Quantum entanglement analysis based on abstract interpretation

Quan tum En tanglemen t Analysis based on Abstract In terpretati on Simon Perdrix Oxford Univ ersity Computing Laboratory simon.perd rix@comlab.ox.ac .uk Abstract. Entangl ement is a non lo cal prop ert y of q uantum states whic h has no class ical counterpart and pla ys a decisiv e role in quantum information theory . S everal protocols, like the telep ortation, are based on quantum entangled states. Moreo ver, any quantum algorithm which does not create entanglemen t can b e efficiently sim ulated on a classical computer. The exact role of the e ntanglemen t is nevertheless not w ell un- derstoo d. S ince an exact analysis of entanglemen t evolution induces an exp onential slo wdown, we consider appro ximative analysis based on the framew ork of abstract in terpretation. In this pap er, a concrete quantum seman tics based on sup erop erators is associated with a simple quantum programming language. The representation of entanglemen t, i.e. the de- sign of the abstract domain is a key issue. A representation of entangle- ment as a partition of th e memory is chosen. An abstract semantics is introduced, and the soundness of the appro ximation is prov en. 1 In tro duction Quantum en tanglement is a non lo cal pro p er t y of quantum mec hanics. The en- tanglement reflects the ability of a quantum system co mpo s ed of sev era l sub- systems, to b e in a state which cannot be decomp osed in to the states o f the subsystems. En tanglement is o ne of the prope r ties o f quantum mech anics which caused Einstein and others to dislike the theo ry . In 1 9 35, Einstein, P o dolsk y , and Rosen for mu lated the EP R paradox [7]. On the other hand, quan tum mechanics has b een highly successful in pro - ducing cor rect exp erimental predictions, and the str ong corre lations asso c ia ted with the phenomenon o f quantum en tanglement have b een observed indee d [2]. Entanglemen t leads to cor relations b etw een subsystems that can b e exploited in information theory (e.g., telep or tation scheme [3]). T he en tanglement plays also a decisive, but not yet w ell-understo o d, role in q uantum co mputation, since any quantum alg orithm c an be e fficient ly simulated on a classical computer when the quantum memory is not e n tang le d during all the computation. As a conse- quence, in teresting quantum algorithms, like Sho r’s algorithm for factoris ation [19], exploit this pheno menon. In order to know what is the amo un t of entanglemen t of a q ua nt um state, several measures of entanglement hav e been in tro duced (see for instance [13 ]). Recent works cons is t in character is ing, in the framework of the one-w ay quantum computation [20], the a mount of en tanglement ne c essary for a universal mo del of quan tum computation. Notice that all these techniques co nsist in a nalysing the entanglemen t of a g iven sta te, starting with its mathematical description. In this pap er , the ent ang lement evo lution during the computation is analysed. The des cription of quantum evolutions is done via a simple quantum prog ram- ming language. The dev elopment of such qua ntum programming la nguages is recent, s e e [1 7,8] for a s ur vey on this topic. An exact a na lysis of entanglemen t evolution induces an ex p o nen tial slow- down of the co mputation. Mo del c hecking techniques hav e b een in tro duced [9] including entanglemen t. Exp onential slowdo wn of suc h analysis is avoided by reducing the domain to stabiliser sta tes (i.e. a subs et of q uantum states that can be efficien tly s im ulated on a clas sical computer). As a consequence, any q uantum progra m that cannot b e efficiently simulated on a cla ssical computer ca nnot b e analysed. Prost and Zerr a ri [1 6] hav e recently intro duced a logical entanglemen t anal- ysis for functional languag es. This logical fra mework allows analy sis of higher- order functions, but do es not provide any static analysis for the quantum pro- grams without annota tion. Moreov er, only pure quantum states are considered. In this pap er, w e int ro duce a nov el appro ach of entanglement analysis ba sed on the framew ork of a bstract interpretation [5 ]. A co ncrete quantum s eman- tics based o n supero per ators is a sso ciated with a simple quantum prog ramming language. The r epresentation of entanglement , i.e. the design o f the abstr act domain is a key issue. A represe ntation of en tanglement as a par tition o f the memory is c hosen. An abstract sema ntics is intro duced, and the soundness of the approximation is prov ed. 2 Basic Not ions and Entangle men t 2.1 Quan tum Computing W e br iefly r ecall the bas ic definitions of quantum computing; please refer to Nielsen and Chuang [13] for a co mplete introduction to the s ub ject. The s tate o f a qua nt um system can be describ ed by a density matrix, i.e. a self a djoint 1 po sitive-semidefinite 2 complex matrix of tr ace 3 less than one. The set of density matrices of dimension n is D n ⊆ C n × n . The ba sic unit of information in quantum computatio n is a qua nt um bit or qubit . The state of a single qubit is des crib ed b y a 2 × 2 density matrix ρ ∈ D 2 . The state of a r e g ister comp osed of n qubits is a 2 n × 2 n density matr ix . If tw o registers A and B are in states ρ A ∈ D 2 n and ρ B ∈ D 2 m , the comp osed s ystem A, B is in state ρ A ⊗ ρ B ∈ D 2 n + m . The ba sic ope rations on quantum sta tes are unitary op erations and measure- men ts. A unitary opera tion maps an n -qubit state to a n n -qubit state, and is 1 M is self adjoint (or Hermitian) if and only if M † = M 2 M is positive-semidefinite if all t he eigen v alues of M are non-negativ e. 3 The t race of M (tr( M )) is the sum of the diagonal elements of M given by a 2 n × 2 n -unitary matrix 4 . If a system in sta te ρ ev olves according to a unitary tr a nsformation U , the res ulting density matrix is U ρU † . The par allel comp osition of tw o unitary transforma tions U A , U B is U A ⊗ U B . The following unitary transformations form an appr oximativ e univ ersa l fam- ily of unitary transformatio ns, i.e. any unitary transformation can b e approxi- mated by compos ing the unitary tra nsformations of the family [13]. H = 1 √ 2  1 1 1 − 1  , T =  1 0 0 e iπ/ 4  , C N ot =     1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0     σ x =  0 1 1 0  , σ y =  0 − i i 0  , σ z =  1 0 0 − 1  A measur e ment is descr ibe d b y a family of pro jecto rs { P x , x ∈ X } satisfying P 2 i = P i , P i P j = 0 if i 6 = j , and P x ∈ X P x = I . A computational basis measur e- men t is { P k , 0 ≤ k < 2 n } , where P k has 0 en tries everywhere exce pt one 1 at row k , column k . The parallel comp osition of t wo measure ments { P x , x ∈ X } , { P ′ y , y ∈ Y } is { P x ⊗ P ′ y , ( x, y ) ∈ X × Y } . According to a proba bilistic interpretation, a mea s urement accor ding to { P x , x ∈ X } of a state ρ pro duces the classical outcome x ∈ X with pro ba - bilit y tr(P x ρ P x ) and tr a nsforms ρ into 1 tr(P x ρ P x ) P x ρ P x . Densit y ma trices is a useful formalism for re presenting probability distr i- butions of quan tum states, since the state ρ of a system whic h is in state ρ 1 (resp. ρ 2 ) with probability p 1 (resp. p 2 ) is ρ = p 1 ρ 1 + p 2 ρ 2 . As a consequence, a measurement according to { P x , x ∈ X } transfo rms ρ into P x ∈ X P x ρ P x . Notice that the s equential comp os itions of tw o measurements (or o f a mea- surement and a unitary transforma tion) is no more a measurement nor a unitary transformatio n, but a sup erop era tor, i.e. a tr ace-decrea sing 5 completely p o s itive 6 linear map. Any quantum e volution can b e desc r ib ed b y a sup er op erator. The a bilit y to initialise any qubit in a given state ρ 0 , to apply an y unitar y transformatio n from a universal family , and to p erform a computational mea- surement a re enough for simulating any s uper op erator . 2.2 En tangleme n t Quantum entanglement is a non lo cal pro pe r ty which ha s no c la ssical counter- part. Intuitiv ely , a quantum state of a system comp os ed of several subsystems is 4 U is unitary if and only if U † U = U U † = I . 5 F is t race decreasing iff tr( F ( ρ )) ≤ tr( ρ ) for an y ρ in the domain of F . Notice that sup eroperators are sometimes defined as trace-perserving maps, how ever trace- decreasing is more suitable in a semantical context, see [18] for details. 6 F is p ositive if F ( ρ ) is p ositive-semidefinite for any p ositiv e ρ in the domain of F . F is completely p ositive if I k ⊗ F is p ositiv e for any k , where I k : C k × k → C k × k is the identity map. entangle d if it cannot b e decomposed into the sta te of its subsystems. A quan tum state which is not entangled is called sep ar able . More precis ely , for a given finite set of qubits Q , let n = | Q | . F or a given partition A, B of Q , a nd a given ρ ∈ D 2 n , ρ is biseparable according to A, B (or ( A, B )-se pa rable for short) if a nd only if there exist K , p k ≥ 0, ρ A k and ρ B k such that ρ = X k ∈ K p k ρ A k ⊗ ρ B k ρ is e ntangled ac cording to the pa rtition A, B if and o nly if ρ is not ( A, B )- separable. Notice that bisepara bilit y provides a very partia l information ab out the en- tanglement of a quantum state, for instance for a 3-qubit state ρ , which is ( { 1 } , { 2 , 3 } )-sepa rable, qubit 2 and qubit 3 may be entangled or not. One w ay to gener alise the bisepa rability is to consider that a quantum state is π -s e parable – wher e π = { Q j , j ∈ J } is a partition of Q – if and only if there exist K , p k ≥ 0, and ρ Q j k such that ρ = X k ∈ K p k   O j ∈ J ρ Q j k   Notice that the s tructure of quantum entanglemen t pr esents s ome in teresting and non trivial proper ties. F or instance there exist some 3-qubit states ρ such that ρ is bi-sepa rable for any bi-partition of the 3 qubits, but no t fully separ a- ble i.e., separable ac cording to the partition {{ 1 } , { 2 } , { 3 } } . As a consequence, for a given quantum s tate, there is no t necessar y a b est r epr esentation of its ent ang lement . 2.3 Standard and diagonal basis F or a given state ρ ∈ D Q and a g iven qubit q ∈ Q , if ρ is ( { q } , Q \ { q } )-separable , then q is separ a ted fro m the rest of the memory . Mor e ov er, such a qubit may be a basis state in the standard basis ( s ) or the diagonal basis ( d ), meaning that the state o f this qubit can b e seen as a ’classica l state’ ac c ording to the corres p o nding basis . More fo r mally , a qubit q of ρ is in the standard basis if there exists p 0 , p 1 ≥ 0, and ρ 0 , ρ 1 ∈ D Q \{ q } such that ρ = p 0 P true q ⊗ ρ 0 + p 1 P false q ⊗ ρ 1 . Equiv alently , q is in the standard basis if and only if P true q ρP false q = P false q ρP true q = 0. A qubit q is in the diag onal basis in ρ if a nd only if q is in the standar d basis in H q ρH q . Notice that some states, lik e the maximally mixed 1-qubit state 1 2 ( P true + P false ) are in both standard and diagonal basis, while o thers a re neither in stan- dard nor diag onal basis like the 1-qubit state T H P true H T . W e int ro duce a function β : D Q → B Q , where B Q = Q → { s , d , ⊤ , ⊥} , such that β ( ρ ) describ es which qubits o f ρ are in the sta ndard or diag onal basis: Definition 1. F or any finite Q , let β : D Q → B Q such that for any ρ ∈ D Q , and any q ∈ Q , β ( ρ ) q =          ⊥ if q is in b oth standar d and diagonal b asis in ρ s if q is in the standar d and not in the diagonal b asis in ρ d if q is in t he diagonal and not in t he standar d b asis in ρ ⊤ otherwise 3 A Quan tum Programming Language Several quantum pr ogra mming langua g es have been introduced recently . F or a complete ov erview see [8]. W e use an imp era tive quantum progr a mming languag e int ro duce d in [15], the synt ax is similar to the language in tro duced by Abra msky [1]. F or the s a ke of simplicit y and in orde r to focus on en tanglement ana lysis, the memory is supposed to be fixed and finite. Moreover, the memory is supposed to be comp osed of qubits only , whereas hybrid memories comp os e d of clas sical and quantum parts ar e often co nsidered. How ever, contrary to the quantum circuit or qua n tum T uring mac hine fra meworks, the a bsence of c lassical memory do e s not av oid the c lassical control of the qua ntum computation since classic a lly- controlled co nditional structures ar e allow ed (see section 3.1.) Definition 2 (Syntax ). F or a given finite s et of symb ols q ∈ Q , a pr o gr am is a p air h C, Q i wher e C is a c ommand define d as fol lows: C ::= skip | C 1 ; C 2 | if q then C 1 else C 2 | while q d o C | H ( q ) | T ( q ) | CNot ( q, q ) Example 1. Quantum entanglemen t b e t ween tw o qubits q 2 and q 3 can b e crea ted for instance by apply ing H and C N ot o n an appropr ia te state. Such a n ent ang led state ca n then b e used to telep orte the state of a third qubit q 1 . The proto c o l of telepo rtation [3] can b e describ ed as h telep ortation , { q 1 , q 2 , q 3 }i , where telep ortation : H ( q 2 ); CNot ( q 2 , q 3 ); CNot ( q 1 , q 2 ); H ( q 1 ); if q 1 then if q 2 then skip else σ x ( q 3 ) else if q 2 then σ z ( q 3 ) else σ y ( q 3 ) The semantics o f this pro gram is given in example 2. 3.1 Concrete Sem an tics Several domains for quantum computation have been in tro duced [12,1,14]. Among them, the domain of sup erop era tors o ver densit y matrice s, int ro duced by Selinger [18] turns o ut to b e one of the mo s t a dapted to qua n tum semantics. Thu s, we introduce a deno ta tional semantics follo wing the work of Selinger. F or a finite set of v aria bles Q = { q 0 , . . . , q n } , let D Q = D 2 | Q | . Q is a set of qubits, the sta te of Q is a density op er ator in D Q . Definition 3 (L¨ owner partial order). F or matric es M and N in C n × n , M ⊑ N if N − M is p ositive-semidefinite. In [18], Selinger pro ved that the pos et ( D Q , ⊑ ) is a complete partial order with 0 as its le a st element. Moreover the p oset of sup erop era tors ov er D Q is a complete partial order as well, with 0 as least element a nd where the partial order ⊑ ′ is defined as F ⊑ ′ G ⇐ ⇒ ∀ k ≥ 0 , ∀ ρ ∈ D k 2 | Q | , ( I k ⊗ F )( ρ ) ⊑ ( I k ⊗ G )( ρ ), where I k : D k → D k is the identit y map. Notice that these co mplete partial orders are no t la ttices (see [18].) W e are now ready to in tro duce the concrete deno tational seman tics which asso ciates with any progra m h C, Q i , a s uper op erator J C K : D Q → D Q . Definition 4 (Deno tational seman tics). J skip K = I J C 1 ; C 2 K = J C 2 K ◦ J C 1 K J U ( q ) K = λρ.U q ρU † q J CNot ( q 1 , q 2 ) K = λρ.C N ot q 1 ,q 2 ρC N ot † q 1 ,q 2 J if q then C 1 else C 2 K = λρ.  J C 1 K (P true q ρ P true q ) + J C 2 K (P false q ρ P false q )  J while q do C K = l fp  λf .λρ.  f ◦ J C K (P true q ρ P true q ) + P false q ρ P false q  = P n ∈ N ( F P false ◦ ( J C K ◦ F P true ) n ) wher e P true =  1 0 0 0  and P false =  0 0 0 1  , F M = λρ.M ρM † , and M q me ans that M is applie d on qubit q . We r efer t he r e ader to an ex t ende d version of t his p ap er fo r t he te chnic al explanatio ns on c ontinuity and c onver genc e. In the absence o f clas sical memory , the class ical control is enco ded into the conditional structur e if q then C 1 else C 2 such that the qubit q is fir st measured according to the co mputational basis . If the first pro jector is a pplied, then the classical outcome is interpreted as true and the command C 1 is applied. Other- wise, the second pro jector is applied, and the command C 2 is p erformed. The classical control appear s in the lo o p while q do C a s well. As a consequence of the classic a l con tro l, non unitary tra ns formations can be implemented: J if q t hen q els e σ x ( q ) K : D { q } → D { q } = λρ. P true J while q d o H ( q ) K : D { q } → D { q } = λρ. P false Notice that the matrices P true and P false , used in definition 4 for describing the computational mea surement { P true , P false } can also b e us e d as density matr ices for describing a q uantum s tate as ab ov e. Moreov er , notice that all the ingredients for approximating an y sup ero pe r a- tors can b e encoded into the languag e: the ability to initialise any qubit in a given state (for instance P true or P false ); an approximative universal family of unitary transformatio n { H , T , C N ot, σ x , σ y , σ z } ; and the co mputational measurement of a qubit q with if q then skip else sk ip . Example 2. The program h telep ortation , { q 1 , q 2 , q 3 }i describ ed in example 1 re - alises the telepor ta tion from q 1 to q 3 , when the qubits q 2 and q 3 are both ini- tialised in state P true : for any ρ ∈ D 2 , J telep ortation K ( ρ ⊗ P true ⊗ P true ) =   1 4 X k,l ∈{ true , f alse } P k ⊗ P l   ⊗ ρ 4 En tanglemen t Analysis What is the ro le of the entanglemen t in quantum information theory? How do es the e ntanglement evolve during a quantum computation? W e conside r the problem o f ana lysing the entanglemen t evolution on a classical co mputer, since no la rge sca le qua ntum computer is av ailable at the momen t. E ntanglemen t analysis using a quantum computer is left to further inv estigations 7 . In the absence of quan tum computer, a n ob vious solution cons is ts in sim- ulating the quan tum co mputation on a classical co mputer. Unfortunately , the classical memory r equired for the sim ulation is e x po nent ially lar g e in the size of the quantum memory of the progr am sim ulated. Mo reov er, the problem SE P of deciding whether a given quantum state ρ is bisepar able or not is NP Hard 8 [10]. F urther more, the input o f the problem SEP is a density matrix, which siz e is exp onential in the num b er of qubits. As a consequence, the solutio n o f a classical simulation is not suitable for an efficient en tanglement analysis. 7 Notice that this is not clear that t h e use of a q uantum computer av oids the use of th e classical computer since there is n o wa y to measure the entanglemen t of a quantum state without transforming the state. 8 F or pure quantum states (i.e. tr( ρ 2 ) = tr( ρ )) , a linear algorithm hav e b een introduced [11] to solv e the sub - problem of finding b iseparabilit y of t he form ( { q 0 , . . . , q k } , { q k +1 , . . . , q n } ) – thus sensitiv e to the ordering of the qub its in the register. Notice that this algorithm is linear in the size of the input which is a den- sit y matrix, th us the algorithm is exp onential in the n umber of qubits. T o tackle this problem, a so lution co nsists in reducing the size of the quan- tum state space b y considering a subspace o f pos s ible states, suc h that there exist a lg orithms to decide whether a state of the subspa c e is en tangled o r no t in a polyno mial time in the n umber of qubits. This solution has bee n developed in [9], b y considering stabiliser states only . Ho wev er, this s olution, whic h ma y b e suitable for some quantum proto co ls, is questiona ble for a nalysing quantum algo - rithms since all the qua nt um prog rams on which s uch an en tanglement a na lysis can b e driven are also efficiently sim ulable o n a classical co mputer. In this pap er, we intro duce a no vel approach w hich co nsists in approximating the entanglement evolution of the quantum memory . This solution is based on the framework of abstract interpretation intro duce d by Co us ot and C o usot [5]. Since a class ical domain for driving a sound and complete ana lysis o f entangle- men t is expone ntially large in the num b er n of qubits, we co nsider an abstract domain of size n and w e introduce a n abstr act semantics whic h leads to a sound approximation of the en tangle ment evolution dur ing the computation. 4.1 Abstract seman tics The entanglement of a quantum state c an b e repres ent ed as a partition of the qubits of the state (see section 2.2), th us a natura l abstract domain is a do ma in comp osed of pa rtitions. Moreov er, for a given state ρ , one ca n add a fla g for eac h qubit q , indica ting whether the state of this qubit is in the standard ba sis s or in the diag onal basis d (see sec tio n 2.3). Definition 5 (Abs tract Domain). F or a finite set of variables Q , let A Q = B Q × Π Q b e an abstr act domain, wher e B Q = Q → { s , d , ⊤ , ⊥} and Π Q is the set of p artitions of Q : Π Q = { π ⊆ ℘ ( Q ) \ { ∅} | [ X ∈ π X = Q an d ( ∀ X, Y ∈ π, X ∩ Y = ∅ or X = Y ) } The abstract doma in A is order ed as follows. Fir st, let ( { s , d , ⊤ , ⊥} , ≤ ) be a po set, wher e ≤ is defined as : ⊥ ≤ s ≤ ⊤ and ⊥ ≤ d ≤ ⊤ . ( B Q , ≤ ) is a poset, where ≤ is defined point wise. Mo r eov er, for any π 1 , π 2 ∈ Π Q , let π 1 ≤ π 2 if π 1 rafines π 2 , i.e . for every blo ck X ∈ π 1 there exists a blo ck Y ∈ π 2 such that X ⊆ Y . Finally , for any ( b 1 , π ) , ( b 2 , π 2 ) ∈ A Q , ( b 1 , π ) ≤ ( b 2 , π 2 ) if b 1 ≤ b 2 and π 1 ≤ π 2 . Prop ositio n 1. F or any fi n ite set Q , ( A Q , ≤ ) is a c omplete p art ial or der, with ⊥ = ( λq . ⊥ , {{ q } , q ∈ Q } ) as le ast element. Pr o of. Every c hain has a s upr emum s ince Q is finite. ⊓ ⊔ Basic op era tions of meet and join are defined on A Q . It turns out that con- trary to D Q , hA Q , ∨ , ∧ , ⊥ , ( λq . ⊤ , { Q } ) i is a la ttice. A r emov al op eratio n on partitions is intro duced as follows: for a giv en parti- tion π = { Q i , i ∈ I } , let π \ q = { Q i \ { q } , i ∈ I } ∪ {{ q }} . Mo r eov er, for an y pair of qubits q 1 , q 2 ∈ Q , let [ q 1 , q 2 ] = {{ q | q ∈ Q \ { q 1 , q 2 }} , { q 1 , q 2 }} . Finally , for any b ∈ B Q , any q 0 , q ∈ Q , a ny k ∈ { s , d , ⊤ , ⊥ } , let b q 0 7→ k q = ( k if q = q 0 b q otherwise W e are now r e ady to define the a bstract semantics o f the lang uage: Definition 6 (Deno tational abstract sem an tics). F or any pr o gr am h C, Q i , let J C K ♮ : A Q → A Q b e defin e d as fol lows: F or any ( b, π ) ∈ A Q , J skip K ♮ ( b, π ) = ( b, π ) J C 1 ; C 2 K ♮ ( b, π ) = J C 2 K ♮ ◦ J C 1 K ♮ ( b, π ) J σ ( q ) K ♮ ( b, π ) = ( b, π ) J H ( q ) K ♮ ( b, π ) = ( b q 7→ d , π ) if b q = s = ( b q 7→ s , π ) if b q = d = ( b, π ) otherwise J T ( q ) K ♮ ( b, π ) = ( b q 7→⊤ , π ) if b q = d = ( b q 7→ s , π ) if b q = ⊥ = ( b, π ) otherwise J CNot ( q 1 , q 2 ) K ♮ ( b, π ) = ( b, π ) if b q 1 = s or b q 2 = d = ( b q 1 7→ s , π ) if b q 1 = ⊥ and b q 2 > ⊥ = ( b q 2 7→ d , π ) if b q 1 > ⊥ and b q 2 = ⊥ = ( b q 1 7→ s ,q 2 7→ d , π ) if b q 1 = ⊥ and b q 2 = ⊥ = ( b q 1 ,q 2 7→⊤ , π ∨ [ q 1 , q 2 ]) otherwise J if q then C 1 else C 2 K ♮ ( b, π ) =  J C 1 K ♮ ( b q 7→ s , π \ q ) ∨ J C 2 K ♮ ( b q 7→ s , π \ q )  J while q do C K ♮ ( b, π ) = lfp  λf .λπ .  f ◦ J C K ♮ ( b q 7→ s , π \ q ) ∨ ( b q 7→ s , π \ q )  = W n ∈ N  F ♮ q ◦ ( J C K ♮ ◦ F ♮ q ) n  wher e F ♮ q = λ ( b, π ) . ( b q 7→ s , π \ q ) . Int uitively , quantum op erations act on e ntanglement a s follows: – A 1-q ubit measurement makes the measured q ubit separable from the res t of the memory . Mo r eov er, the state of the measur ed qubit is in the sta ndard basis. – A 1 - qubit unitary tr a nsformation does not mo dify ent ang lement . An y Pauli op erator σ ∈ { σ x , σ y , σ z } preserves the standard and the diagona l bas is o f the qubits. Ha da mard H tra nsforms a state of the standard ba sis in to a state of the diag onal basis and vice-versa. Finally the pha se T pres erves the standard basis but not the diago nal basis. – The 2 -qubit unitary transformatio n C N ot , applied on q 1 and q 2 may create ent ang lement betw een the qubits o r not. It turns out that if q 1 is in the stan- dard ba s is, or q 2 is in the diagonal basis, then no en tanglement is cr eated and the basis of q 1 and q 2 are preserved. Otherwis e, s ince a sound approximation is desired, C N ot is abstracted in to an op eration w hich creates entanglement. R emark 1. Notice tha t the spac e needed to store a pa rtition of n elements is O ( n ). Moreover, meet, join and r emov al and can be done in either consta n t o r linear time. Example 3. The abstract seman tics of the telepor tation (see example 1) is J telep ortation K ♮ : A { q 1 ,q 2 ,q 3 } → A { q 1 ,q 2 ,q 3 } = λ ( b, π ) . ( b q 1 ,q 2 7→ s ,q 3 7→⊤ , ⊥ ). Thus, for an y 3-qubit s tate, the state of the memor y after the telepo rtation is fully separable. Assume that a fourth qubit q 4 is entangled with q 1 befo re the telep or- tation, whereas q 2 and q 3 are in the s tate P true . So that, the state o f the memory b efore the telep ortation is [ q 1 , q 4 ]-separa ble. The abstract semantics of h telep ortation , { q 1 , q 2 , q 3 , q 4 }i is such that J telep ortation K ♮ ( b, [ q 1 , q 4 ]) = ( b q 1 ,q 2 7→ s ,q 3 7→⊤ , [ q 3 , q 4 ]) Thu s the abstract semantics predicts that q 3 is en tangled with q 4 at the end of the telep or tation, even if q 3 never in teracts with q 4 . Example 4. Consider the pro gram h trap , { q 1 , q 2 }i , where trap = CNot ( q 1 , q 2 ); CNot ( q 1 , q 2 ) Since C N ot is self-inv ers e, J trap K : D { q 1 ,q 2 } → D { q 1 ,q 2 } = λρ.ρ . F or instance, J trap K ( 1 2 ( P true + P false ) ⊗ P true ) = 1 2 ( P true + P false ) ⊗ P true . How ever, if b q 1 = d and b q 2 = s then J trap K ♮ ( b, {{ q 1 } , { q 2 }} ) = ( b q 1 7→⊤ ,q 1 7→⊤ , {{ q 1 , q 2 }} ) Thu s, a ccording to the abstr act semantics, at the end of the computation, q 1 and q 2 are entangled. 4.2 Soundness Example 4 p oints o ut that the abstract semantics is an approximation, so it ma y differ from the entanglemen t evolution of the concrete semantics. How ever, in this section, we prov e the soundness of the abstract interpretation (theorem 1). First, w e define a function β : D Q → B Q such that β ( ρ ) des crib es whic h qubits of ρ a re in the standar d or diago na l basis: Definition 7. F or any finite Q , let β : D Q → B Q such that for any ρ ∈ D Q , and any q ∈ Q , β ( ρ ) q =      s if P true q ρP false q = P false q ρP true q = 0 d if ( P true q + P false q ) ρ ( P true q − P false q ) = ( P true q − P false q ) ρ ( P true q + P false q ) = 0 ⊤ otherwise A natural soundness re lation is then: Definition 8 (So undness relation). F or any fin ite set Q , let σ ∈ ℘ ( D Q , A Q ) b e t he soundness r elation: σ = { ( ρ, ( b, π )) | ρ is π -s ep ar able and β ( ρ ) ≤ b } The appr oximation relation is nothing but the partia l or der ≤ : ( b, π ) is a more precise approximation tha n ( b ′ , π ′ ) if ( b, π ) ≤ ( b ′ , π ′ ). Notice that the abstract soundness assumption is satisfied: if ρ is π - separable and π ≤ π ′ then ρ is π ′ - separable. So, ( ρ, a ) ∈ σ and ( ρ, a ) ≤ ( ρ ′ , a ′ ) imply ( ρ ′ , a ′ ) ∈ σ . How ever, the best approximation is not ensured. Indeed, there exist some 3-qubit states [6,4] which are sepa rable according to any of the 3 bipar titions of their qubits { a, b, c } but whic h ar e not {{ a } , { b } , { c }} -separable. Th us, the best approximation does not exist. How ever, the soundness r elation σ s atisfies the following lemma: Lemma 1. F or any finite s et Q , any ρ 1 , ρ 2 ∈ D Q , and any a 1 , a 2 ∈ A Q , ( ρ 1 , a 1 ) , ( ρ 2 , a 2 ) ∈ σ = ⇒ ( ρ 1 + ρ 2 , π 1 ∨ π 2 ) ∈ σ Moreov er , the abs tract sema n tics is mono tonic according to the approxima- tion rela tion: Lemma 2. F or any c ommand C , J C K ♮ is ≤ -monotonic: for any π 1 , π 2 ∈ A Q , π 1 ≤ π 2 = ⇒ J C K ♮ ( π 1 ) ≤ J C K ♮ ( π 2 ) Pr o of. The pro of is by induction on C . Theorem 1 (Soundness). F or any pr o gr am h C , Q i , any ρ ∈ D Q , and any a ∈ A Q , ( ρ, a ) ∈ σ = ⇒ ( J C K ( ρ ) , J C K ♮ ( a )) ∈ σ Pr o of. The pro of is by induction on C . In other words, if ρ is π -sepa rable and β ( ρ ) ≤ b , then J C K ( ρ ) is π ′ -separa ble and β ( J C K ( ρ )) ≤ b ′ , where ( b ′ , π ′ ) = J C K ♮ ( b, π ). 5 Conclusion and P ersp ect iv es In this pap er, w e ha ve int ro duce d the first q uantum entanglement analysis based on a bstract interpretation. Since a classica l doma in for driving a sound a nd complete ana lysis of entanglement is exponentially lar g e in the num b er o f qubits, an abstract doma in ba sed on par titions has b een int ro duced. Moreov er, since the concrete domain of sup ero per ators is no t a lattice, no Galo is connection can be establis he d b etw een concrete and abstract domains. How ever, despite the absence of best abs traction, the s o undness of the en tanglement ana lysis has bee n prov ed. The abstract domain is not o nly c o mpo sed of par titions of the memory , but also of descr iptions of the qubits whic h are in a basis state a ccording to the stan- dard or diagonal basis. Tha nks to this additional information, the en tanglement analysis is more subtle than an ana lysis of in tera ctions: the C N ot transforma- tion is not an en tangling op eration if the first q ubit is in the s tandard basis or if the seco nd qubit is in the diagona l basis. A p ers p ective, in order to rea ch a mor e pr ecise entanglement analysis, is to int ro duce a more concr ete abstract domain, adding for instance a third basis, since it is k nown that there are three mutually unbiased basis for each qubit. A s imple quan tum imp erative language is considered in this paper . This language is expres s ive enoug h to encode any quantum evolution. Howev er, a per sp ective is to develop such abstra ct interpretation in a mo re gene r al setting allowing hig h-order functions, r e presentation of cla ssical v ariables, o r un b o unded quantum memory . The ob jective is also to provide a practical to o l for analysing ent ang lement evolution o f more sophisticated programs, lik e Shor’s algor ithm for factorisa tion [19]. Another p ersp ective is to consider that a quant um c o mputer is av ailable for driving the ent ang lement a na lysis. 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