Using Cloning to Solve NP Complete Problems

Transcription

1 Using Cloning to olve P Complete Problems Jon. Draopoulos an Teoore. Tomaras Department of Pysics, University of Crete P.O.Box 08, Heralion 7003, Crete, Greece. Tel , Fax ttp:// [email protected] BTRCT ssuming a cloning oracle, satisfiability, wic is an P complete problem, is sown to belong to BPP C an BQP C (epening on te ability of te oracle C to clone eiter a binary ranom variable or a qubit. Te same result is extene in te case of an approximate cloning oracle, tus establising tat P BPP C BQP C an P BPP C BQP C, were C an C are te exact an approximate cloning oracles respectively. ltoug exact cloning is impossible in quantum systems, approximate cloning remains a possibility. However, te best nown metos for approximate cloning (base on unitary evolution o not currently acieve te esire precision levels. n it remains an open question weter tey coul be improve wen non-linear (or non-unitary operators are use. Finally, a straigtforwar attempt to ispense wit cloning, replacing it by unitary evolution, is prove to be impossible.. Introuction Quantum computing is a new an exciting interisciplinary area tat combines computer science an (quantum pysics. s early as 98, Feynman observe tat te straigtforwar simulation of a quantum system on a classical computer (eterministic or probabilistic Turing macine require an exponential slowown an witout any apparent way to spee up te simulation [, ]. He ase weter tat is inerent to quantum systems an e suggeste te esign of computing macines base on quantum teory implying, at te same time, tat suc quantum computers coul peraps compute more efficiently tan classical computers. bout te same time, an aressing te opposite problem, Benioff sowe tat a eterministic Turing macine coul be simulate by te unitary evolution of a quantum process an tus provie te first inication of te strengt of quantum computing [, ]. ubsequently, Deutsc propose a general moel of quantum computation--te quantum Turing macine--wic coul simulate any given quantum system but possibly wit exponential slowown [8]. Berstein an Vazirani improve upon te concept of a quantum Turing macine proposing a univesal quantum Turing macine, wic, as tey prove, coul simulate a broa class of quantum Turing macines wit only a polynomial slowown [4]. In classical computing, logic circuits provie an alternative to Turing macines (actually current computers are buil from integrate circuits an not as Turing macines. Deutsc first propose a similar moel of quantum circuits, wic e

2 calle quantum computational networs, an examine some of teir properties [9]. Yao subsequently prove te polynomial equivalence between quantum Turing macines an quantum circuits (tus proviing a truly universal moel of quantum computation by proving tat an arbitrary quantum Turing macine coul simulate, an be simulate by, a polynomial size quantum circuit [0]. Te moel of quantum computing aving been efine, its computational power must be ientifie. Te following classes of ecision problems ave been efine: P is te class of all ecision problems tat can be solve in polynomial time by a eterministic Turing macine BPP is te class of all ecision problems tat can be solve in polynomial time by a probabilistic Turing macine wit a probability of error boune by /3 for all inputs P is te class of all ecision problems tat can be solve in polynomial time by a non-eterministic Turing macine BQP is te class of all ecision problems tat can be solve in polynomial time by a quantum Turing macine wit a probability of error boune by /3 for all inputs (pparently it is P BPP P. Furtermore, Benniof's result [, ] implies tat P BQP. Deutsc an Josza [] an Bertiaume an Brassar [0, 6] prove tat, relative to certain oracles, tere are computational problems tat can be solve exactly an in polynomial time by quantum Turing macines but cannot be solve polynomially for all inputs by eterministic or probabilistic Turing macines. However, tose problems belong to BPP; an tus te above results o not confer te suppose extra computing power of quantum Turing macines. Bennett et al. [3] prove tat relative to a ranom oracle, it is not true tat P BQP. Bernstein an Vazirani [4] prove tat BPP BQP. Bernstein an Vazirani [4] an imon [8] invente problems tat are not nown to be in BPP but belong to BQP. or gave polynomial time quantum algoritms for te factoring an iscrete log problems [7]. (ote tat not only imon's problem but also factoring an iscrete log belong to P co-p. However, Bennett et al. [3] prove tat relative to a permutation oracle, it is not true tat P co-p BQP. Finally Grover sowe ow to accept te class P relative to any oracle in time O( n/. ( formal analysis of Grover's algoritm appears in [7]. It soul be note tat or's factoring an iscrete log algoritms (wit its implications in cryptograpy an cryptosystems an Grover's atabase searc algoritm are of practical importance. Furtermore, te application of quantum computing for te evelopment of secure cryptograpic communication systems (wic etect unautorize access or guarantee tat information woul not be compromise [5] is obviously of ig commercial value. It is still unnown weter BPP BQP, an weter P BQP. (Te latter is tremenously important since te class P contains a large number of optimization problems wit a broa range of applications from computer science an statistics to engineering, automatic control, integrate circuit esign etc. In tis paper, we provie polynomial algoritms for solving te satisfiability problem, wic is a nown P complete problem, assuming an oracle tat can (eiter exactly or approximately clone a binary ranom variable or a qubit. Terefore, we prove tat P BPP C BQP C an P BPP C BQP C, were C( an C( are te exact an approximate cloning oracles as efine below. uc oracles, wic o

3 not merely provie a Boolean reply to a query but generate an object (suc as a ranom variable or qubit, are calle manufacturing oracles. In section, we escribe te satisfiability problem. In section 3, we provie an algoritm for te case of exact cloning, wile, in section 4, we prove tat te algoritm belongs to BPP C (or BQP C respectively. similar construction is mae for te case of approximate cloning in section 5. In section 6, an attempt is mae to ispense wit cloning an a negative result is sown. Finally, in concluing section 7, te limitations an some practical implications are iscusse.. Te satisfiability problem Boolean variable x is a variable tat can assume only te values true an false, (wic are usually associate wit te numbers an 0 respectively. trut assignment is an assignment of true/false values to a set of Boolean variables. logical expression or Boolean formula is an expression consisting of Boolean variables combine wit logical connectives suc as or, an, not, etc. logical expression can be unsatisfiable (wen it is false for all trut assignments of its variables, satisfiable (wen it is true for at least one assignment of its variables, an tautological (wen it is true for all trut assignments of its variables. For example, assume tat + enotes logical isjunction (or, enotes logical conjunction (an, an a bar over an expression enotes te logical negation of te expression. Ten ( x + x 3 ( x + x + x 4 is a logical expression containing four variables an is true if an only if at least one of x an x 3 is false an at least one of x, x, an x 4 is true. Terefore, it is satisfiable but not tautological. literal is a Boolean variable or its negation, an a clause is a isjunction of literals. Te satisfiability problem is as follows: Given m clauses c 0,..., c m- containing n Boolean variables x 0,..., x n- is te formula c 0 c... c m- satisfiable? atisfiability is nown to be P-complete [6, c.8, teorem 8.]. Finally, if E is a logical expression an a trut assignment E/ is te logical expression tat results after we substitute te variables of E tat appear in wit teir assigne values. For example, if E a + b + c an {afalse, true} ten E/ b + c. 3. Te logical boosting algoritm We assume tat [ ] is te Boolean-to-integer operator (tat is [ϕ], if ϕ is true; [ϕ] 0, if ϕ is false. We enote probability measure by P(, an quantum an, or, an not gates by &,, an ~ respectively. binary ranom variable is a ranom variable tat can be eiter 0 or. qubit, wic is te corresponing unit of quantum information, beaves more or less lie a ranom variable an is escribe by a pair of complex numbers (a, b wose magnitues are te probabilities tat te qubit woul be in state 0 or state (respectively. ince tose are ifferent concepts, for te rest of tis paper, we sall use bot te term binary ranom variable an qubit to escribe situations were eiter object coul be use. Te subsequent results woul tus apply to probabilistic an quantum computing (respectively. 3

4 Te following algoritm assumes a function C(X tat can generate a copy (or clone of any binary ranom variable or qubit X. In oter wors, C(X returns a binary ranom variable or qubit tat is inepenent of X an ientically istribute. Te logical boosting (B( algoritm: Given a binary ranom variable or qubit copy function C(, a satisfiability problem of m clauses c 0,..., c m- an n Boolean variables x 0,..., x n- te algoritm ecies weter c 0 c... c m- is satisfiable as follows:. Create n inepenent binary ranom variables or qubits X 0,..., X n- (corresponing to Boolean variables x 0,..., x n- suc tat P(X i P(X i 0 /, for i 0,,...,n-.. Create m Boolean ranom variables or qubits C 0,..., C m- tat correspon to clauses c 0,..., c m- : If x i,..., x i are te Boolean variables appearing non-negate in c 0 a j an x,..., x 0 b are te Boolean variables appearing negate in c j ten C j X i... X (~ X... (~ X, j 0,,..., m. 0 ia 0 b 3. Create Boolean ranom variable or qubits D 0 as follows: D 0 C 0 &... & C m- 4. Coose n an create Boolean ranom variables or qubits D,..., D as follows: D v D v- C(D v-, v,,,. 5. oo at D. If it is, te initial problem is satisfiable; oterwise assume tat te problem is unsatisfiable. ote tat te algoritm requires &-gates an -gates wit large number of inputs (n coul be very large an m coul be even larger. However, tis situation can be easily remeie since an M-input &-gate or -gate can be constructe as a networ of M K-input &-gates or -gates respectively. K 4. Te probability of error an te time complexity of te B algoritm Teorem. Te B( algoritm asserts correctly tat a given satisfiability problem is satisfiable, an te probability of error P err, in te case it asserts tat te problem is unsatisfiable, is boune: P err < e were e is te Euler number (e , n is te number of Boolean variables, an is te boosting level cosen in te B algoritm. et u be any trut assignment of te Boolean variables x 0,..., x n- u' be te corresponing assignment of values 0 an to qubits X 0,..., X n- be te number of ifferent trut assignments tat satisfy ; tat is, n 4

5 [ / u] u v P(D v 0, v 0,,,, Ten, 0 P( D0 P( D0 / u' P( u' [ / u] n n u' u an since D v an C(D v are inepenent an ientically istribute: v+ P( Dv ( Dv 0 P( Dv 0 P( ( Dv 0 v, v 0,,, - s a result, n n n 0 < n n Te above inequality emerges from te fact tat te sequence am m, m > 0, is strictly increasing an converges to its supremum e. ow, observe tat if D is foun to be, ten P( D > 0 < 0 < P( D0 > 0 > 0 is satisfiable. Tus, te algoritm fins te correct answer in tis case. On te oter an, wen D is foun to be 0 an is unsatisfiable, te algoritm again guesses correctly. Te case of error is only wen D is foun to be 0 an is satisfiable. However, in tat case it is > 0 (or equivalently an tus te probability of error is P err n n e P( D 0 < Q.E.D. e e ltoug is left out as a free parameter in te B( algoritm, it nee not be significantly larger tan n. For example, if n + 6, ten P err <.6 * 0-8. s argue in [4, 4.5.4], te probability of an error in a computer circuit ue to arware malfunction or cosmic raiation is larger tan te above boun. Teorem. ssume a satisfiability problem wit m clauses of n Boolean variables. Furtermore assume tat we implement te B( algoritm using K-input quantum gates (K tat operate in time t K ; tat te copy function C( taes time t C ; an tat te time to create a uniformly istribute qubit is t q. Ten te time complexity of B( is T O( t n + t mlog n + ( t t B ( q K K K + C wic reuces to O( t n + t mlog n t n, wen (as expecte O(n. q K K + C et T i be te time consume by te i-t step of te algoritm (i,, 3, 4, 5. Ten it is: T O(t q n (to create te n qubits T O(t K m log K n (since we can implement an -gate of O(n inputs wit log K n layers of K-input -gates T 3 O(t K log K m (since we can implement an &-gate of m inputs wit log K m layers of K-input &-gates T 4 O((t C + t K (to copy an boost times sequentially T 5 O( (to loo up a given qubit ing te above times togeter Q.E.D. m 5

6 T etting t max{t q, t K, t C }, te time complexity of B( becomes O( t( mlog. ince soul be O(n an m woul usually be muc B ( + K n larger tan n, te complexity reuces to TB ( O( tmlogk n. n because t woul be a constant inepenent of n an m, te complexity of te B( algoritm is TB ( O( m log K n. Given tat satisfiability is P complete, an tat, epening on te nature of te cloning oracle, eiter binary ranom variables or qubits coul be use, te above teorems establis tat P BPP C BQP C. 5. pproximate cloning Unfortunately, as sown in [9], it is not possible to create a cloning function lie C( using linear evolution. ltoug, tere as been so far no suc proof for nonlinear systems, it is not very reasonable to believe--given te alreay proven fact tat cloning leas to solution of P complete problems-- tat non-linear systems capable of exact cloning woul be realizable (in te near future. Te above restrictions lea us to consier approximate cloning: a function C( tat can generate an approximate copy of any binary ranom variable or qubit X. Tat approximate copy soul ave te property tat P(X 0 - P(C(X 0 ε. In tat case, we woul say tat C(X is an ε-approximation of X an tat ε is te approximation egree of C(X. (ε woul normally be a small nonnegative number. Consequently, we can efine te approximate logical boosting algoritm (B( to be te B( algoritm were C( as been replace by C(. We sall now prove tat if ε is exponentially small (wrt n ten te error of B( is trivial an negligible. Teorem 3. Te B( algoritm asserts tat a given satisfiability problem is satisfiable or unsatisfiable wit a probability of error P err, boune: 7( n + 34 P err < max{, ( ε } were n 7 is te number of Boolean variables, is te boosting level cosen in te B algoritm, an ε -n- is te approximation egree of C(X. In orer to prove tis teorem, we must first prove te following two lemmas: emma. If { } is a sequence of real numbers satisfying te property tat n n 7( n an + ( + ε, were ε, n 7, ten it is <. pparently, + ε (te proof, by inuction on, is trivial. Terefore, { } is a ecreasing sequence. ow we sall prove tat n+ < 3 assuming tat n+ 3 an eriving a logical contraiction. Tus, if 3 ten, for every n+, it woul be 3 an: n+ 6

7 < ε ( + 3ε ( + 3ε 0 ( + 3ε { ( + } n n 3 {( ( + } n { } (by efintion of { (since (by inuction on (since (since ( n+ ( (since ( < 0 3 n } - < an ε < e e Te latter inequality, for n+, leas us to te esire contraiction. ow, te prove statement, n+ < 3, by repeate applications of te efining n inequality of { } an te fact tat ε, n 7, leas to te following bouns: n+ 3 < 4 ε, n+ 4 < 4 3ε, n+ 5 < 6 ε, n+ 6 < 56, n+ 6 + ε < 8 ; te last two now being use to complete tis proof: 6 ( + ε < <... + ( n+ 6 < ( n+ 6 7( n 34 Q.E.D. 8 8 emma. If { } is a sequence of real numbers satisfying te property tat 0 an + ( ε, ten it is ( ε (equality ols only for. We sall prove te lemma by inuction on : : 0 ( 0 ε ε (equality ols inee. : ( ε ( ε ( ε > 3ε (equality ols inee. Inuction step: ssume te mentione inequality for, an prove it for +: ε (by efintion of { } ow, it is P + err > P( D P ( { ( ε}{ ε} ( ( err,0 max + ( p + 0 / ε + P { P, P } err,0 err, err, 0 P( D 0 / > err, P( D / 0 P( 0 ε + ( > 0 P( err, p 8 ε > 0 + P( D *56 / -n- (by inuction ypotesis 0 P( 0 Q.E.D. were, P 0 is te probability of error wen is satisfiable, P is te probability of error wen is unsatisfiable, an p is te a-priori probability tat problem is unsatisfiable. Furtermore, efining { v } as in teorem ( v P(D v 0, v 0,,,,, we ave + P(D C(D 0 P(D 0P(C(D 0 ( +ε were -ε ε ε is te approximation error of C(D (at -t step. s a result ( -ε + ( +ε an tus, P err,0 P err, n 0 0 7( n + 34 ( ε (by lemma (by lemma Finally, we can substitute te above in te efinition of P err Q.E.D. n 6 consequence of te above teorem is tat, by selecting n +, ε, we get a trivial error: P < 50 [4, 4.5.4]. Furtermore, as it was sown in te above err 7

8 proof, n 6 n+ 4 < 4 3ε. Terefore, selecting n+4 an ε, we get a probability of error less tan ¼; wic means tat te B(n+4 algoritm is proof of membersip of satisfiability in BQP C (or BQP C, epening on te cloning oracle. However, te fact tat ε is exponentially small (wit respect to n is a very ar constraint. t te time of tis writing an to te best of te autors' nowlege, it is not nown weter suc ig precision approximate cloning is possible. Wen unitary evolution is use, provably optimal approximate cloning, starting from M ( + + qubits an generating M clones at te en, results in fielity (, or equivalently M + M precision n 6 ε M ( + [3]. Tat is very low wen compare to ε, wic is require by te B(n+4 algoritm. Te use of non-linearity may improve upon te situation (since cloning is a non-linear operation but it remains an open researc problem te maximum possible precision an its time complexity. Te fact tat cloning leas to solution of te P-complete problems only stresses its importance. On te oter an, if we consier low precision cloning, i.e. ε O(/p(n, were p(n is a polynomial of n, ten, since 0 can be as large as - -n, tere are no bouns on te probability of error (oter tan te trivial ones. However, it is possible, in tat case, to treat te algoritm in a probably approximately correct fasion; tat is to question weter P ( P err > δ < γ, for some small δ, γ. However, tat analysis is beyon te scope of tis paper an may be presente elsewere. 6. on-cloning logical boosting (fixe point Te uncertainty about ig precision approximate cloning leas to oter consierations: peraps we can transform D to D + ( 0,,, - using unitary evolution an logic. et + is te set of nonnegative real numbers, be te set of complex numbers, t inicate matrix an vector transpose, an z enote te magnitue of z. Eac qubit Q, represente by a vector of two complex numbers [q 0, q ] t ( q 0 P(Q 0, q P(Q, q 0 + q, correspons to a point (or vector in. Furtermore, t n if x [ x0,, x n ], we efine x x x [ ] t n an x x0,, x n n n n Finally, a function f : as a magnitue fixe point (mfp x + if an n only if y ( y x f ( y x. (pparently, if f as an mfp x, ten f ( x x. Te Unitary Boosting lgoritm (UB( is lie B( except tat, instea of using a qubit copy/cloning function, it transforms D to D + using unitary matrix U ( an logical circuit (, i.e. D + ( (U ( D P (, were P ( is a set of normalize p p + ( ( ( internal parameters (,, + p P P U. ote tat, if we esire tat UB( never errs wen it asserts tat is satisfiable, it must be wen 0 ; or, equivalently D [ ] t, 0 wen D [, 0] t 0 -- wic, in turn, means tat te overall transformation of D 0 to D as an mfp at [, 0] t. Tis later property is implie by te conition tat te transformation of D to D + (for all as an mfp at [, 0] t. However, as te following teorem sows, it is impossible to reuce in tat case. 8

9 Teorem 4. If D + (U D H, were D an D + are qubits ( D, D +, H is a unit-lengt vector of complex numbers ( H is te set of internal of ien parameters corresponing to qubits, U is a + + unitary matrix ( + + U, is an arbitrary logical circuit of + inputs, an te overall transformation as an mfp at [,0] t, ten +. If W is a matrix, let W i,j enote te i-t row an j-t column element of W, W j enote t te j-t column of W, an tus ( W i enote te vector correspoing to te i-t row of W. imilarly, if V is a vector, let V i be te i-t element of V. ow let { 0, R, R } be an ortonormal base of, were R, 0 H an form te matrix R0 R X t 0 0 R0 R t t X,,( X is an ortonormal base of +. (pparently, { } ( 0 ow efine UX -, or, equivalently, UX, an D [ a, b ] t. s a result, is unitary an t t UD H X [ a P, b H ] [ a,0,,0, b,0,,0] a 0 + b an if we inicate by T an F te sets of integers (in te interval [0, + -] wose binary representation correspons to logical assignments tat mae to yiel true or false, respectively, ten it woul be, P( ( UD H 0 (by te efinition of, D + 0 a i,0 a + b a + b ( a + b i + b i, i, i, + Y + Y + Y + Y (by te prev.result about UD (werey a b (assuming (assuming Y 0 (since a i,0 + i,0 i, + + H i F i, 0 To complete te proof, we must prove te premises an Y0 (bot of wic result from te fact tat our -U transformation as an mfp at [, 0] t. Inee, wen D [, 0] t t ten ( U[,0] H. Finally, te fact tat U + i F (an tus is unitary implies tat:.,0 +,0 i i i,0 0 i T i, 0 i,0 ( 0 0 i T i T. 0 0,0 +,0 0,0 0 0 i, Y i i i, i i, i T Q.E.D. 9

10 Te above teorem emonstrates tat no combination of suc unitary transformations an logic can solve te P complete problems (probabilistically. Weter tere is a sequence of unitary matrices an logical circuits wic constitute transformations wit no mfp at [, 0] t an result in an algoritm wit small probability of error is still an open researc question. 7. Conclusion P is a very interesting class of problems of extensive an tremenous practical significance. P complete problems (te arest in P seem, at first sigt, to be irrelevant to cloning. However, teir irect relationsip to cloning an terefore te inerent ifficulty of te latter as now been establise. Exact cloning is peraps impossible since it is an operation of infinite precision. pproximate cloning is possible but being a non-linear operation, it is peraps best acieve troug non-linear operations. inear (an unitary approaces offer very limite precision, wic oes not allow for te solution of large P complete problems. It is terefore important to try to evelop cloning metos tat woul employ non-linearity to gain precision. Finally, it is not clear weter it woul be possible to replace cloning in te B( algoritm by a sequence of (unitary an non-linear transformations tat woul result in a final binary ranom variable or qubit wit trivial (or small probability of error. straigtforwar attempt was proven fruitless, but tere is no evience in eiter irection about more elaborate scemes. Te matter is an open researc question. cnowlegements Tis wor was partially supporte by te European Union RT grants HPR-CT an References. P. Benioff, "Quantum mecanical Hamiltonian moels of Turing macines," Journal of tatistical Pysics, vol. 9, pp , 98.. P. Benioff, "Quantum mecanical Hamiltonian moels of Turing macines tat issipate no energy," Pysics Review etters, vol. 48, pp , C. H. Bennett, E. Bernstein, G. Brassar, an U. Vazirani, "trengts an weanesses of quantum computing," IM Journal on Computing, vol. 6, no. 5, pp , E. Bernstein an U. Vazirani, "Quantum complexity teory," Proceeings of 5t CM ymposium on Teory of Computing, pp. -0, Bertiaume an C. Brassar, "Te quantum callenge to structural complexity teroy," Proceeing of 7t IEEE Conference on tructure in Complexity Teory," pp. 3-37, 99. 0

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