Structured quantum search in NP-complete problems using the cumulative density of states

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1 Structured quantum search n NP-complete problems usng the cumulatve densty of states Keth Kastella and Rchard Freelng Keth.Kastella@Verdan.com Freelng@ERI-Int.com Verdan Ann Arbor Research and Development Center P.O. Box 348, Ann Arbor, I (December 3, ) In the multtarget Grover algorthm, we are gven an unstructured N -element lst of objects S contanng a T -element subset τ and functon f, called an oracle, such that f ( S ) = f S τ, otherwse f ( ) =. By usng quantum parallelsm, an element of S τ can be retreved n O( N / T) steps, compared to ON ( / T) for any classcal algorthm. In ths paper, we show that n combnatoral optmzaton problems wth N = 4, the densty of states can be used n conjuncton wth the multtarget Grover algorthm to construct a sequence of oracles that structure the search so that the optmum state s obtaned wth certanty n O (log N) steps. PACS numbers: 3.67, 89.7, 89.8 The last decade has seen great progress n the development of quantum algorthms that explot quantum parallelsm to solve problems much faster than classcal algorthms. Perhaps the best known of these are the Shor factorzaton and the Grover search algorthms. Inspred by these results, a number of researchers have explored applcaton of these methods to NP-complete problems such as decson and combnatoral search. Stmulated by the realzaton that for a completely unstructured problem, the Grover algorthm s optmal, most of ths work has centered on the explotaton of problem structure combned wth the Grover search 3,4. Chen 5 et al assume that a sequence of auxlary oracles can be constructed that mark states that agree wth the target state n the frst j bts. Then they use a recursve argument to show that O (log N) steps are requred to fnd the target state. Ths paper presents an algorthm that a) uses the densty of states to structure quantum search effcently n a combnatoral optmzaton problem and b) exhbts non-recursve constructon of the untary transformatons requred to mplement t. In statstcal physcs, the densty of states characterzes the number of contnuum or dscrete states as a functon of the system energy. In combnatoral optmzaton, ths generalzes naturally to the number of confguratons as a functon of a cost parameter to be optmzed. For a system wth N states K S N labeled S, S,,, the cost of state s the realvalued functon C ( S j ). Defnng = C S c, the cumulatve densty of Q c { ( ) } states s defned here as ν ( j c ) = Qc elements of Q c. Whle our goal s to fnd the unque state that mnmzes C ( S), the densty of states provdes nformaton about the dstrbuton of costs but not about the cost of ndvdual states. Ths s the type of nformaton that s lkely to be avalable to help structure quantum optmzaton problems. Whle not rgorously establshed, many combnatoral optmzaton problems are thought to be selfaveragng 6. Ths means that many propertes of such problems converge to sample ndependent values n the lmt of large N. For example, the travelng salesman problem s to fnd the shortest closed connected path connectng ctes, gven ther par-wse dstance matrx. In ths case there are N = ( )!/ unque tours correspondng to states of the system. Nevertheless, the optmal tour dstance, the system entropy and related quanttes can be obtaned analytcally n the large- N lmt 7. Our algorthm bulds on the Grover search algorthm. To brefly revew the Grover algorthm and defne notaton, we are gven an N -element lst of objects S, =, K,N. Wthn the lst, there s a T -element subset τ and functon f, called an, the number of oracle, such the f ( ) = f τ, otherwse S f ( ) =. The elements of τ are referred to as S S

2 good or target states. The remanng elements are bad. Let P denote the operator wth matrx elements S P =. The operator U ( P / N ) exp( πf ), referred to as the Grover terate, s untary. The untarty of U follows from the observaton that n the bass, P s an N N matrx of ones wth P = NP so UU = ( P/ N ) = (here U s the Hermtan conjugate). In the Grover algorthm, the system s prepared n the state = S and U s appled repeatedly. N In most applcatons, T << N, and t turns out π N that after m teratons of U the 4 T ampltude of the system s nearly n the T S good states and very small n the bad states. In the usual applcaton of the Grover algorthm, the system state s now measured. The outcome of the measurement has nearly unt probablty to be n a good state, so that n ths way, one of the good states s located, wth some small probablty of error. Whle the orgnal Grover algorthm contans a small probablty of error, there have been several varants developed that can provde a good state wth certanty 8,9,. One nterestng such case occurs when T = N / 4, n whch case, the soluton s found wth certanty n only a sngle teraton (.e., one applcaton of U ). The mportant pont for our optmzaton applcaton s that when T = N / 4, one applcaton of U places all of the system ampltude unformly n the target states whle the bad state ampltudes vansh. The other mportant buldng block n the algorthm presented below s the proof that a combnaton of one- and two-bt quantum gates s unversal n the sense that any untary operator on n - Cumulatve Densty of States nu(c) - deal densty realzed densty c s - c Fgure Cumulatve densty of states for a model optmzaton problem wth Raylegh-dstrbuted costs. The deal densty s obtaned from the cumulatve probablty dstrbuton for the costs. The realzed densty s computed from the costs obtaned n a sngle 64-element problem nstances. The parttonng costs c, =,, 3are used to construct a nested sequence of Hlbert spaces that structure the quantum search process.

3 qubts can be constructed usng a fnte number of two-qubt gates. Thus, for the algorthm to be effcently mplementable, t suffces to show that t s emboded as a untary transformaton. Then the procedure of Reck 3 et al. can be used to obtan an explct decomposton of the transformaton. Note that wth perfect knowledge of the densty of states, we also have the cost of the optmal state. Ths could n theory be used to construct an optmzaton algorthm by defnng the oracle to be unty only on the optmal state. However, ths would not be a very effcent algorthm, snce t would requre O ( N ) teratons. As shown below, we can obtan O(log ( N )) complexty by extractng more nformaton from the densty of states. To obtan ths mproved scalng, we use the cumulatve densty of states n two ways. Frst, t s used to construct a sequence of oracles that mark a nested sequence of good states contanng the optmal state. Second, the densty of states enables the constructon of a sequence of projecton operators that decouple the prevous bad subspaces n subsequent processng. In ths way the algorthm concentrates all of the ampltude on a progressvely smaller porton of the Hlbert space. Once the ampltude has been forced to vansh on a bad subspace, t s held at zero durng the remander of the process. The nput assumptons on ths optmzaton algorthm are as follows. Let a system have N = 4 states labeled S, S, K, S N. A real-valued cost functon C(S) s defned on the states. Our goal s to fnd the unque state that mnmzes C (S). In addton to the cost, we post that the densty of states ν (c) for ths system s also provded,.e., for any cost c n the range of C(S) we can compute number of states ν (c) such that CS ( ) c. The frst step of the algorthm s to use construct a cost sequence ν ( c ) = 4 f = I f ( S) Θ( C(S) ) c c, =, =, K, Θ ( x) ν (c) where the satsfes Θ =, x ;, otherwse. f Wrtten as a matrx, f s N N wth 4 ones on ts dagonal and zeros elsewhere. Recallng that S P =, defne, f = f D P = P, + 4 R V = f ) = exp(π, f D R f + ( f ), P = f D f VV and The functons = f Pf Pf = 4 P + ( f ) =. Intalze the = S to =,K, such that. Then defne the sequence of oracles () are a set of real-valued lnear operators on the Hlbert space H wth bass. = f Pf, () (3) (4) (5) The V s are untary, whch can be seen as follows: and system n the superposton state, so j ( S ) Fgure Intermedate and fnal results for the ampltude dstrbuted n structured quantum search usng the densty of states. 3

4 and defne = V, =, K,. Then measurement of the state wll produce the optmal state wth unt probablty. The optmalty of follows from the followng observatons. Denote the range of by R H has dmenson T. Its f - complment V - R - \ R -. On R, D s the usual Grover terate, R ( ) 4 actng on the number T = ν c = of good states. The rato of good states to non-nvarant bad states s T / T = / 4, so that the Grover terate provdes certan convergence after a sngle teraton. The net result s that V moves R - R certanty, whle the ampltude n R - s held fxed at. At the concluson of steps, all of the ampltude s concentrated n the sngle optmal state. Fgures and show ths algorthm appled to a test problem wth N = 64 states, ndexed =, K, 64. The costs are a random draw from a mean Raylegh dstrbuton, pc ( ) = cexp( c / ). Fgure shows a realzaton of the cumulatve densty of states ν () c. For comparson, we show the deal cumulatve densty of states ν deal () c N c xdxex p( x /) )) parttonng costs c determned from ν () c that are used to defne the sequence of oracles (Eq. ()). Fgure shows how the modulus of evolves for the problem nstance of Fgure. Note that the number of computatonal bass states n wth non-vanshng ampltude s = H s nvarant under the probablty ampltude from to wth ( ( = N exp c /. Fgure also shows the reduced by a factor of 4 wth each step. The fnal ampltude s n the optmal state ( = 6 n ths nstance), as desred. In general applcatons the densty of states s unlkely to be known wth certanty. We have performed some anecdotal exploraton of the algorthm behavor when there s some error n the densty of states so that at each stage ( ) 4 ν c s only approxmately correct. Interestngly, n ths case the optmal soluton s stll obtaned wth hgh probablty, although certan convergence s lost. A lkely more vable approach when ( ) 4 ν s to use the approxmate parttonng c sequence of c s to generate nested subsets, and then use quantum countng 4 to provde the exact values of the ν ( ). To estmate ν ( ) to O () accuracy wll c requre a countng regster wth roughly bts. Once the ν ( c ) have been determned, we proceed as before wth a slght modfcaton to the terate, whch s requred to guarantee convergence when the rato of good states to bad states s no longer exactly ¼. As long as the rato s close to ¼, only or teratons are requred to provde assured convergence, and the complexty of the optmzaton process s stll only O(log N ). Each ν ( c ) s evaluated n a countng process that requres about O (log N) work wth a falure rate that can be ( ) made vanshngly small. The number of c O(log N) s held fxed at O(log N), so that the total complexty of the ( ) 3 countng process s O (log N). In summary, we have shown that for combnatoral optmzaton problem wth N = 4 confguratons and a known cumulatve densty of states, ths quantum search algorthm obtans the optmal confguraton n O(log N) steps wth unt probablty. Ths new quantum search methodology appears to have potental for wdespread applcatons n areas such as schedulng, bonformatcs, communcatons, and sgnal processng. ACKNOWLEDGEENTS The authors benefted from stmulatng dscusson wth K. Augustyn, I. arkov and C. Zalka. Ths work was supported by the Verdan Ann Arbor Research and Development Center and by the Ar Force Research Laboratory and Ar Force Offce of Scentfc Research under contract SPO9-96-D- 8. Shor P W, Polynomal-tme algorthms for prme factorzaton and dscrete logarthms on a quantum computer, SIA J. Comp., 6, No. 5, pp , Oct Grover L K, Quantum mechancs helps n searchng for a needle n a haystack, Phys. Rev. Lett. 79, (quant-ph/97633); From Schrodnger s equaton to the quantum search algorthm, Am. J. Phys. 69 (7) July, pp Cerf N J, Grover L K, and Wllams CP, Nested quantum search and NP-complete problems, quantph/98678 c 4

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