Analysis of Deutsch-Jozsa Quantum Algorithm

Transcription

1 Aalysis of Deutsch-Jozsa Quatum Algorithm Zhegju Cao Jeffrey Uhlma Lihua Liu 3 Abstract. Deutsch-Jozsa quatum algorithm is of great importace to quatum computatio. It directly ispired Shor s factorig algorithm. I this ote we remark that Deutsch-Jozsa algorithm has cofused two uitary trasformatios: oe is performed o a pure state the other is performed o a superpositio. I the past decades o costructive specificatio o the essetial uitary operator performed o a superpositio has bee foud. Thus we thik the algorithm eeds more specificatios so as to facilitate the costructio of the wated quatum oracle. Keywords: quatum computig Deutsch-Jozsa algorithm Shor s algorithm superpositio. Itroductio Deutsch-Jozsa algorithm 5 is oe of the first examples of a quatum algorithm that is expoetially faster tha ay possible determiistic classical algorithm. The algorithm has become the corerstoe for quatum computatio ad ispired Grover s algorithm 7 ad Shor s algorithm 3. I this ote we wat to poit out that Deutsch-Jozsa algorithm has cofused two uitary trasformatios: oe is performed o a pure state the other is performed o a superpositio. So far o costructive specificatios o the essetial uitary trasformatio performed o a superpositio have bee foud. We believe this fact reders the algorithm somewhat dubious. Prelimiaries A qubit is a quatum state Ψ of the form Ψ = a 0 + b where the amplitudes a b C such that a + b = 0 ad are basis vectors of the Hilbert space. Two quatum mechaical systems are combied usig the tesor product. For example a system of two Departmet of Mathematics Shaghai Uiversity Shaghai Chia. Departmet of Computer Sciece Uiversity of Missouri Columbia USA. 3 Departmet of Mathematics Shaghai Maritime Uiversity Shaghai 0306 Chia. liulh@shmtu.edu.c

2 qubits Ψ = a 0 + a ad Φ = b 0 + b ca be writte as a ( ) ( ) b a b a b Ψ Φ = = a b a b a b Its shorthad otatio is Ψ Φ. Operatios o a qubit are described by uitary matrices. Of these the most importat is the Hadamard gate H =. Clearly H 0 = ( 0 + ) H = = I. 0 3 Deutsch-Jozsa quatum algorithm Let f : {0 } {0 }. The Deutsch-Jozsa algorithm eeds a quatum oracle computig f(x) from x which does t decohere x. It begis with the + bit state 0. That is the first qubits are each i the state 0 ad the fial qubit is i the state. A Hadamard gate is applied to each qubit to obtai the followig state H (+) : 0 x ( 0 ). () + Suppose that the oracle U f : x y x y f(x) is available where is additio modulo. Applyig the quatum oracle it gives W : x ( 0 ) x ( f(x) f(x) ). () + + For each x f(x) is either 0 or. The state ca be writte as + ( )f(x) x ( 0 ). Igorig the last qubit ad applyig the Hadamard gate to each of the first qubits it gives H : ( ) f(x) x ( ) f(x) ( ) x y y (3) where x y = x 0 y 0 x y x y is the sum of the bitwise product. The above ew superpositio ca be writte as ( ) f(x) ( ) x y y. The probability for measurig the state 0 is ( )f(x).

3 4 Aalysis of Deutsch-Jozsa algorithm The process of Deutsch-Jozsa algorithm ca be described as follows 00 }{{ 0} H (+) x ( 0 ) + W x ( f(x) f(x) ) + igorig the last qubit ad obtaiig the state H ( ) f(x) ( ) f(x) x ( ) x y y observig the state ad obtaiig its probability } 00 {{ 0 }. 4. How to practically costruct the oracle performed o a pure state I Deutsch-Jozsa algorithm the quatum oracle U f : x y x y f(x) must be of the form U f = I V f where I is the idetity matrix ad V f is a uitary matrix. X X X X Suppose that V f =. We have V f y = y = y f(x). If y = 0 X 3 X 4 X 3 X 4 the 0 = ( ) ( 0. It gives X ) X 3 = f(x). Sice f(x) {0 } we obtai X X 3 {0 }. If y = the = ( ) ( 0. It gives X ) X 4 = f(x). Sice f(x) {0 } we obtai X X 4 {0 }. Thus V f is i the set { Clearly to determie V f oe has to ivoke the classical computatioal result f(x). That meas the uitary matrix V f should be further specified as V f(x). The otatio is very useful because it idicates the costructive specificatio of the ivolved uitary matrix. So it is better to rewrite the quatum oracle as U f(x) = I V f(x). Note that the costructio of the oracle depeds essetially o the classical computatioal result f(x). Besides the oracle is performed o the pure state x y. }. 3

4 4. Is it possible to costruct the oracle performed o a superpositio The uitary operator W is performed o the superpositio x ( 0 ) ad keeps the states of the first qubits. Hece it ca be decomposed as W = I Γ where Γ is a uitary matrix. By the descriptio of Deutsch-Jozsa algorithm we have + W = I Γ = U f(x) = I V f(x). That meas oe has to extract a classical computatioal result f(x) from the superpositio + x ( 0 ) i order to costruct the operator W practically. Sice x rus through all values 0 oe has to measure the superpositio so as to obtai a value ˆx. Oce the value ˆx is measured applyig W = I V f(ˆx) to + produce oe state of the followig 0 x ( 0 ) or x ( 0 ) or x ( 0 ) or or x ( 0 ) or + ot the wated state x ( f(x) f(x) ). x x x ( 0 ) will ( 0 ) ( 0 ) All i all it has cofused a quatum oracle performed o a pure state with a quatum oracle performed o a superpositio. We ow wat to ask: is it possible to costruct the wated oracle performed o the superpositio? Fially we would like to stress that oly the Hadamard gate H is applied to each of the first qubits twice. Sice H = I we fid the algorithm always produces the state 00 }{{ 0} χ where χ {0 }. The claim that the probability for the state 0 is ( )f(x) is icorrect. 5 Coclusio We poit out that there are some flaws i Deutsch-Jozsa algorithm. We would like to stress that the costructio of a uitary operator performed o a superpositio must be compatible with tesor product which describes the combiatio of two quatum systems. Some physical 4

5 experimets o Shor s algorithm are criticized for usig less qubits i the secod register ad other deficiecies 3. So far those so-called quatum computers D-wave 6 ad IBM 9 have bee reported to optimize some combiatoric problems oly ot accelerate ay umerical computatios. We thik Deutsch-Jozsa algorithm eeds more specificatios so as to facilitate the costructio of the wated quatum oracle ad check its correctess. 6 Ackowledgemets We thak the Natioal Natural Sciece Foudatio of Chia ( ). Refereces Z.J. Cao ad Z.F. Cao: O Shor s Factorig Algorithm with More Registers ad the Problem to Certify Quatum Computers. IACR Cryptology eprit Archive 04: 7 (04) Z.J. Cao Z.F. Cao ad L.H. Liu: Remarks o Quatum Modular Expoetiatio ad Some Experimetal Demostratios of Shor s Algorithm. IACR Cryptology eprit Archive 04: 88 (04) 3 Z.J. Cao Z.F. Cao ad L.H. Liu: Commet o Demostratios of Shor s Algorithm i the Past Decades. IACR Cryptology eprit Archive 05: 07 (05) 4 A. Dag et al.: Optimisig Matrix Product State Simulatios of Shor s Algorithm arxiv:7.073v (07) 5 D. Deutsch ad R. Jozsa: Rapid solutios of problems by quatum computatio. Proceedigs of the Royal Society of Lodo A (99) 6 D-Wave Systems PDF D-Wave%0000Q%0Tech%0Collateral_07F.pdf 7 L. K. Grover: A fast quatum mechaical algorithm for database search. Proceedigs of the Twety- Eighth Aual ACM Symposium o Theory of Computig. pp. C9 (996) 8 E. Lucero et al.: Computig prime factors with a Josephso phase qubit quatum processor. Nature Physics arxiv: (0) C.Y. Lu et al.: Demostratio of a Compiled Versio of Shor s Quatum Factorig Algorithm Usig Photoic Qubits Physical Review Letters 99 (5): arxiv: (007) B. Layo et al.: Experimetal Demostratio of a Compiled Versio of Shor s Algorithm with Quatum Etaglemet Physical Review Letters 99 (5): arxiv: (007) E. Martí-López et al.: Experimetal realizatio of Shor s quatum factorig algorithm usig qubit recyclig. Nature Photoics. doi:0.038/photo.0.59 (0) 3 P. Shor: Polyomial-time algorithms for prime factorizatio ad discrete logarithms o a quatum computer. SIAM J. Comput. 6 (5): (997) 4 L. Vadersype et al.: Experimetal realizatio of Shor s quatum factorig algorithm usig uclear magetic resoace Nature 44 (6866): arxiv:quat-ph/076 (00) 5