Matrix Representation of Quantum Gates

Transcription

1 Internatonal Journal of Computer Applcatons ( ) Volume 59 - No.8, February 7 Matrx Representaton of Quantum Gates Aradhyamath Poornma Physcs Department Vjayanagar Scence College Hosapete, Karnataka, Inda Naghabhushana N. M. Physcs Department RYM Engneerng College Ballar, Karnataka, Inda Rohtha Ujjnmatad Electroncs and Communcaton Department Proudhadevaraya Insttute of Technology Hosapete, Karnataka, Inda ABSTRACT The feld of quantum computng s growng rapdly and there s a surprsngly large lterature. Research n ths area ncludes the desgn of quantum reversble crcuts and developng quantum algorthms for the models of quantum computng. Ths paper s focused on representng quantum reversble gates n matrx form. In turn these matrces can be used to develop quantum crcuts wth help of K-Map. Also ths paper gves the hstorcal development of quantum algorthms and bascs concepts n quantum compuaton. General Terms Quantum Computer, Quantum Algorthms Keywords Quantum Computaton, Quantum gates, Qubts.. INTRODUCTION In th century Quantum theory s the greatest achevement of scentsts whch provdes a unform frame work for the constructon of varous modern physcal theores. After more than 5 years from ts ncepton, quantum theory marred wth computer scence, another great ntellectual trumph of the th century and the new subject of quantum computaton was born. Todays computer both n theoretcal (Turnng machnes) and Practcal (PCs) are based on classcal physcs. However quantum computaton tells us that the world behaves qute dfferently. A quantum system can be superposton of many dfferent states at the same tme and produces nterference effect durng ts evoluton. Important goals of quantum algorthms are sgnfcantly work faster than any classcal algorthm solvng the same problem. The potental advantage of quantum computers over classcal computers has generated a sgnfcant amount of nterest n quantum computaton, and has resulted n a large number of quantum algorthms not only for dscrete problems, such as nteger factorzaton, but also for computatonal problems n scence and engneerng, such as multvarate ntegraton, path ntegraton, the soluton of ordnary and partal dfferental equatons, egenvalues problems, and numercal lnear algebra problems. In the early 98s, Mann (98) and Feynman (98) ndependently observed that computers bult from quantum mechancal components would be deally suted to smulatng quantum mechancs. Feynman [, ] suggested that constructng computers based on the prncples of quantum mechancs mght enable the quantum systems of nterest to physcsts to be effcently smulated, whereas ths seemed to be very dffcult wth classcal computers. Also he ponted out that accurately and effcently smulatng quantum mechancal systems would be mpossble on a classcal computer, but that a new knd of machne, a computer tself bult of quantum mechancal elements whch obey quantum mechancal laws, mght one day perform effcent smulatons of quantum systems. Classcal computers are nherently unable to smulate such a system usng sub-exponental tme and space complexty due to the exponental growth of the amount of data requred to completely represent a quantum system. Quantum computers, on the other hand, explot the unque, non-classcal propertes of the quantum systems from whch they are bult, allowng them to process exponentally large quanttes of nformaton n only polynomal tme. Quantum computers acheve speedup over classcal computaton by takng advantage of nterference between quantum ampltudes. The models of quantum computaton have ther ancestors from the studes of connectons between physcs and computaton. In 973, to understand the thermodynamcs of classcal computaton Bennet [3] noted that a logcally reversble operaton does not need to dsspate any energy and found that a logcally reversble Turng machne s a theoretcal possblty. Benoff [4, 5, 6] defnes physcal systems n whch the laws of quantum mechancs would lead to the smulaton of classcal Turnng machne, but does not consder the quantum computaton. He constructed a quantum mechancal model of a Turng machne. Hs constructon s the frst quantum mechancal descrpton of computer, but t s not a real quantum computer because the machne may exst n an ntrnscally quantum state between computaton steps, but at the end of each computaton step the tape of the machne always goes back to one of ts classcal states. Another mportant theme n quantum computng has been the development of quantum cryptographc technques, gong back to the work of Bennett and Brassard [7] whch n turn bult on work, not publshed untl several years after ts concepton, by Wesner [8]. Yao [9] showed that quantum crcut model s equvalent to a quantum Turng machne n the sense that they can smulate each other n polynomal tme. Snce then, quantum crcuts has become the most popular model of quantum computaton n whch most of the exstng quantum algorthms are expressed. Synthess of quantum crcuts s crucal for quantum computaton due to the fact that n current technologes t s very dffcult to mplement quantum gates actng on three or more qubts. As early as n 995, t was shown

2 Internatonal Journal of Computer Applcatons ( ) Volume 59 - No.8, February 7 that any quantum gate can be (approxmately) decomposed to a crcut consstng only of the CNOT gates and a small set of sngle qubt gates []. Recently, some more effcent synthess algorthms for quantum crcuts have been found; see for example []. Some authors ntated the studes of smplfcaton and optmzaton of quantum crcuts. The am s to develop methods and technques to reduce the number of quantum gates n a quantum crcut and the depth of a quantum crcut. Due to the dffculty of mplementng large quantum crcuts, ths problem s even more mportant n quantum computaton than n classcal computaton In partcular, Deutsch [, 3] ntroduced the technque of quantum parallelsm based on the superposton prncple n quantum mechancs by whch a quantum Turng machne can encode many nputs on the same tape and perform a calculaton on all the nputs smultaneously. Furthermore, he proposed that quantum computers mght be able to perform certan types of computaton that classcal computers can only perform very neffcently. He nvestgated the possble computatonal power of physcally realzable computers, and formulated a quantum verson of the Turng machne. He defnes quantum Turng machnes (QTM) as the frst model for general quantum computaton, wth the crucal property that superposton of machne states are allowed, and defnes a unversal QTM. He observed that quantum computers rase nterestng problems for the desgn of programmng languages, computng scentsts were slow to respond to ths challenge. Quantum computaton offers the possblty of consderable speedup over classcal computaton by explorng the power of superposton of quantum states. Ths can be llustrated very well by the DeutschJozsa algorthm, whch was desgned n [4]. One of the most strkng advances was made by Shor [5, 6]. By explorng the power of quantum parallelsm, he dscovered a polynomal-tme algorthm on quantum computers for prme factorzaton of whch the best known algorthm on classcal computers s exponental. For a long tme, QC research has been the luxury of just a few academc elte n the world, that s, untl 994 when Shor nvented hs famous prme factorzaton algorthm. He showed n a concrete example that a QC could do much better than a classcal computer. More mportantly, the dffculty n factorng a large number s the bass of the RvestShamrAdleman (RSA) publc key encrypton scheme that s wdely used today. Through Shors algorthm, the QC has suddenly become a real possble threat, and ths algorthm has sparked worldwde nterests n the QC. Shors algorthm s applcable only to a specfc problem. There s an nterestng nterplay between quantum computng and quantum cryptography, n that whle Shors algorthm for nteger factorzaton has the potental to undermne many current cryptosystems, quantum cryptographc systems can be proved secure aganst any form of attack, ncludng attacks whch make use of quantum computng. Quantum search algorthms are devsed by Grover [7,8,9] they are applcable to many problems. Grovers quantum search algorthm solves the problem of unsorted database searchng. Fndng a marked state from an unsorted database requres N searches for a classcal computer. Grovers algorthm fnds a marked tem n only N steps where N s the sze of the database. Grovers algorthm has many applcatons such as decpherng the dgtal encrypton schyeme (DES) encrypton scheme optmzaton. The standard Grover algorthm acheves quadratc speedup over classcal searchng algorthms. Ths algorthm suffers from one problem: the probablty of fndng the marked state may never be exactly. To overcome ths dffculty, one has to generalze the standard Grover algorthm by replacng phase nversons by rotatons of smaller angles so that the search step can be made smaller. The rest of the paper s organzed as follows: Secton hghlghts the basc concepts n quantum computaton. Secton 3 represents all the basc quantum gates n matrx form. Fnally secton 4 gves concluson n bref.. BASIC CONCEPTS IN QUANTUM COMPUTATION The bt s the fundamental concept of classcal computaton and classcal nformaton. Quantum computaton and quantum nformaton are bult upon an analogous concept, the quantum bt or Qubt. Classcal computer s bult from an electrcal crcut contanng wres and logc gates where as quantum computer s bult from a quantum crcut contanng wres and elementary quantum gates. A classcal bt s ether or. Two possble states for Qubts are > and >. The dfference between classcal bts and quantum bts s that a Qubt can be n a state other than > or >. The superposton ψ >of Qubt s a lnear combnaton of these states. ψ a > +a > () Where a s the ampltude of measurng > and a s the ampltude of measurng the value >. a and a are the complex coeffcents satsfy the normalzaton condton a + a. The probablty of observng a sngle possble state from the superposton s obtaned by squarng the absolute value of ts ampltude. The probablty of the Qubt beng n the state > s a and the probablty that the Qubt wll be measured as > s a. The most common bass used n the quantum computng s called computatonal bass. [ [ >, > ] ] But any other orthonormal bass could be used. For example, the bass vector + > > > + > > > The equvalent way of expressng of a Qubt ψ > a > +a > The text book [] s referred for more nformaton on fundamentals of quantum computaton and quantum algorthms. 3. QUANTUM GATES IN MATRIX FORM The matrx representaton of Quantum gate nput >< output () (3)

3 3. Quantum NOT gate n the matrx form >< + >< + + (4) Internatonal Journal of Computer Applcatons ( ) Volume 59 - No.8, February 7 > > >, > > > > > >, > > > 3. Quantum Z gate n matrx form It nverts sgn of > to gve > and leaves > unaltered. For > nput the ouput s > and > nput the output s >. >< + >< Hardmard gate n matrx form For > nput to the Hardmard gate the output s >+ > > nput the output s > >. H > [ > + > + > ] > > (5) and for [ >< + >< + >< >< ] [ [ + + ] ] [ ] + + [ ] + (6) 3.4 Controlled NOT Gate (CNOT) A A B B A CNOT gate has two nput qubts known as control qubt and target qubt respectvely. The crcut representaton for CNOT s shown n the fgure. The top lne represents the control qubt, whle the bottom lne represents the target qubt. The acton of the gate may be descrbed as follows. If the control qubt s set to, then the target qubt s left alone. If the control qubt s set to, then the target qubt s flpped. The truth table of CNOT gate. Input Output > > > > > > > > C NOT nput >< output >< + >< + >< + >< C NOT [ ] [ ] [ ] [ ] (7) 3.5 SWAP Gate n Matrx form A B B A It swaps the states of the two qubts. The swap gate s prepared usng three CNOT gates. The sequence s as follows. The nputs are A, B >. The ouput of frst CNOT gate s A, A B >. Ths s fed to the second CNOT gate and oupt of the second CNOT gate s A (A B), A B > B, A B >. The ouput of thrd CNOT gate s B, B (A B) B, A > Truth Table of Swap gate Input Output > > > > > > > > Matrx Representaton of Swap gate M swap nput >< output >< + >< + >< + >< 3

4 Internatonal Journal of Computer Applcatons ( ) Volume 59 - No.8, February 7 M swap [ ] [ ] [ ] [ ] (8) Toffol gate n Matrx form Any classcal crcut can be replaced by equvalent crcut contanng only reversble elements by makng use of a reversble gate known as the Toffol gate []. The Toffol gate has three nput bts and three output bts as shown n the fgure. Two of the bts that are control bts that are unaffected by the acton of the Toffol gate. The thrd bt s a target bt that s flpped f both control bts are set to, otherwse s left alone. a a (9) The truth table of Toffol gate b b c Input Output a b c a b c > > > > > > > > > > > > > > > > Matrx Representaton of Toffol gate c c ab 3.7 Controlled - U gate: It s a natural extenson of the controlled NOT gate. Such a gate has a sngle control Qubt ndcated by the lne wth the black dot and n target Qubts ndcated by the boxed U. If the control Qubt s set to then nothng happens to the gate U s appled to the target Qubts. 3.8 Fredkn gate : A U P A M T offol nput >< output B F redkngate Q AB AC >< + >< + >< + >< + >< + >< + >< + >< C Truth Table of Fredkn Gate R AC AB 4

5 Internatonal Journal of Computer Applcatons ( ) Volume 59 - No.8, February 7 A B C P Q R Matrx Representaton of Fredkn gate M F redkn nput >< output >< + >< + >< + >< + >< + >< + >< + >< A B C P Q R Matrx Representaton of Peres gate: M P eres nput >< output >< + >< + >< + >< + >< + >< + >< + >< 4. CONCLUSION () In ths paper we studed the pre-hstory of quantum computaton and challenges n the quantum feld. Also we have gven the basc concepts n the quantum computaton. All quantum gates are studed thoroughly and represented them n the matrx form. These matrces are useful n generatng quantum crcuts. We strongly feel that ths paper wll be helpful for the begnners who are dong research n the models of quantum computaton. 3.9 Peres Gate n Matrx form: A P A B Q A B C Truth table of Peres gate: R AB C () 5. REFERENCES [] R. P. Feynman, Smulatng Physcs wth Computers, Internatonal Journal of Theoretcal Physcs, vol., no. 6/7, pp , 98. [] R. P. Feynman, Quantum mechancal computers, Foundaton of Physcs, Vol. 6, pp (986). (Orgnally appeared n optcs news, February 985). [3] C. H. Bennet, Logcal reversblty of computaton, IBM Journal of Research and Development 7 (973) [4] Benoff. P. The computer as a physcal system: A mcroscopc quantum mechancal Hamltonan model of computers as represented by Turnng machnes. Journal of Statstcal Physcs (5): [5] Benoff. P. quantum mechancal Hamltonan model of Turnng machnes Journal of Statstcal Physcs Vol 9, pp (98) [6] Benoff. P. quantum mechancal Hamltonan model of Turnng machnes that dsspate no energy, Physcs Revew letters Vol. 48, pp (98) [7] C. H. Bennett and G. Brassard, n Proc. IEEE Int. Conf. on Computers, Systems, and Sgnal Processng, Bangalore, Inda (984), pp

6 Internatonal Journal of Computer Applcatons ( ) Volume 59 - No.8, February 7 [8] S. Wesner, Conjugate codng, wrtten crca 97 and belatedly publshed n Sact News 5(), pp , 983 [9] A.C. Yao, Quantum crcut complexty, n: Proc. of the 34th Ann. IEEE Symp. on Foundatons of Computer Scence, 993, pp [] A. Barenco, C. Bennet, R. Cleve, D.P. DVncenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smoln, H. Wenfurter, Elementary gates for quantum computaton, Physcal Revew A 5 (995) [] V. V. Shende, A.S. Bullock, I. L. Markov, Synthess of quantum-logc crcuts, IEEE Transactons on Computer- Aded Desgn of Integrated Crcuts and Systems 5 (6). [] Deutsch D. Quantum theory, the Church-Turng prncple and the unversal quantum computer, Proceedngs of the Royal Socety of London A4:977. [3] Deutsch D. Quantum computatonal networks, Proceedngs of the Royal Socety of London, Vol. A45, 739, 989. [4] D. Deutsch, R. Jozsa, Rapd soluton of problems by quantum computaton, Proceedngs of the Royal Socety of London A 439 (99) 553. [5] Shor P. W. Algorthms for quantum computaton: dscrete logarthms and factorng, In Proceedngs of the 35 t h Annual IEEE Symposum on Foundatons of Computer Scence, pages 434. IEEE Press. [6] Shor P. W. Polynomal-Tme Algorthms for Prme Factorzaton and Dscrete Logarthms on a Quantum Computer, [7] Grover L. K. A fast quantum mechancal algorthm for database search, In Proceedngs of the 8 t h Annual ACM Symposum on the Theory of Computaton, pages 9. ACM Press. Also arxv:quant-ph/ [8] Grover L. K. Quantum mechancs helps n searchng for a needle n a haystack, Physcal Revew letters 79(), pp , 997. [9] L. K. Grover, Quantum telecomputaton, arxv:quantph/974. [] M.A. Nelsen, I.L. Chuang, Quantum Computaton and Quantum Informaton, Cambrdge Unversty Press, Cambrdge,. [] T. Toffol, Reversble computng, Tech. Memo MIT/LCS/TM- 5, MIT Lab. For Com. Sc