalue is 1, the performance is about the same as
with the mixing phase function τ(f)=1 f, cost phase in the unweighted case, but still slightly worse. In light
function ρ(f) = f and taking h¯ = 1. Af−ter these steps, oftheseobservations,weusethe unweightedH(0) inthis
the probability to find a solution is P = ψ(j) 2, paper.
soln s| |
with the sum over all solutions s. We compare the adiabatic limit with two other meth-
As a simple choiceforthe evolution,we takePf to vary ods, summarized in Table I. First, for the discrete adia-
linearly from 0 to 1. We exclude the steps with f = 0 baticcasewetake∆independentofnandj,violatingthe
and 1 since they have no effect on P . Specifically, we condition ∆n 0 so Eq. (2) no longer closely approxi-
soln →
take f =h/(j+1) for step h, ranging from 1 to j. mates the continuous evolution and does not necessarily
The expected number of steps required to find a solu- give Psoln 1 as j . In this case, a discrete ver-
→ → ∞
tion is C = j/P , providing a commonly used proxy sionoftheadiabatictheorem,describedintheappendix,
soln
for the computational cost of discrete methods, pending ensures Psoln is close to 1 if ∆ is not too large.
furtherstudyofclockratesfortheunderlyinggateopera- Second, the heuristic method, studied previously [16,
tions andthe ability ofcompilersto eliminate redundant 17], has ∆ = 1/j and forms for τ(f) and ρ(f) that do
operations. As also observed with conventional heuris- not range between 0 and 1. Instead, these phase func-
tics, the cost distribution for random k-SAT is highly tionsmustbe selectedappropriatelyto givegoodperfor-
skewed, so a few instances dominate the mean cost. In- mance. Identifying such choices and characterizing their
stead, we use the median cost to indicate typical behav- performance are major issues for this algorithm, though
ior. The time for the continuous formulation is T =j∆, mean-field approximations based on a few problem pa-
so the adiabatic limit is j∆ . By contrast, in the rameters, e.g., the ratio m/n for k-SAT, can give rea-
→ ∞
discrete formulation, ∆ parameterizes the operators of sonably good choices. This method does not correspond
Eq.(2)ratherthandeterminingthetimerequiredtoper- to the adiabatic limit: P has a limit less than 1 as
soln
form them. j .
→∞
Eq. (2) follows the continuous evolution, ψ(h+1) For all these techniques, expected cost C = j/P is
soln
e−iH(f)∆ψ(h), when ∆ H 0 which holds when ∆ ≈ minimizedforintermediatevaluesofj ratherthantaking
|| || → ≪
1/n [14, 15]. This last condition uses the fact that the j asusedinthe limits listedinTableI. Identifying
→∞
norm H is the largest eigenvalue of H, which is O(n) parameters and phase functions, ρ(f) and τ(f), giving
|| ||
since we consider k-SAT problems with m n. As a minimal cost for a given problem instance depends on
∝
specific choice, we use ∆ = 1/√j. Other scaling choices details of the searchspace structure unlikely to be avail-
∆ = 1/jα with 0 < α < 1 give qualitatively similar be- ablepriortosolvingthatinstance. However,asdescribed
haviors to those reported here while maintaining corre- below, taking j to grow only as a fairly small power of n
3
1 gap
n3 1 gy3
r2
0.5 e
n1
2 0.8 e
n
0 0.5 1
0.2 0.6 f
3/2
n
0.1 0.4
n
l n
o
s
P 0.05 0.2
f
0.2 0.4 0.6 0.8 1
0.02
0.01 FIG.2: Differencebetweeneigenvaluesofthelowest5excited
states and the ground state vs. f for an instance with n =
20, m = 85 and 5 solutions. The inset shows the actual
8 12 16 20 24 eigenvalues, with the gray curve showing the expected cost
n
c in theground state.
h i
FIG.1: Log-logplotofmedianP fortheadiabaticmethod
soln energy
vs.nwiththenumberofstepsj equalton,theintegernear-
est n3/2, n2 and n3 (solid curves, from bottom to top, re- 6
spectively). We use ∆ = 1/√j. For comparison, the dashed
5
curve shows P for the heuristic method using at most n
soln
steps. The error bars show the 95% confidence intervals [18,
4
p. 124] of the medians estimated from the random sample of
instances. Thesameinstancesweresolvedwitheachmethod. 3
Weuse1000instancesforeach nupto20, and500forlarger
n, except only 100 for j =n3 for n 16. 2
≥
1
providesrelativelymodestcosts,onaverage,forproblem
f
sizes feasible to simulate. 0.2 0.4 0.6 0.8 1
FIG.3: Lowesttwoeigenvaluesvs.f foraninstancewithn=
20, m = 85, one solution and a particularly small minimum
III. BEHAVIOR
gap. Thegraycurveshowstheexpectedcost c intheground
state, equal to m/2k = 10.625 at f = 0. Nhoite the abrupt
For the adiabatic method, Fig. 1 shows the median drop at the location of the minimum gap. The ground state
Psoln for various growth rates of the number of steps. for f = 1 is the solution, whose cost is zero, so c 0 as
P 1 as j increases. At least for n < 20, P 1 f 1. h i →
wshoelnn→j = n3, so median costs are O(n3∼), a subssotlann≈tial →
improvement over all known classical methods if it con-
tinues for larger n. However, for smaller powers of n, using sparse matrix techniques [19] for n 20 gives the
≤
P values decrease, but this is only evident for j =n2 median G in the range 0.3 0.5, as illustrated for one
soln
−
for n > 20. This raises the possibility of such a decline, instanceinFig.2,and,moresignificantly,itdoesnotde-
atsomewhatlargern, forlargerj aswell. Providedsuch crease over this range of n. This minimum is not much
a decline only leads to P decreasing as a power of smallerthanothervaluesofg(f). Hence,unlikeforlarge
soln
n, corresponding to a straight line on the log-log plot of n, the cost is not dominated by the minimum gap size
Fig. 1, median costs would still only grow as a power of and so the values of Fig. 1 may not reflect asymptotic
n. The remainder of this section describes the algorithm scaling.
behaviors in more detail. By contrast, Fig. 3 illustrates the behavior of an in-
stance with a small minimum gap. One characterization
of the eigenstates of H(f) is their expected cost, i.e.,
A. Energy Gap c = c(s)φ(a)(f)2 where φ(a)(f) is the ath eigen-
h ia s | s |
vector of H(f). In particular, for a = 1 this gives the
P
Asymptotically, the adiabatic method’s cost is domi- expected cost in the ground state, which we denote sim-
nated by the growth of 1/G2 where G = min g(f) and ply as c . The expected cost in the ground state drops
f
h i
g(f) is the energy gap in H(f), i.e., the difference be- rapidly at the minimum gap location, in contrast to the
tweenthegroundstateeigenvalueandthesmallesthigher smooth behavior for instances with larger gaps (as, for
eigenvalue corresponding to a non-solution. Evaluation example, in Fig. 2). We thus see a difference in behavior
4
of the ground state for instances with small gaps, pre-
5000
sumably representative of typical behavior for larger n,
and the behavior of more typical instances for n 20.
≈
WiththeadiabaticmethodandT sufficientlylarge,the
actual state of the quantum computer after step h, ψ(h),
1000
closely approximates the ground state eigenvector φ(1), adiabatic
up to an irrelevant overallphase. Thus the computation
500
will also show the jump in expected cost. t
s
Detailedquantitativecomparisonofthetypicalbehav- o
c
unstructured
iorsduetosmallminimumgapsandconventionalheuris-
tics requires larger problem sizes. Nevertheless, we can
100
gain some insight from instances with small gaps for
GSAT
n 20, which tend to have high costs for both the
50
≈
quantum methods and conventional heuristics, such as
heuristic
GSAT [20], even when restricting comparison to prob-
lems with the same numbers of variables and solutions.
For the instance shown in Fig. 3, GSAT trials readily 12 14 16 18 20 22 24 26 28
n
reach states with 1 or 2 conflicts, but have a relatively
low chance to find the solution. This behavior, typical
FIG.4: Logplotofmediansearchcostvs.nfortheheuristic
ofconventionalheuristics[21],correspondstotheabrupt
(diamond), unstructured search (box), GSAT with restarts
dropin c ofFig.3. Thusfindingassignmentswithcosts
h i afternsteps(circle) andadiabaticsearch withj =n2 (trian-
below this value dominates the running time of both the
gle). ThevaluesarebasedonthesameinstancesasinFig.1.
quantum and conventionalmethods. These observations
The lines are exponential fits to the unstructured (dashed)
suggest small energy gaps characterize hard problems and adiabatic (solid) methods.
moregenerallythanjustfortheadiabaticmethod,which
may provide useful insights into the nature of search
along with quantities such as the backbone (i.e., vari-
for quantum computer implementations with relatively
ables with the same values in all solutions [13]).
few qubits and limited coherence times which are thus
Simpleproblemsoralgorithmsignoringproblemstruc-
limitedtosmallproblemsandfewsteps. Fig.4compares
ture allow determining the gap for large n [14, 15, 22].
the median values of the expected search costs C. For
This is difficult for random SAT. For instance, although the adiabatic method, using j =n3 gives large costs, far
randomk-SATcorrespondstorandomcostsforH(c) and
higher than those of conventional heuristics and other
theextremeeigenvaluesofrandommatricescanbedeter-
quantum methods. Using just enough steps to achieve
mined when elements are chosen independently [23, 24], moderate values of P reduces cost [7], e.g., j = n2.
soln
the costs of nearby states for SAT instances are highly
Alternatively,foreachn,testingvariousjonasmallsam-
correlatedsincetheylikelyconflictwithmanyofthesame
pleofinstancesindicatesthenumberofstepsrequiredto
clauses. Alternatively, upper [25] and lower [26] bounds
achieve a fixed value of P , e.g., 1/8. In our case, the
soln
for eigenvalues can be based on classes of trial vectors.
latter approach has median costs about 20% lower than
For instance, vectors whose components for state s de-
the former,butwiththe samecostgrowthrate. Because
pend only on c(s) give fairly close upper bounds for the
this improvement is minor compared to the differences
ground state of random 3-SAT, on average, as well as a
with other algorithms shown in the figure, and to avoid
mean-field approximation for the heuristic method [16].
theadditionalvariabilityduetoestimatingj fromasam-
However, simple lower bounds for higher energy states ple of instances, we simply take j = n2 to illustrate the
are below the upper bound for the ground state for
adiabatic method.
some values of f, and so do not give useful estimates
ThefigurealsoshowsGrover’sunstructuredsearch[27]
for G. Furthermore, typicalsoluble instances have expo-
(withoutpriorknowledgeofthenumberofsolutions[28])
nentially many solutions (although still anexponentially
and the conventional heuristic GSAT [20]. Unlike the
small fraction of all states). Thus a full analysis of per-
quantum methods, conventionalheuristics canfinish im-
formance based on energy values must also consider the
mediately when a solution is found rather than waiting
behavior of the many eigenvalues corresponding to solu-
until all j steps are completed. For comparisonwith the
tions, which can be complicated, as illustrated in Fig. 2.
differentchoicesofj inFig.1,themediancostsatn=20
for j = n, n3/2, n2 and n3 are, respectively, 1741, 879,
1010 and 8222. The unstructured search cost grows as
B. Search Cost e0.32n. The exponential fit to the adiabatic method is
e0.13n. This fit gives a residual about half as large as
Even if n<20 does not identify asymptotic behavior, that from a power-law fit. The growth rate is about the
this range of∼feasible simulations allows comparing algo- same as that of GSAT.
rithm costs. Such comparisons are particularly relevant Fig.4 showsthe heuristic,using atmostnsteps,gives
5
150 p(f) = 1.92708f 2.66179f2+ 1.73471f3. This cubic
−
issimilartothe functionalformoptimizingtheadiabatic
methodforunstructuredsearch[15,22]. Fig.5showsthe
100 resulting cost reduction. Hence, tuning the algorithm to
the problem ensemble is beneficial, as also suggested by
GSAT
a mean-field analysis of the heuristic [16].
70
The simulations also show these quantum algorithms
have a large performance variance among instances with
ost 50 given n and m, and no single choice for ρ and τ is best
c
heuristic forallprobleminstances. Thus portfolios[30]combining
a variety of such choices can give further improvements.
cubic
30 linear
IV. CONCLUSION
20
In summary, for random SAT, the adiabatic method
15 improves on unstructured search and provides a gen-
12 14 16 18 20 22 24 26 28 30
n eral technique to exploit readily computed properties
of hard search problems through the choice of Hamil-
FIG. 5: Log plot of median search cost vs. n for GSAT tonians. However nonadiabatic-limit algorithms require
(circle), the heuristic method (diamond), both of which are fewer steps, comparable to GSAT, and appear to have
alsoshowninFig.4,andtwoversionsofthediscreteadiabatic slower cost growth. As a caveat, small energy gaps ap-
method: ∆ = 1.2 with linear phase functions (triangle) and
peartobeassociatedwithinstancesdifficulttosolvewith
the cubic polynomial variation with f (gray box) described
both quantum and classical methods. Thus the simula-
in the text. The lines are exponential fits to GSAT (dashed)
tionresultspresentedhere,basedonfairlysmallproblem
andthetwodiscreteadiabatic quantummethods. Thefigure
sizes for which most instances have fairly large energy
uses thesame instances as Fig. 4.
gaps, may not reveal the asymptotic scaling of the typi-
calsearchcostforhardrandomSAT problems. Evaluat-
ingthebehaviorofthesealgorithmsand,moregenerally,
low costs due to its fairly high values for P shown in
soln
identifying better ways to use state costs in quantum al-
Fig. 1. The constant∆ scaling for the discrete adiabatic
gorithms remain open questions.
method also gives large P values for j n. Thus
soln
both ∆ = 1/j and ∆ independent of j make≈better use Quantum computers with only a moderate number of
of quantum coherence in the discrete formulation than qubits could testalgorithms beyondthe rangeofsimula-
the continuous adiabatic limit (with 1/j ∆ 1/n) tors,andhenceprovideusefulinsightseveniftheproblem
≪ ≪ sizes are still readily solved by conventional heuristics.
for hard random 3-SAT. These behaviors are shown in
Fig. 5. Such studies could help addressthe question ofwhether,
with suitable tuning based on readily evaluated average
Because these quantum methods and GSAT consistof
properties of search states, the ability to operate on the
aseriesofindependenttrials,theycanbecombinedwith
entire search space allows quantum computers to effec-
amplitude amplification to give an additional quadratic
tivelyexploitweakcorrelationsamongstatecostsinways
performance improvement [29]. However, this is only a
classical machines cannot.
significantbenefitwhenP is fairlysmall,whichisnot
soln
the case for the heuristic and GSAT methods for these
problem sizes.
Acknowledgments
For the adiabatic method, taking ρ(f) and τ(f) in
Eq. (2) to vary according to g(f)2 reduces costs [15, 22].
This concentrates steps at values of f close to the min- I have benefited from discussions with Rob Schreiber
imum gap. While g(f) is costly to evaluate for SAT in- and Wim van Dam. I thank Miles Deegan and the HP
stances, using average values of g(f) based on a sample High Performance Computing Expertise Center for pro-
of instances gives some benefit. E.g., for j = n2, P viding computational resources for the simulations.
soln
increases from around 0.4 shown in Fig. 1 to a range of
0.5 0.6 but this does not appear to reduce the cost’s
−
growth rate. APPENDIX A: DISCRETE ADIABATIC
Similar improvement occurs with constant ∆. Opti- BEHAVIOR
mizing τ and ρ separately for each step on a sample of
instances gives values close to a cubic polynomial in f. When ∆ is held constant, the steps of Eq. (2) do not
Restricting attention to such polynomials, for a set of approximate the continuous evolution induced by H(f),
100 n = 12 instances the best performance was with and hence ψ(h) does not closely follow the ground state
∆ = 1.31275, ρ(f) = p(f), τ(f) = 1 p(f) where of H(f) when T . Nevertheless, ψ(h) does closely
− → ∞
6
follow an eigenstate of the unitary operator involved in 2
Eq. (2). This discrete version of the adiabatic theorem
ensures good performance of the algorithm provided the 1
continuous change in the eigenvector takes the initial
y
ground state into the final one, rather than into some g 0
r
other eigenvector. e
n
e
-1
1. The Discrete Adiabatic Limit
-2
Considerasmoothlychangingsequenceofunitaryma- 0 0.2 0.4 0.6 0.8 1
f
trices U(f) defined for 0 f 1 and vectors ψ(h+1) =
U(f)ψ(h) with f =h/j fo≤r h=≤0,...,j 1. Let e−iθr(f)
and eˆ (f) be the rth eigenvalue and (no−rmalized) eigen- FIG.6: Energyvaluesθr(f)correspondingtothetwoeigen-
r values of U(f) vs. f for ∆ = 1 (gray) and 4 (black). The
vector of U(f).
values are defined only up to a multiple of 2π, and we take
We start with ψ(0) equal to the eigenvector eˆ1(0) of π < θ π. The ground states of H(0) and H(c) corre-
− ≤
U(0), which we assume to be nondegenerate for sim- spond to θ(0) = 0 and θ(1) = 0, respectively. The values
plicity. Provided the difference between eigenvalues is for ∆ = 1 are close to those of the combined Hamiltonian
boundedawayfromzero,forsufficientlylargej,ψ(j) will H(f)=(1 f)H(0)+fH(c). However, the ∆=4 values do
−
beclosetoaneigenvectorofU(1). Toseethisletǫ=1/j not remain close to those of H(f)∆.
andexpandψ(h) = c (f)Λ (f)eˆ (f)intheeigenbasis
r r r r
of U(f) where
P Psoln
1
h−1
Λ (h/j) exp i θ (k/j)
r ≡ − r ! 0.8
k=0
X
0.6
Firstorderperturbationtheorygivesthechangeinthe
cr values during one step to be O(ǫ). After j steps, 0.4
it might appear that these changes could build up to
O(ǫj) = O(1). However, this is not the case due to the 0.2
rapid variation in phases when j is large. Specifically,
steps
the changes in coefficients for r =1 are 10 20 30 40 50
6
dc
r
df =P1,r(f)Φr(f) (A1) FIG. 7: Psoln vs. j for ∆ = 1 (gray) and 4 (black). For
comparison, the dashed curve uses ∆ = 1/√j corresponding
to thecontinuousadiabatic limit.
where Ps,r(h/j) e−ijΘs,r(f),
≡
h−1
1
2. An Example
Θ (f) (θ (k/j) θ (k/j))
s,r s r
≡ j −
k=0
X
An important caveat in applying this result to quan-
and
tum algorithms is that while j suffices to ensure
→ ∞
ψ(h) closely follows the evolution of an eigenvector of
r dU/df 1
Φr ≡ eh−|iθr e−|iθi1 U(f),thisevolutionmaynotleadtothedesiredeigenvec-
− tor of U(1), i.e., correspondingto solutions to the search
Since c (0)=0, Eq. (A1) gives problem. This is because the eigenvalues of U(f) lie
r
on the unit circle in the complex plane and can “wrap
f around” as ∆ increases. Hence, in addition to ensur-
cr(f)= e−ijΘ1,r(κ)Φr(κ)dκ ing the eigenvalue gap does not get too small, good per-
Z0 formance also requires selecting appropriate ∆. Alter-
As j increases, the integrand oscillates increasingly natively, one could start from a different eigenvector of
rapidlysotheintegralgoestozeroasj byapplying U(0),whichwouldbeusefulifonecoulddeterminewhich
the Riemann-Lebesgue lemma since dΘ→∞/df = θ θ eigenvector maps to the solutions.
1,r 1 r
−
is nonzeroand Φ (f) isboundedfor allf andr =1,by Oneguaranteeofavoidingthisproblemisthatnoneof
r
| | 6
the assumption of no level crossing. Hence c (f) 0 so the eigenvalues of U(f) wrap around the unit circle, i.e.,
r
ψ(j) approaches eˆ (1), up to an overall phase fac→tor, as ∆ H 0, corresponding to the continuous adiabatic
1
|| || →
j . limit. Simulations show performance remains good for
→∞
7
100 so U(f) = e−iH(0)(1−f)∆e−iH(c)f∆. Fig. 6 shows the be-
havior of the two eigenvalues of U(f) for two values of
10-1 discrete ∆. For ∆ = 4 the initial ground state eigenvector, with
eigenvalue 1, evolves into the 2nd eigenvector of U(1)
n rather than the eigenvectorcorresponding to the ground
l
Pso10-2 state of H(c).
continuous
Fig.7showstheconsequenceofthisbehavior: when∆
10-3 is too large, ψ(h) follows the evolving eigenvector to the
heuristic unstructured wrong state when f =1, giving P 0 as j . As
soln
→ →∞
anotherobservationfromthisfigure,P (j)exhibitsos-
100 101 102 103 104 105 soln
steps cillations (though they are quite small for ∆=1). With
appropriatephasechoices,theseoscillationscanbequite
FIG. 8: P vs. j for several search methods solving a 20- large, allowing P to approach 1 with only a modest
soln soln
variable 3-SAT instance with 85 clauses and 5 solutions, the numberofsteps, evenwhenP approaches0 forlarger
soln
same instance used in Fig. 2. The gray curve is the discrete j. This observation is the basis of the heuristic method.
adiabatic method with ∆ = 1, the thick black curve is for
∆ = 1/√j corresponding to the continuous adiabatic limit. To see the consequence of this behavior for search,
For comparison, the thin black curve is the heuristic, with Fig. 8 compares the behavior of several search methods.
∆ = 1/j, and the dashed curve is for unstructured search In this case ∆ = 1 is sufficiently large that the initial
(showing only the first period of its sinusoidal oscillation on eigenstate of the unitary operator evolves into a nonso-
this log-log plot).
lutioneigenstate. Thus asthe numberofsteps increases,
theprobabilitytofindasolutiongoestozero,aswiththe
large ∆ case in Fig. 7. Nevertheless, for smaller j, most
moderate values of j even if ∆ does not go to zero, pro-
of the amplitude “crosses”the gapto another eigenstate
vided∆isbelowsomethresholdvalue. Forhardrandom
that does evolve to a solution state. Consequently, this
3-SATproblemswithj n,thisthresholdappearstobe
∝ discreteadiabaticmethodgivesloweroverallsearchcost,
somewhat larger than 1.
using a moderate number of steps, than the continuous
Toillustratetheseremarksconsiderthen=1example
adiabatic method (which has P 1 as j ). By
soln
→ → ∞
contrast, for ∆ larger than 2 or so P always remains
1 1 1 0 0 soln
H(0) = − , H(c) = small.
2 1 1 0 2
(cid:18)− (cid:19) (cid:18) (cid:19)
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