Linear-algebraic lambda-calculus

0.16 MB·

by  Pablo Arrighi

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Linear-algebraic -calculus
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Pablo Arrighi∗ GillesDowek†
a
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Abstract
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2 Witha view towards models of quantum computation and/or the inter-
pretationoflinearlogic,wedefineafunctionallanguagewhereallfunctions

1 arelinearoperatorsbyconstruction. Asmallstepoperationalsemantic(and
v henceaninterpreter/simulator)isprovidedforthislanguageintheformofa
0 termrewritesystem. Thelinear-algebraicλ-calculusherebyconstructedis
5 linearinadifferent(yetrelated)sensetothat,say,ofthelinearλ-calculus.
1 These various notions of linearity are discussed in the context of quantum
1
programminglanguages.
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0 1 Introduction
/
h
p
Quantumcomputationlacksa convenientmodelofcomputation. To thisday its
-
t algorithmsare expressed in terms of quantumcircuits, but their descriptionsal-
n
waysseemastonishinglyremotefromthetasktheydoaccomplish[12].Moreover
a
u universality is only provided via the notion of uniform family of circuits [28].
q QuantumTuring machinessolve this latter point, yet they are even less suitable
: as a programming language [6]. Another approach is to enclose quantum cir-
v
i cuitswithin a classicalimperative-stylecontrolstructure[20] — butwe wish to
X
avoid this duality, in an attempt to bring programscloser to their specifications.
r Functional-style controlstructure, on the other hand, seem to mergewith quan-
a
tum evolution descriptions in a unifying manner. With a view towards models
ofquantumcomputation,wedescribeafunctionallanguageforexpressinglinear
operators,andlinearoperatorsonly.
Wearecareful,however,nottoburyourpresentationofthislanguageoflinear
operatorswithintoomanyquantumcomputation-specificconsiderations.Theaim
istoreachanaudienceoflogiciansalso,aswesuspectastrongconnectionwith
issuesofcomputationalinterpretationsoflinearlogic.
∗InstitutGaspardMonge, 5BdDescartes, Champs-sur-Marne, 77574Marne-la-Valle´e Cedex2,
France,[email protected].
†E´cole polytechnique and INRIA, LIX, E´cole polytechnique, 91128 Palaiseau Cedex, France,
[email protected].
Weprovideasemanticforthelanguageintheformofatermrewritesystem
[11]. These consist in a finite set of rules l r, each interpreted as follows:
−→
“Any term t containing a subterm σl in position p (i.e. t = t[σl] ) should be
p
rewritten into a term t containingσr in positionp, with all the rest unchanged
′
(i.e. t = t[σr] )”. Here σ denotes a variable substitution. The minimalist in-
′ p
terpretationoftherulesmakestermrewritesystems(TRS)extremelysuitablefor
describing the behavior of a computer languages unambiguously — so long as
theorderinwhichthereductionsoccurdoesnotmattertotheendresult(aprop-
ertynamedconfluence). Moreover,becausel r maybeseen asanoriented
−→
version of equation l = r, the TRS provides both an operational semantic (an
interpreter/simulatorforthelanguage)andanaxiomaticsemantic(anequational
theoryinwhichtoprovepropertiesaboutthelanguage).
We begin with a simple language for vectors containing constants for base
vectors, addition and product by a scalar. On terms of this language we define
a rewrite system reducing any term expressing a vector to a linear combination
of base vectors [4]. We have also proposed in [4] an extension to a language
containingatensorialproductoperation(Section2).
Suchalanguagemustrelyonalanguageandrewritesystemforscalars. This
raisestheproblemoftheconditionalrewritingrequiredfordivision,whichwecan
circumvent,basing quantumcomputationuponthe ring of diadicfloats together
with 1 and imaginarynumberi (Section 3). Moregenerally, it should be said
√2
thatalanguageoflinearoperatorsdoesnotneeddivision.
ModerndaysfunctionallanguagessuchasCaml,Haskelletc. arebasedupon
twobasicevaluationmechanisms: matching,whichprovidesconditionalbranch-
ingbyinspectionofvalues;andsomeavataroftheλ-calculus. Thefirstmecha-
nism is obtainedas we extendthe term rewritesystem to handlelinear maps—
themselvesdenotedassuperpositionsofbipartitestates,e.g.
(true(cid:3)false+false(cid:3)true) false true.
∗
∗ −→
Applications are therefore analogous to contractions in tensorial calculus: this
approachoffersanelegantparadigmtorepresentquantumoperationsasquantum
states(Section4).
Thesecondmechanismisobtainedthroughanimplementationofλ-termsviade
Bruijn indices, a scheme whereby variablesare encodedas integers referringto
theirbinders,e.g.
λx.(λy.(x y))isencodedasL(L(var(1) var(0))).
⊗ ⊗
2
The question of the interpretation of terms such as λx.(x x) is lengthily ad-
⊗
dressedaswedrawadistinctionbetweencloningandcopying. Thesemanticof
ourcalculusforbidsonlytheformer,non-linearoperation,byenforcingahigher
priorityof the addition’sdistributivity oversubstitution (Section 5). Thisis fol-
lowedbyashortexampletakenfromourimplementation(Section6).
Erasure on the other hand remains allowed in our calculus, because we do
not restrict ourselves to unitary operations. Whilst we discuss possible well-
formedness conditions to implement this restriction (a crucial one for quantum
computation), the claim here is to have provided a “linear” λ-calculus, in the
sense of linear algebra. We discuss the various notions of “linearity” used in
quantum programminglanguages, such as the one by Van Tonder [23] (Section
7).
2 Vectorial spaces
We seektorepresentquantumprograms,theirinputvectors,theiroutputvectors
and their applications as terms of a first-order language. Moreover we seek to
providerulessuchthatthetermformedbytheapplicationofaquantumprogram
onto its input vector should reduce to its output vector. Several terms may be
usedtoexpressoneoutputvector,asaconsequencewemustensurethattheseall
reducetooneunique,normalform,uponwhichthereisnothingmoretocompute.
Themostnaturalnormalformtoaimforisthatofalinearcombinationofthebase
vectors,i.e. thecomputationfinishesoncewehavethecoordinatesoftheoutput
vector.
We start with the languageof vectorialspaces, i.e. a two-sortedlanguageL
havingsortK forscalarsandsortE forvectors—togetherwith: twoconstants
0 and1ofsortK; a constant0 ofsortE; two binarysymbols+ and ofrank
×
K,K,K ; a binary symbol + (also) of rank E,E,E ; and a binary symbol .
h i h i
ofrank K,E,E . In[4]wedescribedatermrewritesystemreducinganyterm
h i
expressing a vector into a linear combination of base vectors. The term rewrite
systemdevelops
4.(false+true) 4.false+4.true
−→
butfactorizes
4.false+6.false (4+6).false.
−→
accordingtotherulesinfigure1. SuchaTRSarisesasweorientsixoftheeight
equationsaxiomatizingvectorialspaces. Onlythosetwoaxiomscorrespondingto
associativityandcommutativityofvectoradditionareleftaside,becauseweuse
3
rewritingmoduloAC(+). Moreoverwe needto addthreemorerulesforconflu-
ence.
Figure1: VECTORIALSPACES
λ.(u+v) λ.u+λ.v
−→
λ.u+µ.u (λ+µ).u
−→
λ.(µ.u) (λ µ).u
−→ ×
u+0 u
−→
1.u u
−→
0.u 0
−→
λ.u+u (λ+1).u
−→
u+u (1+1).u
−→
λ.0 0
−→
with+anACsymbol.
Buttheserewriterulesdonottakeintoaccountcomputationonscalars. The
lattermustbeaddedbymixinginanotherrewritesystemS,rewritingscalartoa
normalform.
Definition1 (Scalarrewrite system) A scalar rewrite system is a rewrite system
onalanguagecontainingatleastthesymbols+, ,0and1,suchthat:
×
S isterminatingandgroundconfluent,
•
forallclosedtermsλ,µandν,thepairofterms
•
– 0+λandλ,
– 0 λand0,
×
– 1 λandλ,
×
– λ (µ+ν)and(λ µ)+(λ ν),
× × ×
– (λ+µ)+ν andλ+(µ+ν),
– λ+µandµ+λ,
– (λ µ) ν andλ (µ ν),
× × × ×
– λ µandµ λ
× ×
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havethesamenormalforms,
0and1arenormalterms.
•
Thefollowingpropositionscanbefoundin[4].
Proposition1 Foranyscalarrewrite systemS,therewrite systemR S ister-
∪
minatingandgroundconfluent.
Proposition2 Iftisanormalclosetermwhoseconstantsareamongstx1,...,xn.
Thetermtis0orithastheformλ1xi1+...λkxik+xik+1+xik+l wheretheindices
i1,...,ik+l aredistinctandtheλk’sareneither0nor1.
NotethatthealgorithmdefinedbyRisrelativelycommonincomputing,for
presentinganyvectorasalinearcombinationofbasevectors. Butitdoesinfact
definevectorialspaces,asanymathematicalstructurevalidatingthealgorithm.In
thissensewehaveprovidedacomputationaldefinitionofvectorialspaces.
Furthermorenotethatthesupportfortensorproductsiseasilyaddedintothe
TRS,throughthesixrulesgiveninfigure2. Proposition1remainstruewhenRis
extendedwiththosesixadditionalrules,whilstproposition2nowyieldsnormal
formsfortermsinE E oftheform0or
⊗
λ1xi1 ⊗yj1 +...+λkxik ⊗yjk +xik+1 ⊗yjk+1 +...+xik+l ⊗yjk+l,
where the pairs of indices i1,j1 ,..., ik+l,jk+l are distinct and the λk’s are
h i h i
neither0nor1[4].
Figure2: VECTORIALSPACES:TENSORS
(u+v) w u w+v w
⊗ −→ ⊗ ⊗
(λ.u) v λ.(u v)
⊗ −→ ⊗
u (v+w) u v+u w
⊗ −→ ⊗ ⊗
u (λ.v) λ.(u v)
⊗ −→ ⊗
0 u 0
⊗ −→
u 0 0
⊗ −→
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3 The field of quantum computing
Fields are not easily implemented as term rewrite systems, because of the con-
ditionalrewritingrequiredforthedivisionbyzero. Inthe previoussectionsuch
problems were avoided by simply assuming a TRS for scalars having a certain
numberofproperties,butiftheobjectiveistolaythegroundforformalquantum
programminglanguages,thenwemustprovidesucha TRS. Thepresentsection
brieflyoutlineshowthisisachieved.
3.1 Background
Weseektomodelquantumcomputationasaformalrewritesystemuponafinite
set of symbols. Since the complexnumbersare uncountable,we musttherefore
depart from using the whole of C as the field K of our vectorial space. Such
considerations are commonplace in computation theory, and were successfully
addressedwith theprovisionofthefirstrigorousdefinitionofaquantumTuring
machine[6]. In shortthequantumTuringmachinesarebroughtas anextension
ofprobabilisticTuringmachines
Q:headstates, Σ:alphabet,
h
δ:transitionfunction, q ,q :start,endstate
o f
i
whose transition functions are no longer valued over the efficiently computable
positivereals(probabilities)
δ :Q Σ (Q Σ Left, Right R˜+)
× −→ × ×{ }→
butovertheefficientlycomputablecomplexnumbers(amplitudes)
δ :Q Σ (Q Σ Left, Right C˜).
× −→ × ×{ }→
Inbothcasesδisconstrainedtobeaunitfunction(probabilities/squaredmodulus
summingtoone),andforthequantumTuringmachineδ isadditionallyrequired
toinduceaunitaryglobalevolution.Awell-knownresultofcomplexitytheoryis
thatprobabilisticTuringmachinesremain aspowerfulwhenthe transition func-
tion δ is further restricted to take values in the set 0,1,1 . The result in [6]
{ 2 }
is analogous: quantumTuringmachinesremainas powerfulwhenthe transition
functionδisfurtherrestrictedtotakevaluesintheset 1, 1 ,0, 1 ,1 .Later
{− −√2 √2 }
it was shown in [2], and independentlyin [22] that no irrationalnumberis nec-
essary,i.e. δ mayberestrictedtotakevaluesintheset 1, 8, 3,0,3,8,1
{− −5 −5 5 5 }
withoutlossofpowerforthequantumTuringmachine.
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Inthecircuitmodelofquantumcomputationtheemphasiswasplacedonthe
abilityto approximateanyunitarytransformfroma finite setofgates. Thisline
ofresearch(cf. [21][15]tocitea few)hassofarculminatedwith[7],wherethe
followingset
1 0 0 0
0 1 0 0
CNOT =  (1)
0 0 0 1
 0 0 1 0 
 
 1 1 
1 0
H = √2 √2 P =
1 1 0 eiπ/4
√2 −√2 ! (cid:18) (cid:19)
wasproventobeuniversalintheabovestrictsense. Aweakerrequirementfora
setofgatesistheabilitytosimulateanyunitarytransform,anotionwhichisalso
referred to as encoded universality — since a computationon n qubits may for
instance berepresentedasa computationon n+1 “realbits”, througha simple
mapping.Arecentpapershowsthatthegate
1 0 0 0
0 1 0 0
G= ,
0 0 a b
−
 0 0 b a 
 
 
with either a = b = 1 , or a = 3 and b = 8, has this property [19]. Do
√2 5 5
appreciatehowtheresultfallsintolinewiththoseregardingthequantumTuring
machine.
Definition2 Wecallcomputationalscalars,anddenoteK˜ theringformedbythe
additiveandmultiplicativeclosureofthecomplexnumbers 1,1, 1 ,i .
{− √2 }
Oncewehaveshownthatthecomputationalscalarsarithmeticscanbeperformed
byaTRS,itwillbesufficienttoexpressthebasicgates(1)inourformalismtoim-
mediatelyobtainthemoretraditionalnotionofquantumcomputationuniversality.
Henceourchoice.
3.2 Rules
We beginbyimplementingnaturalnumbersandunsignedbinarynumbers. That
suchTRScanbemadegroundconfluentandterminatingarenowwell-established
results[9][27]. Thisplacesusinapositiontobuildupdiadicfloatsoutofasign,
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Figure3: DIADICFLOATS
fl(s,n::0,S(p)) fl(s,n,p)
−→
fl(neg,0,p) fl(pos,0,p)
−→
fl(s,0,S(p)) fl(s,0,zeron)
−→
.
.
.
fl(pos,m1,e1)timesffl(neg,m2,e2) fl(neg,m1timesbm2,addn(e1,e2))
−→
fl(neg,m1,e1)timesffl(pos,m2,e2) fl(neg,m1timesbm2,addn(e1,e2))
−→
.
.
.
anunsignedbinarynumberandanexponent,e.g. fl(neg,1,S(zeron))istostand
for 1,asexemplifiedinfigure3.
−2
ReachedthispointitsufficestonoticethatK˜,i.e. diadicfloatstogetherwith
imaginary number i and real number 1 , can be viewed as a four-dimensional
√2
module upon diadic floats. Indeed any such number could be represented as a
linearcombinationoftheform:
1 i
α.1+β. +γ.i+δ. .
√2 √2
As a consequence we can reuse the results of section 2 to implement computa-
tionalscalarsandtheiradditions.Computationalscalarsmultiplicationthenneeds
tobedefined,wedosomoduloACinfigure4. Noticethatweoverloadthesym-
bol for multiplicationof diadic floats and for multiplicationof computational
×
scalars.
WeconjecturethatthisTRSisgroundconfluentandterminating,buthavenot
yetaformalproofforthisassertion.
Notice we haveneverdefined a division operation. This is because only the
ringpropertiesofthesenumbersarerequiredforexpressinglinearoperations:we
placeourselvesupona“module”ratherthatafullvectorialspace.
4 Matching construct
Wenowturntothedefinitionofthematchingconstructsinourlanguage. Aswe
shallsee,theseconstructsarenothingelsethanareformulationoftherulesforthe
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Figure4: SCALARMULTIPLICATION
1 v v
× −→
1 1
fl(pos,1,S(zeron)).1
√2 × √2 −→
1 i
i
√2 × −→ √2
1 i
fl(pos,1,S(zeron)).i
√2 × √2 −→
i i fl(neg,1,zeron).1
× −→
i 1
i fl(neg,1,zeron).
× √2 −→ √2
i i ..
fl(neg,1,S(zeron)).1.
√2 × √2 −→
(λ.u) v λ.(u v)
× −→ ×
(t+u) v t v+u v
× −→ × ×
with anACsymbol.
×
tensorproduct.
4.1 Notations
Your typical functional language (Haskell, ML...) will always have “match-
ing”constructs(forbranching).Forinstance,hereisapieceofCaml:
let rec not b = match b with
| false -> true
| true -> false ;;
We wish to provide such constructs in our linear-algebraic calculus. Strangely
enough these matching constructs are very close to the tensorial product con-
structs.
Mathematicians and physicist in this field would write linear maps instead:
NOT = true false + false true. Howeverherethe false and true maybe
| ih | | ih | h | h |
viewed as patterns, waiting to be compared to the input vector through a scalar
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product.Thuswechoosetoreconcilebothworldsandwrite:
NOT=false(cid:3)true+true(cid:3)false.
Anexpression(t(cid:3)u)appliedtoavectorvwillthenreduceinto(t v).u,with
thescalarproduct. Inthissense(t(cid:3)u) vdoesreturnuinso far•astoverlap•s
∗
withv. Moreformaljustifications,andformalrewriterulesfollowinthenexttwo
subsections. For now we give the reduction steps involvedin the application of
thephasegateP uponthevectortrue,asamotivatingexamplefortheserules:
1 1
(false(cid:3)false)+true(cid:3)( +i ).true true
√2 √2 ∗
(cid:18) (cid:19)
1 1
(false(cid:3)false) true+ true(cid:3)( +i ).true true
∗
−→ ∗ √2 √2 ∗
(cid:0) 1 1 (cid:1)
∗ (false true).false+(true true). ( +i ).true
−→ • • √2 √2
1 1 (cid:0) (cid:1)
∗ 0.false+( +i ).true
−→ √2 √2
1 1
∗ ( +i ).true.
−→ √2 √2
Allofthethreegatesformingauniversalsetforquantumcomputationaretrivially
expressedastermsinthisnotation:
CNOT =(false false)(cid:3)(false false)
⊗ ⊗
+(false true)(cid:3)(false true)
⊗ ⊗
+(true false)(cid:3)(true true)
⊗ ⊗
+(true true)(cid:3)(true false)
⊗ ⊗
1 1
H = false(cid:3) .(false+true) + true(cid:3) .(false true)
√2 √2 −
(cid:18) (cid:19) (cid:18) (cid:19)
1 1
P =(false(cid:3)false)+ true(cid:3)( +i ).true .
√2 √2
(cid:18) (cid:19)
4.2 Rules
Since (cid:3) is just another type of tensor product, bilinearity applies (see figure 5.
Noticetheconjugationoftheλscalar,denotedλ,easilyimplementedintheTRS).
Otherthanitsleft-hand-sideantilinearity,theparticularityof(cid:3)isthereductionit
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