Solving SAT and MaxSAT with a Quantum Annealer: Foundations and a Preliminary Report

17 Pages·2017·0.37 MB·English

Solving SAT and MaxSAT with a Quantum Annealer:
Foundations and a Preliminary Report
ZhengbingBian1,FabianChudak1,WilliamMacready1,AidanRoy1,
RobertoSebastiani2,andStefanoVarotti2
1 D-WaveSystemsInc.,Burnaby,Canada
2 DISI,UniversityofTrento,Italy
Abstract. Quantumannealers(QA)arespecializedquantumcomputersthatmin-
imizeobjectivefunctionsoverdiscretevariablesbyphysicallyexploitingquan-
tum effects. Current QA platforms allow for the optimization of quadratic ob-
jectivesdefinedoverbinaryvariables,thatis,theysolvequadraticunconstrained
binaryoptimization(QUBO)problems.Inthelastdecade,QAsystemsasimple-
mented by D-Wave have scaled with Moore-like growth. Current architectures
provide2048sparsely-connectedqubits,andcontinuedexponentialgrowthisan-
ticipated.
We explore the feasibility of such architectures for solving SAT and MaxSAT
problems as QA systems scale. We develop techniques for effectively encod-
ingSATandMaxSATintoQUBOcompatiblewithsparseQAarchitectures.We
providethetheoreticalfoundationsforthismapping,andpresentencodingtech-
niquesthatcombineofflineSatisfiabilityandOptimizationModuloTheorieswith
on-the-flyplacementandrouting.Preliminaryempiricaltestsonacurrentgener-
ation2048-qubitD-Wavesystemsupportthefeasibilityoftheapproach.
WeprovidedetailsonourSMTmodeloftheSAT-encodingprobleminthehopes
thatfurtherresearchmayimproveuponthescalabilityofthisapplicationofSMT
technology. Further, these models generate hard SMT problems which may be
usefulasbenchmarksforsolvers.
1 Introduction
QuantumAnnealing(QA)isaspecializedformofcomputationthatusesquantumme-
chanicaleffectstoefficientlysamplelow-energyconfigurationsofparticularcostfunc-
tionsonbinaryvariables.Currently,thelargestQAsystemheuristicallyminimizesan
Isingcostfunctiongivenby
(cid:88) (cid:88)
E(z)=def h z + J z z (1)
i i ij i j
i∈V (i,j)∈E
argmin E(z). (2)
z∈{−1,1}|V|
whereG = (V,E)isanundirectedgraphofallowedvariableinteractions.Isingmod-
elsareequivalenttoQuadraticUnconstrainedBinaryOptimization(QUBO)problems,
whichuse{0,1}-valuedvariablesratherthan±1-valuedvariables.1 Thedecisionver-
sionoftheIsingproblemonmostgraphsGisNP-complete.
Theory suggests that quantum annealing may solve some optimization problems
fasterthanstate-of-the-artalgorithms[18].Quantumeffectssuchastunnelingandsu-
perpositionprovideQAwithnovelmechanismsforescapinglocalminima,therebypo-
tentiallyavoidingsuboptimalsolutionscommonlyfoundbyclassicalalgorithmsbased
onbit-flipoperations(suchasWalkSAT).PracticalQAsystemsarenotguaranteedtore-
turnoptimalsolutions;however,theD-Waveprocessorhasbeenshowntooutperforma
rangeofclassicalalgorithmsoncertainproblemsdesignedtomatchitshardwarestruc-
ture[16,21].Theseresultsalsoprovideguidanceaboutthekindsofenergylandscapes
onwhichQAisexpectedtoperformwell.
Our ultimate goal is to exploit QA as an engine for solving SAT and other NP-
hard problem instances which are relatively small but hard enough to be out of the
reachofstate-of-the-artsolvers(e.g.,SATproblemscomingfromcryptanalysis).SAT
istheproblemofdecidingthesatisfiabilityofarbitraryformulasonatomicpropositions,
typicallywritteninconjunctivenormalform.MaxSAT isanoptimizationextensionof
SAT,inwhicheachclauseisgivenapositivepenaltyiftheclauseisnotsatisfied,and
anassignmentminimizingthesumofthepenaltiesissought.
In principle, converting SAT to optimization of an Ising cost function is straight-
forward.However,practicalQAsystemssuchastheD-Wave2000Qoffersparsecon-
nectivitybetweenvariables.TheconnectivitygraphGofcurrentD-Waveprocessorsis
showninFigure1,andiscalledtheChimeragraph.Further,becausetheIsingmodelis
solvedonaphysical,analogdevice,itissubjecttoengineeringlimitations.TheD-Wave
2000Qsystemcurrentlyrequiresh ∈[−2,2]andJ ∈[−1,1]andtherearelimitson
i ij
the precision to which these parameters may be specified. Parameter imprecisions act
assmalladditivenoisesourcesonparametervalues,andarisefromoperatingquantum
mechanicalsystemsinreal-worldenvironments.Thesereal-worldpracticalitiesneces-
sitateacarefullydefinedSAT-to-Isingencoding.
ThesepracticalconstraintsgenerateachallengingproblembecausetheSATencod-
ing must be done both effectively (i.e., in a way that uses only the limited number of
qubitsandconnectionsavailablewithintheQAarchitecture,whileoptimizingperfor-
manceoftheQAalgorithm),andefficiently(i.e.,usingalimitedcomputationalbudget
forcomputingtheencoding).Inthispaper,weformalizethisproblemandprovideprac-
ticalalgorithms.
A direct formulation of the encoding problem results in a large system of linear
inequalitiesovercontinuous-andBoolean-valuedvariables.Thissystemcanbeeffec-
tivelyaddressedwithSatisfiabilityorOptimizationModuloTheory(SMT/OMT)[3,28]
solvers.SatisfiabilityModulotheTheoryofLinearRationalArithmetic(SMT(LRA))
[3]istheproblemofdecidingthesatisfiabilityofarbitraryformulasonatomicpropo-
sitions and constraints in linear arithmetic over the rationals. Optimization Modulo
theTheoryofLinearRationalArithmetic(OMT(LRA))[28]extendsSMT(LRA)by
searching solutions which optimize some LRA objective(s). Efficient OMT(LRA)
solverslikeOPTIMATHSAT[29]allowforhandlingformulaswiththousandsofBool-
eanandrationalvariables[28].
1Thetransformationbetweenz ∈{−1,1}andx ∈{0,1}isz =2x −1.
i i i i
2
Fig.1. Example of the Chimera topology: the hardware graph for system of 72 qubits in a 3-
by-3gridoftiles.(D-Wave2000Qsystemshave2048qubitsina16-by-16grid.)Thistopology
consistsofalatticeofstrongly-connectedcomponentsof8qubits,calledtiles.Eachtileconsists
of a complete bipartite graph between two sets of four qubits. One set, the “vertical” set, is
connectedtothetilesaboveandbelow;theotherset,the“horizontal”set,isconnectedtothetiles
totheleftandtotheright.Noticethateachqubitisconnectedwithatmostsixotherqubits.In
otherwords,eachvariablez intheIsingmodel(1)hasatmost6non-zeroJ interactionswith
i ij
othervariables.
This monolithic linear programming approach to encoding typically requires the
introduction of additional ancillary Boolean variables, and the resultant SMT/OMT
problemmaybecomputationallyharderthantheoriginalproblem.Incontrast,alarge
BooleanformulacanbescalablyconvertedintoanIsingmodelbydecomposingitinto
subformulae,convertingeachsubformulaintoanIsingmodel(perhapswithintroduc-
tion of additional fresh variables), and linking variables from different subformulae.
Unfortunately,inpracticethisdecomposition-basedapproachrequiresmanyauxiliary
variablesandconnections,whichareincompatiblewiththesparseconnectivityrestric-
tionsimposedbyQAarchitectures.
Tocopewiththesedifficulties,weproposeamixedapproach,whichcombines(i)
novel SMT/OMT-based techniques to produce off-line encodings of commonly-used
Booleansubfunctions,with(ii)theusageoffunctioninstantiationandplacement-and-
routingtechniquestocombineandplaceon-the-flytheencodedfunctionalitieswithin
theQAarchitecture.
3
We have implemented prototype encoders on top of the SMT/OMT tool OPTI-
MATHSAT[29].Asaproofofconcept,wepresentsomepreliminaryempiricalevalua-
tion, in which we have executed encoded SAT and MaxSAT problems on a
D-Wave2000Qsystem.Althoughpreliminary,theresultsconfirmthefeasibilityofthe
approach.Westressthefactthatthispaperisnotsupposedtopresentacomparisonwith
respecttostate-of-the-artofclassiccomputing.Rather,thisisintendedasapreliminary
assessmentofthechallengesandpotentialofQAtoimpactSATandMaxSATsolving.
The rest of the paper is organized as follows. Section 2 presents the theoretical
foundations of this work; Section 3 describes our mixed approach to cope with this
problem; Section4presentsapreliminaryempiricalevaluation;Section5hintsfuture
developments. A longer and more detailed version of this paper, including a section
thatdescribesrelatedwork,isavailableonline[7].
2 Foundations
LetF(x)beaBooleanfunctiononasetofninputBooleanvariablesx=def {x ,...,x }.
1 n
We represent Boolean value ⊥ with −1 and (cid:62) with +1, so that we can assume that
eachx ∈{−1,1}.SupposefirstthatwehaveaQAsystemwithnqubitsdefinedona
i
hardwaregraphG=(V,E),e.g.,Gcanbeanyn-vertexsubgraphoftheChimeragraph
ofFig.1.Furthermore,assumethatthestateofeachqubitz correspondstothevalue
i
ofvariablex ,i = 1,...,n = |V|.OnewaytodeterminewhetherF(x)issatisfiable
i
using the QA system is to find an energy function as in (1) whose ground states z
correspondwiththesatisfiableassignmentsxofF(x).Forinstance,ifF(x)=def x ⊕x ,
1 2
since F(x) = (cid:62) if and only if x +x = 0, the Ising model (z +z )2 in a graph
1 2 1 2
containing2qubitsjoinedbyanedgehasgroundstates(+1,−1)and(−1,+1),thatis,
thesatisfiableassignmentsofF.
BecausetheenergyE(z)in(1)isrestrictedtoquadratictermsandgraphGistyp-
ically sparse, the number of functions F(x) that can be solved with this approach is
limited. To deal with this difficulty, we can use a larger QA system with a number of
additionalqubits,sayh,representingancillaryBooleanvariables(orancillasforshort)
a=def {a ,...,a },sothat|V|=n+h.Avariableplacementisamappingofthen+h
1 h
inputandancillaryvariablesintothequbitsofV.SinceGisnotacompletegraph,dif-
ferentvariableplacementswillproduceenergyfunctionswithdifferentproperties.We
useIsingencodingtorefertotheh andJ parametersin(1)thatareprovidedtothe
i ij
QAhardwaretogetherwithavariableplacementofthevariables.ThegapofanIsing
encodingisthe energydifferencebetween groundstates(i.e.,satisfiable assignments)
andanyotherstates(i.e.,unsatisfiableassignments).Animportantobservationfrom[5]
isthatthelargerthegapthebetterthesuccessratesoftheQAprocess.Theencoding
problemforF(x)istofindanIsingencodingwithmaximumgap.
The encoding problem is typically over-constrained. In fact, the Ising model (1)
hastodiscriminatebetweenmsatisfiableassignmentsandkunsatisfiableassignments,
withm+k = 2n,whereasthenumberofdegreesoffreedomisgivenbythenumber
oftheh andJ parameters,whichintheChimeraarchitecturegrowsasO(n+h).
i ij
Inthissection,weassumethataBooleanfunctionF(x)isgivenandthathqubits
areusedforancillaryvariablesa.
4
2.1 PenaltyFunctions
Here we assume that a variable placement is given, placing x∪a into the subgraph
G. Thus, we can identify each variable z representing the binary value of the qubit
j
associatedwiththejthvertexinV witheitheranx ora variable,writingz=x∪a.
k (cid:96)
ThenwedefinepenaltyfunctionP (x,a|θ)astheIsingmodel:
F
(cid:88) (cid:88)
P (x,a|θ)=def θ + θ z + θ z z , (3)
F 0 i i ij i j
i∈V (i,j)∈E
(cid:40)
=0 ifF(x)=(cid:62)
withthepropertythat ∀x min P (x,a|θ) (4)
{a} F
≥g ifF(x)=⊥
min
where θ ∈ (−∞,+∞) (“offset”), θ ∈ [−2,2] (“biases’) and θ ∈ [−1,1] (“cou-
0 i ij
plings”),s.t.z ,z ∈ z,andg > 0(“gap”)arerational-valuedparameters.Notice
i j min
that a penalty function separates models from counter-models by an energy gap of at
leastg .WecallP (x,a|θ)anexactpenaltyfunctioniffitverifiesastrongerversion
min F
of(4)inwhichthecondition“≥ g ”issubstitutedwith“= g ”.Tosimplifythe
min min
notationwewillassumethatθ =0when(i,j)(cid:54)∈E,anduseP (x|θ)whena=∅.
ij F
TheQAhardwareisusedtominimizetheIsingmodeldefinedbypenaltyfunction
P (x,a|θ). By (4), a returned value of P (x,a|θ) = 0 implies that F is satisfiable.
F F
However,ifP (x,a|θ) ≥ g ,sinceQAdoesnotguaranteeoptimality,thereisstill
F min
achancethatF issatisfiable.Nevertheless,thelargerg is,thelesslikelythisfalse
min
negativecaseoccurs.
Thefollowingexamplesshowthatancillaryvariablesareneeded,evenwhenGisa
completegraph.
Example1. The equivalence between two variables, F(x) =def (x ↔ x ), can be en-
1 2
codedwithoutancillasbymeansofasinglecouplingbetweentwoconnectedvertices,
with zero biases: P (x|θ) =def 1−x x , so that g = 2. In fact, P (x|θ) = 0 if
F 1 2 min F
x ,x have the same value; P (x|θ) = 2 otherwise. Notice that P (x|θ) is also an
1 2 F F
exactpenaltyfunction.PenaltyP (x|θ)iscalledachainoflength2.
F
Example2. ConsidertheANDfunctionF(x)=def x ↔(x ∧x ).Ifx , x , x could
3 1 2 1 2 3
beallconnectedina3-clique,thenF(x)couldbeencodedwithoutancillasbysetting
P (x|θ) = 3 − 1x − 1x +x + 1x x −x x −x x ,sothatg = 2. Since
F 2 2 1 2 2 3 2 1 2 1 3 2 3 min
theChimeragraphhasnocliques,sothattheaboveANDfunctionneeds(atleast)one
ancilla a to be encoded as: P (x,a|θ) = 5 − 1x − 1x +x + 1x x −x x −
F 2 2 1 2 2 3 2 1 2 1 3
x a−x a,whichstillhasgapg =2andisembeddedasinFigure 2(a).
2 3 min
Example3. Consider the XOR function F(x) =def x ↔ (x ⊕ x ). Even within a
3 1 2
3-clique, F(x) has no ancilla-free encoding. Within the Chimera graph, F(x) can be
encoded with three ancillas a ,a ,a as: P (x,a|θ) = 5+x +a −a +x a −
1 2 3 F 3 2 3 1 1
x a −x a −x a −x a −x a +x a −x a ,whichhasgapg = 2andis
1 2 1 3 2 1 2 2 2 3 3 2 3 3 min
embeddedasinFigure 2(b).
5
(a) x ↔(x ∧x ) (b) x ↔(x ⊕x ) (c) x ↔(x ∧(x ⊕x ))
3 1 2 3 1 2 4 3 1 2
withoneancilla. withthreeancillas. obtainedbycombining2(b)and2(a).
Fig.2.MappingswithintheChimeragraph,penaltyfunctionsuseonlycolorededges.2(c)com-
bines2(a)and2(b)usingchainedproxyvariablesy,y(cid:48).Theresultingpenaltyfunctionisobtained
byrewritingx ↔(x ∧(x ⊕x ))intoitsequi-satisfiableformula(x ↔(x ∧y(cid:48)))∧(y(cid:48) ↔
4 3 1 2 4 3
y)∧(y↔(x ⊕x )).
1 2
2.2 PropertiesofPenaltyFunctionsandProblemDecomposition
Afterdeterminingavariableplacement,findingthevaluesfortheθsimplicitlyrequires
solvingasetofequationswhosesizegrowswiththenumberofmodelsofF(x)plusa
numberofinequalitieswhosesizegrowswiththenumberofcounter-modelsofF(x).
Thus, the θs must satisfy a number of linear constraints that grows exponentially in
n. Since the θs grow approximately as 4(n + h), the number of ancillary variables
neededtosatisfy(4)canalsogrowveryrapidly.Thisseriouslylimitsthescalabilityof
asolutionmethodbasedon(3)-(4).Weaddressthisissuebyshowinghowtoconstruct
penalty functions by combining smaller penalty functions, albeit at the expense of a
reducedgap.
Thefollowingtwopropertiescanbeeasilyderivedfromthedefinition.
Property1. LetF∗(x)=def F(x ,...,x ,¬x ,x ,...,x )forsomeindexr.Assume
1 r−1 r r+1 n
avariableplacementofxintoV s.t.P (x,a|θ)isapenaltyfunctionforF(x)ofgap
F
g . Then P (x,a|θ) = P (x,a|θ∗), where θ∗ is defined as follows for every
min F∗ F
z ,z ∈x,a:
i j
(cid:26) (cid:26)
−θ ifz =x −θ ifz =x orz =x
θ∗ = i i r θ∗ = ij i r j r
i θ otherwise; ij θ otherwise.
i ij
Noticethatsincethepreviouslydefinedboundsoverθ(namelyθ ∈ [−2,2]andθ ∈
i ij
[−1,1])aresymmetric,ifθisinrangethenθ∗isaswell.
Two Boolean functions that become equivalent by permuting or negating some of
their variables are called NPN-equivalent [14]. Thus, given the penalty function for a
Booleanformula,anyotherNPNequivalentformulacanbeencodedtriviallybyapply-
ingProperty1.NoticethatcheckingNPNequivalenceisahardproblemintheory,but
itisfastinpracticeforsmalln(i.e.,lessthan16[20]).
Property2. Let F(x) = (cid:86)K F (xk) be Boolean formula such that x = ∪ xk, the
k=1 k k
xksmaybenon-disjoint,andeachsub-formulaF hasapenaltyfunctionP (xk,ak|θk)
k Fk
6
withminimumgapgk wherea = ∪ ak andtheaksarealldisjoint.Givenalistw
min k k
ofpositiverationalvaluessuchthat,foreveryz ,z ∈x∪(cid:83)K ak:
i j k=1
K K
(cid:88) (cid:88)
θ =def w θk ∈[−2,2], θ =def w θk ∈[−1,1], (5)
i k i ij k ij
k=1 k=1
thenapenaltyfunctionforF(x)canbeobtainedas:
K
(cid:88)
P (x,a1...aK|θ)= w P (xk,ak|θk). (6)
F k Fk
k=1
Thisnewpenaltyfunctionevaluatestozeroifandonlyifallitssummandsdo,andoth-
erwiseitisatleastg =minK w gk .Thus,ingeneral,the(weighted)sumofthe
min k=1 k min
penaltyfunctionsofasetofformulasrepresentsapenaltyfunctionfortheconjunction
oftheformulas.
A formula F(x) can be decomposed (e.g., by a Tseitin transformation) into an
equivalently-satisfiableoneF∗(x,y):
m−1
(cid:94)
F∗(x,y)=def (y ↔F (xi,yi))∧F (xm,ym), (7)
i i m
i=1
wheretheF sareBooleanfunctionswhichdecomposetheoriginalformulaF(x),and
i
they sarefreshBooleanvariableseachlabelingthecorrespondingF .ByProperty2,
i i
this allows us to decompose F(x) into multiple F (xi,yi) that can be encoded sepa-
i
ratelyandrecombined.TheproblemistochooseBooleanfunctionsF (xi,yi)whose
i
penaltyfunctionsareeasytocompute,havealargeenoughgap,andwhosecombination
keepsthegapofthepenaltyfunctionfortheoriginalfunctionaslargeaspossible.
Summingpenaltyfunctionswithsharedvariablesmaycauseproblemswithparam-
eterranges:penaltyfunctionsthatsharetermsmaysumupbiasesorcouplingsresulting
inout-of-rangevalues.Usingweights,Property2canhelptomitigatethis,butalsoitis
likelythattheg ofthefinalpenaltyfunctionbecomessmall.
min
Wecancopewiththisproblembymappingsharedvariablesintodistinctqubitsand
thenlinkingthemtogether.ConsideragainF(x)=(cid:86)K F (xk)asinProperty2.We
k=1 k
rewriteitintoitsequi-satisfiableformula
K
F∗(x∗)=def (cid:94) F (xk∗)∧ (cid:94) (x k∗ ↔x k(cid:48)∗) (8)
k i i
k=1 xi∈xk∩xk(cid:48)
k,k(cid:48)∈[1..K],k<k(cid:48)
wherex∗ =∪ xk∗andthexk∗arealldisjoint.Also,asinProperty2,assumewehave
k
P (xk∗,ak|θk) for each k with disjoint ak. If there is an edge between every two
Fk
copies of the same variable x , we can write a penalty function in the following way
i
(usingthepenaltyofExample1):
K
P (x∗,a|θ)=(cid:88)P (xk∗,ak|θk)+ (cid:88) (1−x k∗x k(cid:48)∗), (9)
F∗ Fk i i
k=1 xi∈xk∩xk(cid:48)
k,k(cid:48)∈[1..K],k<k
7
andtheθsstaywithinvalidrangebecausethexk∗sarealldisjoint.Thus,wecanrepre-
sentasinglevariablex withaseriesofqubitsconnectedbystrongcouplings(1−z z(cid:48)).
i i i
Figure 2(c) illustrates a simple example. Two observations are at hand. First, the gap
g ofP (x∗,a|θ)isatleastmin(minK w gk ,2),sinceeach(1−z z(cid:48))penalty
min F∗ k=1 k min i i
has agap of 2. Second,not allcopies of x need tobe directlyadjacent to obtainthis
i
bound:itsufficestousetheedgesofatreeconnectingallcopies.Moregenerally,that
treemaycontainadditionalqubitstofacilitateconnectedness.Atreeconnectingallthe
copiesofavariablex iscalledachainandisthesubjectofthenextsection.
i
2.3 EmbeddingintoChimeraArchitecture
Theprocessofrepresentingasinglevariablex byacollectionofqubitsconnectedin
i
chainsofstrongcouplingsisknownasembedding,inreferencetotheminorembedding
problem of graph theory [12,13]. More precisely, suppose we have a penalty function
basedongraphG(sox andx areadjacentiffθ (cid:54)=0)andaQAhardwaregraphH.
i j ij
AminorembeddingofGinH isafunctionΦ:VG →2VH suchthat:
– foreachG-vertexx ,thesubgraphinducedbyΦ(x )isconnected;
i i
– foralldistinctG-verticesx andx ,Φ(x )andΦ(x )aredisjoint;
i j i j
– foreachedge(x ,x )inG,thereisatleastoneedgebetweenΦ(x )andΦ(x ).
i j i j
TheimageΦ(x )ofaG-vertexisachain,andthesetofqubitsinachainareconstrained
i
tobeequalusing(1−z z(cid:48))couplingsasinFigure2(c).
i i
Embeddinggenericgraphsisacomputationallydifficultproblem[2],althoughcer-
tainstructuredproblemgraphsmaybeeasilyembeddedintheChimeratopology[8,34]
and heuristic algorithms may also be used [9]. A reasonable goal in embedding is to
minimizethesizesofthechains,asquantumannealingbecomeslesseffectiveasmore
qubitsareincludedinchains[22].
A different approach to use QA for finding models for F, global embedding, is
based on first finding a penalty function on a complete graph G on n+h variables,
and secondly, embedding G into a hardware graph H using chains (e.g., using [8]).
Following[5],globalembeddingsusuallyneedfewerqubitsthanthemethodspresented
in this paper; however, the final gap of the penalty function obtained in this way is
generallysmalleranddifficulttocomputeexactly.
3 SolvingtheEncodingProblem
3.1 EncodingSmallBooleanFunctions
Computing Penalty Functions via SMT/OMT(LRA). Given x =def {x ,...,x },
1 n
a =def {a ,...,a },F(x)asinSection2.1,avariableplacementinaChimerasubgraph
1 h
s.t. z = x∪a, and some gap g > 0, the problem of finding a penalty function
min
8
P (x,a|θ)asin(3)reducestosolvingthefollowingSMT(LRA)problem:
F
(cid:94) (cid:94)
Φ(θ)=def (−2≤θ )∧(θ ≤2)∧ (−1≤θ )∧(θ ≤1) (10)
i i ij ij
zi∈x,a zi,zj∈x,a
i<j
(cid:94) (cid:95)
∧ (P (x,a|θ)=0) (11)
F
{x∈{−1,1}n|F(x)=(cid:62)} a∈{−1,1}h
(cid:94) (cid:94)
∧ (P (x,a|θ)≥0) (12)
F
{x∈{−1,1}n|F(x)=(cid:62)} a∈{−1,1}h
(cid:94) (cid:94)
∧ (P (x,a|θ)≥g ). (13)
F min
{x∈{−1,1}n|F(x)=⊥} a∈{−1,1}h
Consequently,theproblemoffindingthepenaltyfunctionP (x,a|θ)thatmaximizes
F
thegapg reducestosolvingtheOMT(LRA)maximizationproblem(cid:104)Φ(θ),g (cid:105).
min min
Intuitively:(10)statestherangesoftheθ;(11)and(12)statethat,foreveryxsatis-
fyingF(x),P (x,a|θ)mustbezeroforatleastone“minimum”aandnonnegativefor
F
alltheothers;(13)statesthatforeveryxnotsatisfyingF(x),P (x,a|θ)mustgreater
F
orequalthanthegap.Consequently,ifthevaluesoftheθ inP (x,a|θ)satisfyΦ(θ),
F
thenP (x,a|θ)complieswith(4).
F
NoticethatΦ(θ)growsexponentiallywith|x|+|a|,andnolongercontainsBoolean
atoms.Noticealsothat,ifa=∅,theOMT(LRA)maximizationproblem(cid:104)Φ(θ),g (cid:105)
min
reducestoalinearprogrambecausethedisjunctionsin(11)disappear.
ToforceP (x,a|θ)tobeanexactpenaltyfunction,weconjointoΦ(θ)thefollow-
F
ing:
(cid:94) (cid:95)
...∧ (P (x,a|θ)=g ). (14)
F min
{x∈{−1,1}n|F(x)=⊥} a∈{−1,1}h
Here,(14)forcesP (x,a|θ)tobeexactlyequaltothegapforatleastone“minimum”
F
a.Exactpenaltyfunctionscanbeusedtoencode(weighted)MaxSATinstances.Sup-
posewepartitionaformulaintotheconjunctionofitssoftconstraintsC eachofweight
i
w ≥0.ThenforeachC wefindanexactpenaltyfunctionof(10)-(14)forC imposing
i i i
agapg proportionaltow ,andwecombinetheresultasinProperty2.
i i
ImprovingEfficiencyandScalabilityusingVariableElimination Asbefore,assume
that the variable placement is fixed and consider the SMT/OMT(LRA) formulation
(10)-(13).Noticetheexponentialdependencyonthenumberofhiddenvariablesh.For
practical purposes, this typically implies a limit on h of about 10. Here, we describe
an alternative formulation whose size dependence on h is O(h2tw), where tw is the
treewidth of the subgraph of G spanned by the qubits corresponding to the ancillary
variables,G .FortheChimeragraph,evenwhenhisaslargeas32,twisatmost8and
a
thereforestilloftractablesize.
Thecruxofthereformulationisbasedontheuseofthevariableeliminationtech-
nique [15] to solve an Ising problem on G . This method is a form of dynamic pro-
a
gramming,storingtablesinmemorydescribingallpossibleoutcomestotheproblem.
9
Whenthetreewidthistw,thereisavariableeliminationorderguaranteeingthateachta-
blecontainsatmostO(2tw)entries.Ratherthanusingnumericaltables,ourformulation
replaceseachofitsentrieswithacontinuousvariableconstrainedbylinearinequalities.
Inprinciple,weneedtoparametricallysolveanIsingproblemforeachx ∈ {−1,1}n,
generating O(2nh2tw) continuous variables. However, by the sequential nature of the
variableeliminationprocess,manyofthesecontinuousvariablesareequal,leadingtoa
reduced(asmuchasanorderofmagnitudesmaller)andstrengthenedSMTformulation.
See[5]formoredetails.
PlacingVariables&ComputingPenaltyFunctionsviaSMT/OMT(LRIA∪UF).
TheformulaΦ(θ)in(10)-(14)canbebuiltonlyafteravariableplacement,sothateach
variable z ∈ x ∪ a has been previously placed in some vertex v ∈ V. There are
j j
manysuchplacements.Forexample,ifn+h = 8andwewanttoencodethepenalty
function into a 8-qubit Chimera tile, then we have 8! = 40320 candidate placements.
Exploiting symmetry and the automorphism group of G, one can show that most of
theseplacementsareequivalent.
Alternatively,wecancombinethegenerationofthepenaltyfunctionwithanauto-
maticvariableplacementbymeansofSMT/OMT(LRIA∪UF),LRIA∪UF being
thecombinedtheoriesoflineararithmeticoverrationalsandintegersplusuninterpreted
functionsymbols.Thisworksasfollows.
Supposewewanttoproducethepenaltyfunctionofsomerelativelysmallfunction
(e.g.,son+h≤8,whichfitsintoasingleChimeratile).Weindexthen+hvertices
inthesetV intowhichwewanttoplacethevariablesasV =def {1,...,n+h},andwe
introduceasetofn+hintegervariablesv =def {v ,...,v }s.t.eachv ∈ V repre-
1 n+h j
sents(theindexof)thevertexintowhichz isplaced.(Forexample,“v = 5”means
j 3
that variable z is placed in vertex #5.) Then we add the standard SMT constraint
3
Distinct(v ,...,v ) to the formula to guarantee the injectivity of the map. Then,
1 n+h
instead of using variables θ and θ for biases and couplings, we introduce the un-
i ij
interpreted function symbols b : V (cid:55)−→ Q (“bias”) and c : V × V (cid:55)−→ Q (“cou-
pling”),sothatwecanrewriteeachbiasθ asb(v )andeachcouplingθ asc(v ,v )
j j ij i j
s.tv ,v ∈[1,..,n+h]andDistinct(v ,...,v ).
i j 1 n+h
ThisrewritestheSMT(LRA)problem(10)-(13)intotheSMT/OMT(LRIA∪
UF)problem(15)-(26).Equation(19)mustbeusediffweneedanexactpenaltyfunc-
tion. (Notice that (22) is necessary because we could have c(v ,v ) s.t. v > v .) By
i j i j
solving(cid:104)Φ(θ ,b,c,v),g (cid:105)wenotonlyfindthebestvaluesofthebiasesbandcou-
0 min
plersc,butalsothebestplacementvofthevariablesinto(theindexesof)thequbits.
3.2 EncodingLargerBooleanFunctions
As pointed out in Section 2.2, encoding large Boolean functions using the SMT for-
mulationsoftheprevioussectioniscomputationallyintractable,soothermethodsmust
beused.One sensibleapproachistopre-computea libraryofencodedBooleanfunc-
tions and decompose a larger Boolean function F(x) into a set of pre-encoded ones
(cid:86)K F (xk). The penalty models P (xk,ak|θk) for these pre-encoded functions
k=1 k Fk
maythenbecombinedusingchainsasdescribedinSection2.3.Thisschemaisshown
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