On the completeness of quantum computation models

On the completeness of quan tum computation mo dels Pablo Arrighi ´ Ecole no rmale sup´ erieure de Lyon LIP , 46 all´ ee d’Italie, 6 9008 Lyon, F rance and Universit ´ e de Gr enoble LIG, 220 r ue de la chimie, 3 8400 Sa in t Martin d’H` eres, F rance . pablo. arrig hi@imag.fr Gilles Dowek ´ Ecole p olytechnique and INRIA, LIX, ´ Ecole p olytechnique, 9112 8 Palaiseau Cedex, F rance. gilles .dowe k@polytech nique.edu Abstract. T he notion of computability is stable ( i .e. indep endent of the choice of an indexing) o ver infinite-dimensional ve ctor spaces pro- vided they ha ve a finite “tensorial dimension”. S uch vector spaces with a finite tensorial dimension p ermit to define an absolute notion of com- pleteness for quantum compu tation mo dels and give a precise meaning to the Churc h-T uring thesis in the framew ork of quantum theory . 1 In tro duction In cla s sical computing, a lg orithms are sometimes expre s sed by bo olean circuits. E.g. the algo rithm mapping the b o o leans x and y to and ( x, not ( y )) can be ex- pressed by a circuit for med with an and g ate and a not gate. A typical result ab out s uch c ir cuit mo dels is that a given set of gates is complete, i.e. that all functions from { 0 , 1 } n to { 0 , 1 } p can b e e x pressed with a circuit with n inputs and p outputs. Y et, cir cuit mo dels are limited b ecause an algor ithm must take as input and r eturn a s o utput a fix e d num ber of b o olean v alues, and not, for instance, an arbitrar y natural num ber or sequence of b o olean v alues of arbitrar y length. Thus, other co mputation mode ls , such a s T uring machines, λ -calculus, cellular automata, . . . hav e been designed. Such mo dels cannot expres s all func- tions from N to N , but a typical completeness result is that they can expr ess all c omputable functions. Thus, befor e stating and pr oving s uch a r esult, we need to define a subset of the set of the functions from N to N : the set of c omputable functions. In quantum c o mputing, b o th t yp es of mo dels exist. Most algo rithms ar e ex- pressed using the quantum circuit mo de l, but more evolv ed mo dels, allowing to express alg orithms working with unbounded size da ta exist also, e.g. quantum T uring machines [9,7,13], quantum λ -calc uli [1,17,18,2,3], quantum cellula r au- tomata [16,6,5]. T o define a notion of completeness for these quantum mo dels o f computation, w e must first define a set of computable quantu m functions. Quantum functions a re linea r functions from a vector s pace to itself, and when this vector spa ce is finite or countable ( i.e. when it is a vector s pace of finite or countable dimension ov er a finite o r countable field), the notion of computability can b e trans p o rted fro m the natural num be r s to this vector space, with the use of a n indexing . The robustness of the notion of computability over the natural num b ers has bee n pointed out for long, and is em b o died b y the Church-T uring thesis. In con- trast, it is well-kno wn that the notion of computability o ver an arbitrary count- able set dep ends o n the choice of an indexing: the co mpo sition of a n indexing with a no n computable function yields a nother indexing that defines a different set of c omputable functions [10,8]. Thus, the robustness o f the computability ov er the na tural num ber s do es not dir e ctly car ry through to countable vector spaces and the na ive no tion of co mpleteness for s uch a qua n tum computational mo del is r elative to the c hoice of an indexing. On the o ther hand, a vector space is not just a set, but a set equipp ed with an addition and a multiplication by a s calar. More g enerally , sets are rar ely used in isolation, but c ome in algebraic structures, equippe d w ith o p er ations. As their na me suggests, these op erations should b e computable. Thus, in alge- braic str uctures, we can res tr ict to the so called admissible indexings that make these op era tions computable [14]. In many ca ses, restr ic ting to such admissible indexings makes the set of computable functions indep endent of the choice of the indexing. In such a cas e, computability is said to b e stable ov er the consider ed algebraic structures [19]. Unfortunately , we show in this pap er that, although computability is stable ov er finite-dimensional vector spa c es, it is not stable ov er infinite-dimensio nal ones, suc h as those we use to define quantum functions on unbounded size data . Indeed, any non-computable per mu tation of the vectors of a bas is changes the set of computable functions witho ut affecting the computability of the o p er ations. Thu s, mak ing the op e rations of the vector space computable is not enoug h to hav e a stable notion of co mputabilit y . This result may b e co nsidered as worrying bec ause infinite-dimensional vector spac es a re rather common in physics: they are the usual so r t of space in which one defines so me dynamics , whether quantum or classical. As one may wish to study the ability of a physical ma chin e or a dynamics, to co mpute, o ne would like to hav e a stable notion o f co mputabilit y on such a spa ce, and not a no tion rela tive to the choice of an indexing. The fact that, in quantum theory , one do es not directly obser ve the state vector but rather the induced probability distribution ov er measurement o utcomes do es not circumv ent this problem: as w as p o inted out [11], uncomputable amplitudes can still be read out pr o babilistically . F or tunately , we show that if we a dd a tensor pro duct as a primitive o p eration of the algebr aic str ucture, a nd restrict to indexing s that make als o this op er ation computable, then computability may b e stable despite the infinite dimension of 2 the space. This theo r em is obtained fr o m a nov el g eneral result on extensions of algebraic structures which preserve stable computability . 2 Stable computabilit y W e a ssume familia rity with the no tion of a computable function ov er the natural nu mbers, as pres ent ed for instance in [15,12]. As usual, this notion is tr ansp orted to finite trees lab eled in a finite set thro ugh the use o f an indexing. T o define this indexing, we first ass o ciate a n index p f q to ea ch element f of the set of lab els and we de fine the indexing of the tree f ( t 1 , . . . , t l ) as p f ( t 1 , . . . , t l ) q = p f q ; p t 1 q ; . . . ; p t l q ; 0 whe r e ; is a computable o ne-to-one mapping from N 2 to N \ { 0 } , e.g. n ; p = ( n + p )( n + p + 1) / 2 + n + 1. The indexing of this set of tree s is the partial function i mapping the index p t q to the term t . Notice that this indexing is indep endent of the choice of the indexing of the lab els, that is if i 1 and i 2 are tw o indexings of the same set o f trees built on different indexings of the lab els, ther e exists a c o mputable function h s uch that i 2 = i 1 ◦ h . When A is a s e t of natural num bers, we say that A is effe ctively enumer able if it is either empt y or the imag e of a total computable function from N to N . This is equiv a lent to the fact that A is the ima ge o f a par tial co mputable function, o r again to the fact that it is the domain of partia l co mputable function. W e shall use the following result several times. Prop ositio n 1 (In v erse). If A and B ar e effe ctively enumer able sets and h is a p artial c omputable function such that for al l x in A , h is define d at x and h ( x ) is in B , and h is surje ctive, i.e. for e ach y in B ther e exists a x in A such that h ( x ) = y , then h has a c omputable right inverse, i.e. t her e exists a p artial c omputable function k su ch that for al l y in B , k is define d at y , h is define d at k ( y ) and h ( k ( y )) = y . Pr o of. As the set A is effectiv ely en umerable, it is either empty , in which ca se B also is empty and the re s ult is trivia l, or it is the ima ge of a total recursive function g , we define k 1 ( y ) to b e the leas t z such that h ( g ( z )) = y and k ( y ) = g ( k 1 ( y )) and we hav e h ( k ( y )) = h ( g ( k 1 ( y ))) = y . 2.1 Indexings Definition 1 (Indexing). An indexing of a set E is a p artial funct ion i fr om N t o E , such that – dom ( i ) is effe ct ively enumer able, – i is surje ctive, – ther e exists a c omputable function eq fr om dom ( i ) × dom ( i ) to { 0 , 1 } such that e q ( x, y ) = 1 if i ( x ) = i ( y ) and e q ( x, y ) = 0 otherwise. An indexing of a family of sets E 1 , . . . , E n is a family of indexings i 1 , . . . , i n of the sets E 1 , . . . , E n . 3 Notice that this definition is slightly more gener al than the usual one (see for instance [1 9], Definition 2.2 .2) as we only require dom ( i ) to b e effectiv ely enum era ble, instead of requiring it to b e decidable. Requir ing full dec idability of dom ( i ) is not needed in this pa per and would b e an obstac le to Prop o sition 11. Definition 2 (Computable function o v er a set equipp ed with an in- dexing). L et E 1 , . . . , E m , E m +1 b e a family of sets and i 1 , . . . , i m , i m +1 an in- dexing of t his family. A function fr om E 1 × . . . × E m to E m +1 is said to b e computable relatively to this indexing if ther e exists a c omputable function ˆ f fr om dom ( i 1 ) × . . . × dom ( i m ) to dom ( i m +1 ) such t hat for al l x 1 in dom ( i 1 ) , . . . , x m in dom ( i m ) , ˆ f ( x 1 , . . . , x m ) is in dom ( i m +1 ) and i m +1 ( ˆ f ( x 1 , . . . , x m )) = f ( i 1 ( x 1 ) , . . . , i m ( x m )) . Definition 3 (Admissible i ndexing of an algebraic s tructure [14]). An indexing of an algebr aic st r u ctur e h E 1 , . . . , E n , op 1 , . . . , op p i is an indexing of the family E 1 , . . . , E n . Su ch an indexing is said t o b e admissible if the op er ations op 1 , . . . , op p ar e c omputable r elatively to the indexings of their domains and c o- domain. 2.2 Stability Definition 4 (Stable computabili t y [19]). Computability is said t o b e sta- ble over a structur e h E 1 , . . . , E n , op 1 , . . . , op p i if t her e exists an admissible in- dexing i 1 , . . . , i n of this st ructur e and for al l admissible indexings j 1 , . . . , j n and j ′ 1 , . . . , j ′ n of this s t ructur e, ther e exist c omputable functions h 1 fr om dom ( j ′ 1 ) to dom ( j 1 ) , . . . , h n fr om dom ( j ′ n ) t o dom ( j n ) su ch that j ′ 1 = j 1 ◦ h 1 , . . . , j ′ n = j n ◦ h n . W e now w ant to prov e that the notion of computable function ov er a struc tur e with stable computability is indep endent o f the choice of the indexing. T o state this in full g e nerality , w e c o nsider a structure h E 1 , . . . , E n , op 1 , . . . , op p i with stable computability , and a family of sets F 1 , . . . , F m , F m +1 some of which are among E 1 , . . . , E n , in which case v arious admissible indexing s ca n b e co nsidered for these sets. W e prove that the set of co mputable functions from F 1 , . . . , F m to F m +1 is indep endent of thes e choices of admissible indexings. Prop ositio n 2. L et h E 1 , . . . , E n , op 1 , . . . , op p i b e a st ructur e with stable c om- putability. Then, for al l famili es of sets F 1 , . . . , F m +1 and indexings i 1 , . . . , i m +1 and i ′ 1 , . . . , i ′ m +1 of this family, such that for al l k either i k = i ′ k or F k is some E l , and i k and i ′ k ar e two indexings of E l extr acte d fr om t wo admissible indexings of the structu r e h E 1 , . . . , E n , op 1 , . . . , op p i , the set of c omputable fun ct ions fr om F 1 , . . . , F m to F m +1 e quipp e d with i 1 , . . . , i m +1 and i ′ 1 , . . . , i ′ m +1 is t he same. Pr o of. Ther e exis t computable functions h 1 , . . . , h m , h m +1 such that i ′ 1 = i 1 ◦ h 1 , . . . , i ′ m = i m ◦ h m , i ′ m +1 = i m +1 ◦ h m +1 . Let h ′ 1 , . . . , h ′ m , h ′ m +1 be right inverses of h 1 , . . . , h m , h m +1 . 4 If there exists a computable function ˆ f such that i m +1 ( ˆ f ( x 1 , . . . , x m )) = f ( i 1 ( x 1 ) , . . . , i m ( x m )), the function ˜ f mapping x 1 , . . . , x m to h ′ m +1 ( ˆ f ( h 1 ( x 1 ) , . . . , h m ( x m ))) is computable and i ′ m +1 ( ˜ f ( x 1 , . . . , x m ) = f ( i ′ 1 ( x 1 ) , . . . , i ′ m ( x m ))). Con- versely , if there exists a function ˜ f such that i ′ m +1 ( ˜ f ( x 1 , . . . , x m )) = f ( i ′ 1 ( x 1 ) , . . . , i ′ m ( x m )), the function ˆ f mapping x 1 , . . . , x m to h m +1 ( ˜ f ( h ′ 1 ( x 1 ) , . . . , h ′ m ( x m ))) is computable and i m +1 ( ˆ f ( x 1 , . . . , x m )) = f ( i 1 ( x 1 ) , . . . , i m ( x m )). W e can also prov e the conv erse . Prop ositio n 3. L et h E 1 , . . . , E n , op 1 , . . . , op p i b e a structur e. If for al l families of sets F 1 , . . . , F m +1 and indexings i 1 , . . . , i m +1 and i ′ 1 , . . . , i ′ m +1 of this family, such that for al l k either i k = i ′ k or F k is some E l , i k and i ′ k ar e extr acte d fr om two admissible indexings of the struct ur e h E 1 , . . . , E n , op 1 , . . . , op p i the set of c omput able functions fr om F 1 , . . . , F m to F m +1 e quipp e d with i 1 , . . . , i m +1 and with i ′ 1 , . . . , i ′ m +1 is the same. Then, c omputability is st able over the st ructur e h E 1 , . . . , E n , op 1 , . . . , op p i . Pr o of. Let j 1 , . . . , j n and j ′ 1 , . . . , j ′ n be tw o admissible indexings o f h E 1 , . . . , E n , op 1 , . . . , op p i . The identit y from E k to E k is computable r elatively to j k , j k . Our hypothesis tells us that it a lso co mputable relatively to j ′ k , j k , i.e. that there exists a computable function h k , from dom ( j ′ k ) to dom ( j k ), such that fo r all x , j k ( h k ( x )) = j ′ k ( x ) i.e. j k ◦ h k = j ′ k . 3 Relativ e finite generation W e now want to prov e that computability is stable ov er finitely generated struc- tures. In tuitively , a structure is finitely generated if all its elements can b e con- structed with the op eratio ns of the structure from a finite num ber of elements a 0 , . . . , a d − 1 . F or ins ta nce, the str ucture h N , S i is finitely gene r ated b ecaus e all natural num b ers can b e generated with the succes sor function from the num- ber 0. This prop erty can also b e stated as the fact that for each element b of the structure there exists a term t , i.e. a finite tree lab eled with the element s a 0 , . . . , a d − 1 and the op er a tions of the structur e, such that the denotation J t K of t is b . Y et, in this definition, we must take into acco unt t wo other ele ments. First, in or der to prove tha t computability is stable, we must add the condi- tion that equa lit y of deno tations is decidable o n terms. This lea ds us to s o me- times consider not the full se t o f terms T generated by the o pe rations of the structure, but just a n effectively enumerable subset T of this set, which is cho- sen large enough to contain a term denoting each ele ment o f the structure. This is in order to a void situations where the equality o f deno ta tions ma y b e un- decidable, or mor e a t lea st mo r e difficult to prov e decidable, o n the full set of terms. Secondly , when a structure has several domains, s uch as a vector space that has a domain for sc alars and a domain for v ectors, we wan t, in so me cases, to a ssume first that some substructure, e.g. the field of the sc alars, has stable computability , and still b e able to state something alike finite generation fo r the 5 other domains, e.g. the do main of the vectors. Thus, we sha ll consider structur es h E 1 , . . . , E m , E m +1 , . . . , E m + n , op 1 , . . . , op p , op p +1 , . . . , op p + q i with tw o kinds of domains, assume that the structure h E 1 , . . . , E m , op 1 , . . . , op p i has stable com- putabilit y , and co nsider ter ms that generate the elements o f E m +1 , . . . , E m + n with the op er ations op p +1 , . . . , op p + q . In these terms, the elements of E 1 , . . . , E m are ex pressed by their index which is itself expressed using the symbo ls 0 and S , in order to k eep the langua g e finite. The cas e m = 0 is the usual defini- tion of finite generation, whereas in the case m > 0 w e say that the alg ebraic structure h E m +1 , . . . , E m + n , op p +1 , . . . , op p + q i is finitely generated relatively to h E 1 , . . . , E m , op 1 , . . . , op p i , whic h in turn may or may no t be finitely g enerated. Definition 5 (T erms, denotation). L et E 1 , . . . , E m b e sets e quipp e d with in- dexings i 1 , . . . , i m , E m +1 , . . . , E m + n b e sets, A = { a 0 , . . . , a d − 1 } b e a finite s et of elements of E m +1 , . . . , E m + n , and op p +1 , . . . , op p + q b e op er ations whose ar- guments ar e in E 1 , . . . , E m , E m +1 , . . . , E m + n but whose values ar e in E m +1 , . . . , E m + n . The set T k of terms o f sort E k is inductively define d as fol lows – if the natur al n umb er x is an element of dom ( i k ) (for k in 1 , . . . , m ), then S x (0) is a term of s ort E k , – if a is an element of A and E k , then a is a term of sort E k , – if t 1 is a term of sort E k 1 , . . . , t l is an term of sort E k l and op is one of the functions fr om op p +1 , . . . , op p + q fr om E k 1 × . . . × E k l to E k l +1 then op ( t 1 , . . . , t l ) is a term of sort E k l +1 . The denotation of a term of sort E k is t he element of E k define d as fol lows – J S x (0) K E k = i k ( x ) , – J a K E k = a , – J op ( t 1 , . . . , t l ) K E k l +1 = op ( J t 1 K E k 1 , . . . , J t l K E k l ) . As a term is a tr e e lab ele d in the finite set a 0 , . . . , a d − 1 , op p +1 , . . . , op p + q , 0 , S , we c an asso ciate an index p t q t o e ach term t in a c anonic al way. Definition 6 (Finite ge n e rativ e set). L et E 1 , . . . , E m b e a family of sets e quipp e d with indexings i 1 , . . . , i m and let h E m +1 , . . . , E m + n , op p +1 , . . . , op p + q i b e a s tructur e. A finite set A of elements of the sets E m +1 , . . . , E m + n is said to b e a finite g enerative set of t he s tructur e h E m +1 , . . . , E m + n , op p +1 , . . . , op p + q i r elatively to E 1 , . . . , E m , i 1 , . . . , i m , if ther e ex ist effe ctively enumer able subset s T m +1 , . . . , T m + n of the sets T m +1 , . . . , T m + n of terms of sort E m +1 , . . . , E m + n , such that for e ach element b of a set E k ther e exists a term t in T k such that b = J t K E k and, for e ach k , ther e exists a c omputable function eq k such that for al l t and u in T k , eq k ( p t q , p u q ) = 1 if J t K = J u K and eq k ( p t q , p u q ) = 0 otherwise. Definition 7 (Finitely g enerated). L et E 1 , . . . , E m b e a family of sets e quipp e d with indexings i 1 , . . . , i m . The structur e h E m +1 , . . . , E m + n , op p +1 , . . . , op p + q i is said t o b e finit ely gener ate d r elatively to E 1 , . . . , E m , i 1 , . . . , i m , if it has a fin ite gener ative set r elatively to E 1 , . . . , E m , i 1 , . . . , i m . 6 R emark 1. This notion o f finite genera tion generalizes to arbitra ry structures the notion of finite-dimensio nal vector space. More gener a lly , we ca n define the dimension of a structure as the minimal ca rdinal of a finite generative set of this structure. Theorem 1. Le t h E 1 , . . . , E m , op 1 , . . . , op p i b e a structu r e with st able c omputabil- ity and s 1 , . . . , s m b e an admissible indexing of this stru ct ur e. The n, if the structur e h E m +1 , . . . , E m + n , op p +1 , . . . , op p + q i is finit ely gener ate d r elatively to E 1 , . . . , E m , s 1 , . . . , s m , then c omput ability is st able over t he structu r e h E 1 , . . . , E m , E m +1 , . . . , E m + n , op 1 , . . . , op p , op p +1 , . . . , op p + q i . Pr o of. W e first prove tha t there exists an admissible indexing of this structure. If k is an e le men t o f 1 , . . . , m , we let i k be the function mapping the index p S n (0) q to J S n (0) K E k = s k ( n ). By co nstruction, the domain of i k is effectively enum era ble, i k is surjective, and equality is decidable. If k is an element of m + 1 , . . . , m + n , we let i k be the function mapping the index p t q of the term t o f T k to J t K E k . The domain of i k is the set o f indices of elements of T k , thus it is effectively enumerable, for ev ery ele ment b of E k , ther e exists a term t of T k such tha t b = J t K E k , th us the function i k is sur jective. Fina lly , equa lit y is decidable b y hypothesis. Thus, i 1 , . . . , i m + n is an indexing of the structure h E 1 , . . . , E m + n , op 1 , . . . , op p + q i . Let us prov e it is admissible. Let op be one of the functions op 1 , . . . , op p . As s 1 , . . . , s m is an admissi- ble indexing of h E 1 , . . . , E m , op 1 , . . . , op p i , there exists a function ˆ op such that s l +1 ( ˆ op ( x 1 , . . . , x l )) = op ( s 1 ( x 1 ) , . . . , s l ( x l )). W e define the function ˜ op as the function mapping p S x 1 (0) q , . . . , p S x l (0) q to p S ˆ op ( x 1 ,...,x l ) (0) q . Let y 1 be an el- ement o f dom ( i k 1 ), . . . , y l be an element of dom ( i k l ), ther e exists x 1 , . . . , x l such that y 1 = p S x 1 (0) q , . . . , y l = p S x l (0) q and we hav e i k l +1 ( ˜ op ( y 1 , . . . , y l )) = i k l +1 ( ˜ op ( p S x 1 (0) q , . . . , p S x l (0) q )) = i k l +1 ( p S ˆ op ( x 1 ,...,x l ) (0) q ) = s k l +1 ( ˆ op ( x 1 , . . . , x l )) = op ( s k 1 ( x 1 ) , . . . , s k l ( x l )) = op ( i k 1 ( p S x 1 (0) q ) , . . . , i k l ( p S x l (0) q )) = op ( i k 1 ( y 1 ) , . . . , i k l ( y l )). Thus, the op er ations op 1 , . . . , op p are computable. Now, if op is one of the functions op p +1 , . . . , op p + q , we let ˜ op b e the computable func- tion mapping x 1 , . . . , x l to p op q ; x 1 ; . . . , x l ; 0 . Notice that this function maps the indices p t 1 q , . . . , p t l q to the index p op ( t 1 , . . . , t l ) q . Let x 1 , . . . , x l be ele- men ts of dom ( i k 1 ) , . . . , dom ( i k l ). Ther e exists terms t 1 , . . . , t l such that x 1 = p t 1 q , . . . , x l = p t l q . W e hav e i k l +1 ( ˜ op ( x 1 , . . . , x l )) = i k l +1 ( ˜ op ( p t 1 q , . . . , p t l q )) = i k l +1 ( p op ( t 1 , . . . , t l ) q ) = J op ( t 1 , . . . , t l ) K = op ( J t 1 K , . . . , J t l K ) = op ( i k 1 ( p t 1 q ) , . . . , i k l ( p t l q )) = op ( i k 1 ( x 1 ) , . . . , i k l ( x l )). Thus, the op era tions op p +1 , . . . , op p + q are computable too . Hence, the indexing i 1 , . . . , i m + n is a dmissible. If j 1 , . . . , j m + n is an arbitra r y admissible indexing o f the structure h E 1 , . . . , E m + n , op 1 , . . . , op p + q i with computable functions ˜ op 1 , . . . , ˜ op p + q . The index ing j 1 , . . . , j m is an admiss ible indexing of the s tructure h E 1 , . . . , E m , op 1 , . . . , op p i . Thu s, ther e exists computable functions g 1 , . . . , g m such that s k = j k ◦ g k . F or k in 1 , . . . , m , we define the computable function h k as the function mapping p S x (0) q to g k ( x ). F or k in m + 1 , . . . , m + n , we define the computable functions h k by induction: 7 – the v alue of h k on the index of a consta nt a o f A is the lea st x such that j k ( x ) = a , – the v a lue of h k on the index p op ( t 1 , . . . , t l ) q is ˜ op ( h k 1 ( p t 1 q ) , . . . , h k l ( p t l q )). Then, we prove b y induction over term structure that, for all ter ms t of sort E k , j k ( h k ( p t q )) = i k ( p t q ). If t has the form S x (0) then w e have j k ( h k ( p t q )) = j k ( h k ( p S x (0) q )) = j k ( g k ( x )) = s k ( x ) = i k ( p S x (0) q = i k ( t ). If t is a c on- stant a , w e have j k ( h k ( p a q )) = a = J a K E k = i k ( p a q ) and if t has the form op ( t 1 , . . . , t l ), for some op er ation op , we have j k l +1 ( h k l +1 ( p op ( t 1 , . . . , t l ) q )) = j k l +1 ( ˜ op ( h k 1 ( p t 1 q ) , . . . , h k l ( p t l q ))) = op ( j k 1 ( h k 1 ( p t 1 q )) , . . . , j k l ( h k l ( p t l q ))) = op ( i k 1 ( p t 1 q ) , . . . , i k l ( p t l q )) = op ( J t 1 K , . . . , J t l K ) = J op ( t 1 , . . . , t l ) K = i k l +1 ( p op ( t 1 , . . . , t l ) q ). Thus, for all elemen ts of the domain o f i k , j k ( h k ( x )) = i k ( x ), i.e. j k ◦ h k = i k . Finally , if j 1 , . . . , j m + n and j ′ 1 , . . . , j ′ m + n are t wo indexings of the struc- ture h E 1 , . . . , E m + n , op 1 , . . . , op p + q i . There exists functions h 1 , . . . , h m + n and h ′ 1 , . . . , h ′ m + n such that for all k , j k ◦ h k = i k and j ′ k ◦ h ′ k = i k . Thus, for all k , j k ◦ h k = j ′ k ◦ h ′ k . Let h ′′ 1 , . . . , h ′′ m + n be rig ht inv erses of h ′ 1 , . . . , h ′ m + n W e hav e j ′ k = j k ◦ ( h k ◦ h ′′ k ). Hence, computability is s table ov er the structur e h E 1 , . . . , E m + n , op 1 , . . . , op p + q i . As co rollarie s , w e get stable computability for well-known cas es. Prop ositio n 4 (Natural n um b ers). Computability is s t able over the st ruc- tur es h N , S i and h N , + i . Pr o of. Co nsider the set of ter ms in the language 0 , S . E ach natur a l num ber is denoted by a term. Mo r eov er, as each natura l num b er is deno ted b y a unique term, equality o f deno tations is trivia l. Th us, the struc tur e h N , S i is finitely generated and computability is stable ov er this structure. Consider the set of terms in the lang uage 0 , 1 , + and its subsets of terms of the form 1 + (1 + . . . + (1 + 0 ) . . . ). E a ch natural num b er is denoted b y a term. Moreov er, as each natural num ber is deno ted by a unique ter m, e q uality of denotatio ns is triv ia l. Thus, the struc tur e h N , + i is finitely generated a nd computability is stable ov er this structure . R emark 2. F or the s tr ucture h N , + i we co uld als o hav e considere d all the terms in the languag e 0 , 1 , +, in whic h case we would have had to pr ovide an algor ithm to test the equa lity of denotations of tw o such terms. Prop ositio n 5 (Rational num b ers). Computability is stable over the struc- tur e h Q , + , − , × , / i . Pr o of. The structure h Q , + , − , × , / i is finitely genera ted. C o nsider all ter ms of the form ( p − q ) / (1 + r ) where p , q and r hav e the form 1 + (1 + . . . + (1 + 0) . . . ), and either p or q is 0, if b o th ar e then r = 0 also, and p − q and 1 + r are rela tively prime otherwis e. Each r ational num b er is denoted by a term. Mor eov er, as each rational num ber is denoted by a unique term, equa lit y of denotations is trivial. Thu s, the structur e h Q , + , − , × , / i is finitely generated and computability is stable ov er this structure. 8 Prop ositio n 6. Computability is stable over the structure h Q , + , ×i . Pr o of. W e prove that the indexings a dmissible for h Q , + , × i and h Q , + , − , × , / i are the same. If the indexing i is admissible for h Q , + , − , × , / i then it is o b viously admissible for h Q , + , ×i . Conv ersely , if i is a dmissible for h Q , + , ×i , then there exist computable functions ˆ + and ˆ × , from N 2 to N , such tha t i ( x ˆ + y ) = i ( x ) + i ( y ) and i ( x ˆ × y ) = i ( x ) × i ( y ). The set dom ( i ) is non empty and r ecursively enum era ble, thus it is the image of a computable function g . L e t n a nd p b e tw o natural num b ers, we define n ˆ − p = g ( z ) where z is the least natural num ber such that p ˆ + g ( z ) = n . W e hav e i ( x ˆ − y ) = i ( x ) − i ( y ). W e build the function ˆ / in a similar way . Thus, i is admissible for h Q , + , − , × , / i . R emark 3. Computability is stable ov er the structure h Q , + , ×i , but this struc- ture is not finitely gener a ted. Indeed, co nsider a finite num ber o f rational num- ber s a 0 = p 0 /q 0 , . . . , a d − 1 = p d − 1 /q d − 1 and call q a common m ultiple of the denominators q 0 , . . . , q d − 1 . All n umbers that are the denotation of a term in the language a 0 , . . . , a d − 1 , + , × hav e the form p/q k for some p a nd k . The num b ers q + 1 and q are r e la tively prime, thus q + 1 is no t a divisor of q k for any k , the nu mber 1 / ( q + 1 ) do es not hav e the form p/ q k , and it cannot be expressed by a term. Thus, finite genera tion is a sufficient condition for stable computability , but it is not a necessary one . The sta bilit y o f computability ov er the structure h Q , + , ×i can b e explained by the fact that, a lthough subtr a ction and divisio n are no t o pe r ations of the structure, they can b e e ffectively defined from these op eratio ns . Such a structure is said to b e effe ctively gener ate d . There a re several wa ys to define this notion. F or instance, [20] pro po ses a definition based on a simple imp era tive progra mming language with the op er- ations o f the algebra ic structure as pr imitiv e. A mor e abstr act definition is the existence of a language, that needs not b e the language of terms built with op er - ations on the s tructure, but may , fo r instance, b e a progra mming la nguage, and a denota tion function a s so ciating an element of the alg e br aic structure to each expression of the language, verifying the following prop erties: – the set of well-formed expressio ns is effectively enumerable, – each element is the denotation of some expressio n, – equality of denotations is decidable, – an express io n denoting op ( a 1 , . . . , a n ) can b e built fr om ones denoting a 1 , . . . , a n , – for any admis s ible indexing, an index of a can b e computed fro m an expres- sion denoting a . Replacing expressions b y their indice s , w e get this wa y exactly the definition of stable co mputability . Thus, stability of computability a nd effective generation are trivially equiv alen t in this approach. 9 4 V ector spaces Prop ositio n 7 (V ector spaces). If c omputability is stable over the field h K, + , ×i and t he ve ctor sp ac e h K, E , + , × , + , . i is finite-dimensional, then c omputabil- ity is stable over h K, E , + , × , + , . i . Pr o of. Let s be an indexing of the field h K , + , × i and e 0 , . . . , e d − 1 be a basis of E . Consider the set o f terms of the form ( S λ 0 (0) .e 0 +( S λ 1 (0) .e 1 +( . . . S λ d − 1 (0) .e d − 1 ))) where λ 0 , . . . , λ d − 1 are in dom ( s ). The denotation of such a term is the vector s ( λ 0 ) .e 0 + s ( λ 1 ) .e 1 + . . . + s ( λ d − 1 ) .e d − 1 . As the function s is surjective and e 0 , . . . , e d − 1 is a basis of E , each element of E is the deno tation of such a term. Moreov er, equality of de no tations ca n b e decided on scalar s, it can b e decided on such terms. Thus, the structure h E , + , . i is finitely generated rela tively to h K, + , ×i a nd co mputabilit y is stable ov er h K, E , + , × , + , . i . Prop ositio n 8. Computability is not stable over the structure h N i . Pr o of. Let f b e a non computable o ne-to-one function from N to N , for instance the function mapping 2 n to 2 n + 1 and 2 n + 1 to 2 n if n ∈ U a nd 2 n to 2 n and 2 n + 1 to 2 n + 1 otherwise , where U is any undecida ble set. The function f and the iden tity are b oth admissible indexings o f the structure h N i . If computability were sta ble over this structure, then there would exist a computable function h such that f = id ◦ h . Thus, f = h would b e computable, which is contradictory . Prop ositio n 9. Computability is stable over the ve ctor sp ac e h K , E , + , × , + , . i if and only if it is stable over the fi eld h K , + , ×i and h K, E , + , × , + , . i is a fin ite- dimensional ve ctor sp ac e. Pr o of. If the dimension of E is not co un table, then the set E itself is not count- able, and hence co mputabilit y is not stable ov er this structur e. If the dimension of E is finite o r co unt able and c omputability is not stable ov er the field h K, + , ×i , then it is not stable over the vector spac e h K, E , + , × , + , . i either. Indeed, if s a nd s ′ are tw o no n equiv ale n t indexings of h K , + , × i , and e 0 , e 1 , . . . is a finite or countable basis of E , then, we let i b e the function ma p- ping p ( S λ 0 (0) .e 0 + ( S λ 1 (0) .e 1 + ( . . . S λ n (0) .e n ))) q where λ 0 , λ 1 , . . . , λ n are in dom ( s ) to s ( λ 0 ) .e 0 + s ( λ 1 ) .e 1 + . . . + s ( λ n ) .e n and i ′ be the function ma pping p ( S λ 0 (0) .e 0 + ( S λ 1 (0) .e 1 + ( . . . S λ n (0) .e n ))) q where λ 0 , λ 1 , . . . , λ n are in dom ( s ′ ) to s ′ ( λ 0 ) .e 0 + s ′ ( λ 1 ) .e 1 + . . . + s ′ ( λ n ) .e n , and s, i and s ′ , i ′ are tw o non e quiv a le n t indexings of h K, E , + , × , + , . i . W e ha ve prov ed in Prop osition 7 that if computabilit y is stable ov er the field h K , + , × i and the vector space h K, E , + , × , + , . i is finite- dimensional, then computability is stable over this s pace. Thus, a ll that remains to b e proved is that if co mputabilit y is s table ov er the field h K , + , ×i a nd the vector s pa ce h K, E , + , × , + , . i ha s a countably infinite dimension, then computability is not stable ov er this vector space. Let s b e an indexing of h K, + , ×i , e 0 , e 1 , . . . b e a basis of E , and i b e the func- tion mapping p ( S λ 0 (0) .e 0 + ( S λ 1 (0) .e 1 + ( . . . S λ n (0) .e n ))) q where λ 0 , λ 1 , . . . λ n 10 are in dom ( s ) to s ( λ 0 ) .e 0 + s ( λ 1 ) .e 1 + . . . + s ( λ n ) .e n . As the function s is surjective and e 0 , e 1 , . . . is a ba sis, the function i is sur jective. As there exist computable functions ˆ + and ˆ × a nd eq o n sc a lars, we ca n build computable functions ˆ + and ˆ . and e q on such terms. Thus, the function i is an a dmissible index ing of E . Then, let f b e a non co mputable one-to- one function from N to N . Let φ b e the one- to-one linear function fro m E to E mapping the basis vector e p to e f ( p ) for all p . As φ is a one-to- one ma pping betw een E and itse lf, φ ◦ i is a surjection from dom ( i ) to E . As φ is line a r, φ ( i ( u ˆ + v )) = φ ( i ( u ) + i ( v )) = φ ( i ( u )) + φ ( i ( v )) and φ ( i ( λ ˆ . u )) = φ ( s ( λ ) .i ( u )) = s ( λ ) .φ ( i ( u )). Thus, φ ◦ i is an admiss ible indexing of E . And if computability were sta ble ov er h K, E , + , × , + , . i , there would exists a c omputable function g from dom ( i ) to dom ( i ) such that φ ◦ i = i ◦ g . Let z b e any num be r such that s ( z ) is the scalar 0 a nd u b e any num ber such that s ( u ) is the scalar 1. Let B b e the computable function, from N to N , mapping p to p S z (0) .e 0 + . . . + S z (0) .e p − 1 + S u (0) .e p q . W e have i ( B ( p )) = e p . Let C b e the partia l computable function from N to N mapping the index p ( S λ 0 (0) .e 0 +( S λ 1 (0) .e 1 +( . . . S λ n (0) .e n ))) q to the least p such that eq ( λ p , u ) = 1. If i ( x ) = e p then C ( x ) = p . Let h be the computable function C ◦ g ◦ B . Let p be an arbitr ary natural n umber. W e hav e i ( g ( B ( p ))) = φ ( i ( B ( p ))) = φ ( e p ) = e f ( p ) . Thu s, h ( p ) = C ( g ( B ( p ))) = f ( p ). Thus, f = h would be computable, which is contradictory . 5 Field extensions Using Prop ositio n 7, we get tha t if co mputabilit y is sta ble ov er the field h K , + , ×i and the field h L, + , × i is a n extensio n of finite degree of this field, then com- putabilit y is stable over the structure h K , L, + , × , + , . i where . is the pro duct o f an e le ment of L by a n element o f K . W e can easily pr ov e that computability is a lso stable o ver the structure h K, L, + , × , + , ., ×i where the multiplication of L is added. Prop ositio n 10 (Field extension of finite deg ree). If c omputability is sta- ble over the field h K , + , ×i , and the field h L, + , × i is an extension of finite de gr e e of h K, + , ×i , t hen c omputability is stable over the structur e h K , L , + , × , + , ., ×i . Pr o of. Let s b e a n indexing of the field h K, + , ×i , e 0 , . . . , e d − 1 be a basis of L , and i b e the function mapping p ( S λ 0 (0) .e 0 + ( S λ 1 (0) .e 1 + ( . . . S λ d − 1 (0) .e d − 1 ))) q , where λ 0 , . . . , λ d − 1 are in dom ( s ), to s ( λ 0 ) .e 0 + s ( λ 1 ) .e 1 + . . . + s ( λ d − 1 ) .e d − 1 . As the function s is sur jective and e 0 , . . . e d − 1 is a basis of E , the function i is surjective. As there exis t computable functions ˆ + and ˆ × and e q on elements of K , we c a n build computable functions ˆ + and ˆ . and eq on s uch terms. Thus, s , i , is an admissible indexing o f the s tructure h K, L, + , × , + , . i . Let us prov e that it is also an admis s ible indexing of the structur e h K , L , + , × , + , ., ×i . The elements e p × e q of L can b e written in a unique wa y Σ r m p,q,r e r . Let µ p,q,r be natural num ber s such that s ( µ p,q,r ) = m p,q,r . Let ˆ × L be the co m- putable function mapping p ( S λ 0 (0) .e 0 + ( S λ 1 (0) .e 1 + ( . . . S λ d − 1 (0) .e d − 1 ))) q and p ( S λ ′ 0 (0) .e 0 + ( S λ ′ 1 (0) .e 1 + ( . . . S λ ′ d − 1 (0) .e d − 1 ))) q to p ( S λ ′′ 0 (0) .e 0 + ( S λ ′′ 1 (0) .e 1 + 11 ( . . . S λ ′′ d − 1 (0) .e d − 1 ))) q where λ ′′ r = ˆ Σ p,q λ p ˆ × K λ ′ q ˆ × K µ p,q,r . It is routine to chec k that if x and y are in the dom ( i ), then i ( x ˆ × L y ) = i ( x ) × i ( y ). Th us, s, i is an admissible indexing of the s tructure h K, L, + , × , + , ., ×i a s well. Finally , if s, j and s ′ , j ′ are tw o admissible indexing s of h K , L , + , × , + , ., ×i , then they are obviously admissible index ings o f h K , L , + , × , + , . i a s well, thus there e xists computable functions h a nd k s uch that s ′ = s ◦ h and j ′ = j ◦ k . Now, we would like to prov e that computability is stable over the field h L, + , × i . B ut, proving this r esult seems to requir e an extra h yp othesis, that K is an effectively enum era ble subset of L , i.e. that if i is a n admissible indexing of L , we ca n e n umerate the indices of the elements of K . As this is alwa ys the case when K is the field Q , and such ex tensions of finite degree of the field Q a re the fields used in q ua nt um computing, we shall restric t to this particular case. Prop ositio n 11 (Exension of finite de g ree of the field Q ). If the field h L, + , × i is an extension of finite de gr e e of t he field h Q , + , ×i , then c omputability is st able over h L, + , ×i . Pr o of. W e have prov ed in Pr op osition 10, the existence o f an admissible indexing s, i o f the structure h K , L , + , × , + , ., ×i . The function i is obviously an admissible indexing of the field h L, + , ×i . Now consider a n arbitrar y admissible indexing j o f this structure and let us prov e that j − 1 ( Q ) is effectively enumerable. The set dom ( j ) is recur sively enu- merable, thus let g b e a genera ting function, we define ˆ − as the function mapping n and p to g ( z ) where z is the lea st natur a l num b er suc h that p ˆ + g ( z ) = n , and we define ˆ / in a similar way . Let z b e a natura l num ber suc h that j ( z ) = 0 and u be a natural n umber suc h that j ( u ) = 1. Let J be the computable function fro m N to dom ( j ) defined by J (0) = z and J ( p + 1) = J ( p ) ˆ + u . Let f be the c om- putable function, from N 3 to N , mapping p , q and r to ( J ( p ) ˆ − J ( q )) ˆ / ( J ( r ) ˆ + u ). The set j − 1 ( Q ) is the image of this function. Thus, it is effectively enu mera ble. Let j a nd j ′ be tw o indexings of h L, + , ×i . The functions j | j − 1 ( Q ) , j are an indexing of h Q , L, + , × , + , ., × i and j ′ | j ′− 1 ( Q ) , j ′ also. Th us, using Prop ositio n 10, there e xists a computable function k such that j ′ = j ◦ k . 6 T ensor spaces In Pro p o sition 9, we hav e shown that infinite-dimensiona l vector s paces do not hav e a s table no tion of computability . Intuit ively , the lack stable computability for infinite-dimensional vector spaces happens for the same reason as it do es for h N i , as ca n b e seen fr om the pr o ofs o f Pro po sition 8 and 9. This is b eca use neither algebra ic structures ar e effectively generated, i.e. there is an infinity of elements (the natural num b ers in one cas e, the vectors of a basis in the other) which are unrelated from one another. In the cas e o f the natura l n umbers this can b e fixed b y req uiring that the successor b e computable, i.e. by co nsidering the structure h N , S i . On the set of finite sequences o f element s taken in a finite s et, the problem would be fixed in 12 a similar way by a dding the c ons op eration, that adds an element at the head of the list. On an infinite set of trees, we would add the op er ation that builds the tree f ( t 1 , . . . , t n ) fro m f and t 1 , . . . , t n , . . . These o pe r ations expres s that, although these data t yp es are infinite, they are finitely generated. In the sa me way , in quantum computing, an infinite data type co mes with some structure, i.e. the vector space used to describ e these da ta has a basis that is finitely gener ated with the tensor pro duct. Typically , the basis vectors hav e the form b 1 ⊗ . . . ⊗ b n where each b i is either | 0 i or | 1 i . Definition 8 (T ensor space). A tensor space h K , E , + , × , + , ., ⊗i is a ve ctor sp ac e with an extr a op er ation ⊗ that is a biline ar fun ct ion fr om E × E to E . Theorem 2. Le t h K , E , + , × , + , ., ⊗i b e a tensor sp ac e such that c omputability is stable over t he fi eld h K , + , ×i and ther e ex ists a fi nite subset A of E such that the set of ve ctors of the form b 1 ⊗ ( b 2 ⊗ . . . ⊗ ( b n − 1 ⊗ b n ) . . . ) , for b 1 , . . . , b n in A , is a b asis of E . Then, c omputability is stable over the struct u r e h K , E , + , × , + , ., ⊗i . Pr o of. Let s b e a n indexing of the field h K , + , ×i . Co nsider the ter ms of the for m ( S λ 0 (0) . ( b 0 1 ⊗ . . . ⊗ b 0 n 0 )+ ( S λ 1 (0) . ( b 1 1 ⊗ . . . ⊗ b 1 n 1 )+ . . . where λ 0 , λ 1 , . . . are in dom ( s ) and b 0 1 , b 0 2 , . . . are in A . The denotation of such a term is the vector s ( λ 0 ) . ( b 0 1 ⊗ . . . ⊗ b 0 n 0 ) + s ( λ 1 ) . ( b 1 1 ⊗ . . . ⊗ b 1 n 1 ) + . . . As s is s urjective and the vectors b 0 1 ⊗ . . . ⊗ b 0 n 0 , . . . form a ba sis, every element of E is the denotation of a term. As equality of denotations can b e decided on scalar s, it can b e decided on terms. Thus, the structure h E , + , ., ⊗i is finitely g enerated rela tively to the field h K , + , × i . As computability is stable ov er this field, it is stable ov er h K , E , + , × , + , ., ⊗i . 7 Conclusion The robustness of the no tion of computability over the natura l num b ers has bee n p ointed out for lo ng. But it has als o be en p ointed out that this robustness do es not extend to other countable do mains, where computability is relative to the c hoice of an indexing. W e hav e s hown tha t this robustness is, in fact, shar ed by many alge br aic str uc tur es: all the extensio ns of finite degr ee of the field of rationals ( e.g. Q [ √ 2] a nd Q [ i, √ 2]), a ll the finite- dimens io nal vector spac es ov er such a field, and all tensor spa ces ov er such a field that are finite-dimensio nal (as tensor spaces) even if they a re infinite-dimensional (as v ector spaces). F or the vector spac e s used in quantum computing, it is no t the dimens io n a s a vector space that ma tter s, but the dimension as a tensor space. Indeed, the vector space op er ations handle the sup erp osition principle, but the finite gener ation of the data types is handled by the tensor pr o duct. Finite-dimensio nal tensor space ov er ex tensions of finite degr ee of the field of ra tionals are probably sufficient to express all quantum algorithms. 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