MOJ MOJES

Ecology & Environmental Sciences
Opinion
Volume 2 Issue 1 - 2017
A Generalization of the Bernstein-Vazirani Algorithm
Nagata K1*, Resconi G2, Nakamura T3, Batle J4, Abdalla S5 and Farouk F6
1Department of Physics, Korea Advanced Institute of Science and Technology, Korea
2Department of Mathematics and Physics, Catholic University, Italy
3Department of Information and Computer Science, Keio University, Japan
4Departament de Fisica, Universidad de les Illes Balears, Spain
5Department of Physics, King Abdulaziz University Jeddah, Saudi Arabia
6Information Technology Department, Al-Zahra College for Women, Egypt
Received: February 02, 2017 | Published: February 09, 2017
*Corresponding author: Nagata K, Department of Physics, Advanced Institute of Science and Technology, Daejeon 305-701, Korea, Email: E-mail:
Citation: Nagata K, Resconi G, Nakamura T, Batle J, Abdalla S. et al. (2017) A Generalization of the Bernstein-Vazirani Algorithm. MOJ Eco Environ Sci 2(1): 00010. DOI: 10.15406/mojes.2017.02.00010

Opinion

In this short contribution we present a generalization of the Bernstein-Vazirani algorithm. Givenhe set of real values { a 1 , a 2 , a 3 ,.... a N } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiWaaO qaaKqzGeGaamyyaSWaaSbaaeaajugWaiaaigdaaSqabaqcLbsacaGG SaGaamyyaSWaaSbaaeaajugWaiaaikdaaSqabaqcLbsacaGGSaGaam yyaSWaaSbaaeaajugWaiaaiodaaSqabaqcLbsacaGGSaGaaiOlaiaa c6cacaGGUaGaaiOlaiaadggajuaGdaWgaaqaaKqzadGaamOtaaqcfa yabaaakiaawUhacaGL9baaaaa@4D16@  and the function g: R → {0, 1}, we shall determinethe following values { g( a 1 ), g( a 2 ),g( a 3 ),.........,g( a N ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaiWaaO qaaKqzGeGaam4zaKqbaoaabmaakeaajugibiaadggalmaaBaaameaa caaIXaaabeaaaOGaayjkaiaawMcaaKqzGeGaaiilaabaaaaaaaaape GaaiiOaiaacEgajuaGdaqadaGcbaqcLbsacaWGHbqcfa4aaSbaaeaa jugWaiaaikdaaKqbagqaaaGccaGLOaGaayzkaaqcLbsacaGGSaGaam 4zaKqbaoaabmaakeaajugibiaadggajuaGdaWgaaqaaKqzadGaaG4m aaqcfayabaaakiaawIcacaGLPaaajugibiaacYcacaGGUaGaaiOlai aac6cacaGGUaGaaiOlaiaac6cacaGGUaGaaiOlaiaac6cacaGGSaGa am4zaKqbaoaabmaakeaajugibiaadggajuaGdaWgaaqaaKqzadGaam OtaaqcfayabaaakiaawIcacaGLPaaaa8aacaGL7bGaayzFaaaaaa@61FF@ simultaneously. We are thus generalizing theBernstein-Vazirani algorithm. We recover the usual algorithm when g:   a i a i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGNb GaaiOoaabaaaaaaaaapeGaaiiOaiaacckacaWGHbqcfa4aaSbaaeaa jugWaiaadMgaaKqbagqaaKqzGeGaeyOKH4QaamyyaKqbaoaaBaaaba qcLbmacaWGPbaajuaGbeaaaaa@4591@ The speed ofdetermining the N values will be shown to outperform the classical case by a factor of N.PACS numbers: 03.67.Lx(Quantum computation architectures and implementations), 03.67.Ac(Quantum algorithms, protocols, and simulations).

Quantum mechanics [1-6] provides exact and frequently remarkably accurate numerical predictions, as reported for over a century. There has been a remarkable link in recent decades between the quantum theory and information theory, given rise to the rich field of quantum information theory (QIT), which novel proposals that outperform classical tasks or simply have no classical counterpart [6].One case that involves both quantum theory and information theory that can be found in the foundations o fthe quantum theory is the Leggett-type non-local variables theory [7], which has been experimentally explored [8-10]. These experiments report that quantum theory does not accept a Leggett-type non-local variables interpretation, although some controversy remains around the conclusions and interpretations of the experimental outcomes [11-13]. Applications of QIT also include the implementation of quantum algorithms. One such case is provided for stance by the Deutsch’s problem [14], first experimentally realized on a nuclear magnetic resonance proof-of principle quantum computer [15]. Implementation of the Deutsch-Jozsa algorithm on an ion-trap quantum computer has also been achieved [16]. There have been, as well, several other attempts to use single-photon two qubitstates for quantum computing. Oliveira et a implemented the Deutsch’s algorithm with polarization and transverse electromagnetic spatial modes as qubits [17]. Other achievements also include single-photon Bell states preparation and measurement [18], a decoherence free implementation of Deutsch’s algorithm using single photon and using two logical qubits [19] and, more recently,a one-way quantum computing implementation of the algorithm[20].These achievements involving the Deutsch-Jozsa algorithm are very well related to the so called Bernstein-Vazirani algorithm [21,22], which can be considered as an extended version of the previous one. After these two algorithms, Simon’s algorithm [23] was discovered, among others. There has been a experimental implementation of a quantum algorithm that solves the Bernstein-Vazirani parity problem without entanglement [24]. Additionally, fiber-optics implementations of the Deutsch-Jozsa and Bernstein-Vazirani quantum algorithms with three qubits have been realized [25]. Also, a variant of the algorithm for quantum learning being robust against noise has been introduced [26], as well as a quantum algorithm for approximating the influences of Boolean functions and its applications [27]. The Bernstein-Vaziranial growth has been also versatile in quantum key distribution [28] and transport implementation with ion qubits [29]. In the present work we shall extend the Bernstein-Vazirani algorithm a bit further. In what follows, we present in full detail our generalization and some conclusions are drawn. Let us suppose that we are given the following sequence of real values

a 1 , a 2 , a 3 ,...., a N . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyyaK qbaoaaBaaabaGaaGymaaqabaqcLbsacaGGSaGaamyyaOWaaSbaaSqa aiaaikdaaeqaaKqzGeGaaiilaiaadggakmaaBaaaleaacaaIZaaabe aajugibiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaaiilaiaadgga kmaaBaaaleaacaWGobaabeaajugibiaac6caaaa@492F@                                                      (1)

Let us now introduce the function

g :  R  { 0,1 }. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaa baaaaaaaaapeGaaiiOaiaacQdacaGGGcGaaiiOaiaackfacqGHsgIR caGGGcGaaiiOaOWaaiWaaeaajugibiaaicdacaGGSaGaaGymaaGcca GL7bGaayzFaaqcLbsacaGGUaaaaa@4944@                                                        (2)

Our goal is to determine the following values:

g( a 1 ),g( a 2 ),g( a 3 ),....,g( a N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaO WaaeWaaeaajugibiaadggakmaaBaaaleaajugWaiaaigdaaSqabaaa kiaawIcacaGLPaaajugibiaacYcacaGGNbGcdaqadaqaaKqzGeGaam yyaKqbaoaaBaaabaGaaGOmaaqabaaakiaawIcacaGLPaaajugibiaa cYcacaGGNbGcdaqadaqaaKqzGeGaamyyaOWaaSbaaSqaaKqzadGaaG 4maaWcbeaaaOGaayjkaiaawMcaaKqzGeGaaiilaiaac6cacaGGUaGa aiOlaiaac6cacaGGSaGaam4zaOWaaeWaaeaajugibiaadggakmaaBa aaleaajugWaiaad6eaaSqabaaakiaawIcacaGLPaaaaaa@57F7@                                          (3)

Recall that in the classical case, we need N queries, that is, N separate evaluations of the function (2). In our quantum algorithm, we shall require a single query.  Suppose now that we introduce another function

f :  { 0,1 } N { 0,1 }, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaa baaaaaaaaapeGaaiiOaiaacQdacaGGGcGcdaGadaqaaKqzGeGaaGim aiaacYcacaaIXaaakiaawUhacaGL9baadaahaaWcbeqaaiaad6eaaa qcLbsacqGHsgIRkmaacmaabaqcLbsacaaIWaGaaiilaiaaigdaaOGa ay5Eaiaaw2haaKqzGeGaaiilaaaa@4B87@                                                            (4)

which is a function with a N-bit domain and a 1-bitrange. We construct the following function:

f ( x )=g( a ) . x= i=1  N g( a i ) x i ( mod2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaa baaaaaaaaapeGaaiiOaOWaaeWaaeaacaWG4baacaGLOaGaayzkaaGa eyypa0Jaam4zamaabmaabaGaamyyaaGaayjkaiaawMcaaiaacckaca GGUaGaaiiOaiaadIhacqGH9aqpdaaeWbqaaiaadEgadaqadaqaaiaa dggadaWgaaWcbaqcLbmacaWGPbaaleqaaaGccaGLOaGaayzkaaaale aacaWGPbGaeyypa0JaaGymaaqaaiaacckacaWGobaaniabggHiLdGc caWG4bWaaSbaaSqaaKqzadGaamyAaaWcbeaakmaabmaabaGaciyBai aac+gacaGGKbGaaGOmaaGaayjkaiaawMcaaaaa@5BFD@     

=g( a 1 ) x 1  g( a 2 ) x 2  g( a 3 ) x 3  ....  g( a N ) x N , x i      { 0,1 } N ,  g( a i )    { 0,1 },  a i     R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcLbsacq GH9aqpcaWGNbGcdaqadaqaaKqzGeGaamyyaKqbaoaaBaaaleaajugW aiaaigdaaSqabaaakiaawIcacaGLPaaajugibiaadIhajuaGdaWgaa qaaiaaigdaaeqaaKqzGeGaeyyLIumeaaaaaaaaa8qacaGGGcGaam4z aOWaaeWaaeaajugibiaadggajuaGdaWgaaqaaKqzadGaaGOmaaqcfa yabaaakiaawIcacaGLPaaajugibiaadIhajuaGdaWgaaWcbaqcLbma caaIYaaaleqaaKqzGeGaeyyLIuSaaiiOaiaadEgakmaabmaabaqcLb sacaWGHbGcdaWgaaWcbaqcLbmacaaIZaaaleqaaaGccaGLOaGaayzk aaqcLbsacaWG4bqcfa4aaSbaaeaacaaIZaaabeaajugibiaacckacq GHvksXcaGGUaGaaiOlaiaac6cacaGGUaGaeyyLIuSaaiiOaiaaccka caWGNbGcdaqadaqaaKqzGeGaamyyaKqbaoaaBaaabaGaamOtaaqaba aakiaawIcacaGLPaaajugibiaadIhajuaGdaWgaaqaaiaad6eaaeqa aKqzGeGaaiilaaGcbaqcLbsapaGaamiEaKqba+qadaWgaaqaaiaacM gaaeqaaKqzGeGaaiiOaiaacckacqGHiiIZcaGGGcGaaiiOaOWaaiWa aeaajugibiaaicdacaGGSaGaaGymaaGccaGL7bGaayzFaaWaaWbaaS qabeaajugWaiaac6eaaaqcLbsacaGGSaGaaiiOaiaacckacaGGNbGc daqadaqaaKqzGeGaamyyaKqbaoaaBaaabaWaaSbaaeaajugWaiaadM gaaKqbagqaaaqabaaakiaawIcacaGLPaaajugibiaacckacaGGGcGa eyicI4SaaiiOaiaacckakmaacmaabaGaaGimaiaacYcacaaIXaaaca GL7bGaayzFaaGaaiilaiaacckacaGGGcqcLbsapaGaamyyaKqzadWd biaacMgakmaaBaaaleaacaGGGcaabeaakiaacckacqGHiiIZcaGGGc GaaiOuaaaaaa@A7F8@           (5)                                                           

where ai is a real value. Here g( a ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgadaqada qaaiaadggaaiaawIcacaGLPaaaaaa@3B95@  symbolizes

g( a 1 )g( a 2 )....g( a N ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaO WaaeWaaeaajugibiaadggajuaGdaWgaaqaaiaaigdaaeqaaaGccaGL OaGaayzkaaqcLbsacaWGNbGcdaqadaqaaKqzGeGaamyyaKqbaoaaBa aabaGaaGOmaaqabaaakiaawIcacaGLPaaajugibiaac6cacaGGUaGa aiOlaiaac6cacaWGNbGcdaqadaqaaKqzGeGaamyyaKqbaoaaBaaaba GaamOtaaqabaaakiaawIcacaGLPaaajugibiaac6caaaa@4E41@                                                                         (6)

Let us follow the quantum states through the algorithm. The input state is

ψ 1   = N |1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiYdK NcdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeaeaaaaaaaaa8qacaGG GcGaaiiOaOWaaaGaaeaajugib8aacqGH9aqpaOWdbiaawQYiaKqzGe GaeyyLIuCcfa4aaWbaaeqabaGaamOtaaaajugibiaacYhakmaaaiaa baqcLbsacaaIXaaakiaawQYiaaaa@4A5D@                                                                         (7)

After the Hadamard transform on the state, we have

ψ 1 = x { 0,1 } N |x 2 N [ | 0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiYdK NcdaaacaqcfayaaOWaaSbaaKqbGeaajugibiaaigdaaKqbagqaaaGa ayPkJaqcLbsacqGH9aqpkmaaqafajuaGbaGcdaWcaaqcfayaaKqzGe GaaiiFaOWaaaGaaKqbagaajugibiaadIhaaKqbakaawQYiaaqaaOWa aOaaaKqbagaajugibiaaikdaaKqbagqaaOWaaWbaaKqbagqajuaiba qcLbsacaWGobaaaaaaaKqbGeaajugWaiaadIhacqGHiiIZjuaGdaGa daqcfasaaKqzadGaaGimaiaacYcacaaIXaaajuaicaGL7bGaayzFaa qcfa4aaWbaaeqabaqcLbmacaWGobaaaaqcfayabKqzGeGaeyyeIuoa kmaadmaajuaGbaGcdaWcaaqcfayaaKqzGeGaaiiFaOWaaaGaaKqbag aakmaaaiaajuaGbaqcLbsacaaIWaaajuaGcaGLQmcajugibiabgkHi TiaacYhacaaIXaaajuaGcaGLQmcaaeaakmaakaaajuaGbaqcLbsaca aIYaaajuaGbeaaaaaacaGLBbGaayzxaaaaaa@6B1F@                                                           (8)

Next, the function f is evaluated using

Uf:|x,y|x,yf ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyvai aadAgacaGG6aGaaiiFaiaadIhacaGGSaGcdaaacaqaaKqzGeGaaiyE aaGccaGLQmcajugibiabgkziUkaacYhacaWG4bGaaiilaiaadMhacq GHvksXcaWGMbGcdaaacaqaamaabmaabaqcLbsacaWG4baakiaawIca caGLPaaaaiaawQYiaaaa@4DB2@                                                                (9)

Givening

|ψ 2 =± x ( 1 ) f( x ) |x 2 N [ |0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiiFai abeI8a5LqbaoaaaiaabaWcdaWgaaadbaWaaSbaaeaadaWgaaqaaiaa ikdaaeqaaaqabaaabeaaaOGaayPkJaqcLbsacqGH9aqpcqGHXcqSju aGdaaeqbqaamaalaaabaWaaeWaaeaacqGHsislcaaIXaaacaGLOaGa ayzkaaWaaWbaaeqabaGaamOzamaabmaabaGaamiEaaGaayjkaiaawM caaaaacaGG8bWaaaGaaeaacaWG4baacaGLQmcaaeaadaGcaaqaaiaa ikdaaeqaamaaCaaabeqaaiaad6eaaaaaaaqaaKqzGeGaamiEaaqcfa yabKqzGeGaeyyeIuoajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqzGeGa aiiFaKqbaoaaaiaakeaajugibiaaicdaaOGaayPkJaqcLbsacqGHsi slcaGG8bqcfa4aaaGaaOqaaKqzGeGaaGymaaGccaGLQmcaaeaajuaG daGcaaGcbaqcLbsacaaIYaaaleqaaaaajuaGdaahaaWcbeqaaaaaaO Gaay5waiaaw2faaaaa@5FF0@                                                     (10)

After the Hadamard transform, using the previous equation and (10) we can now evaluate |ψ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiiFai abeI8a5LqbaoaaaiaabaWcdaWgaaadbaGaaG4maaqabaaakiaawQYi aaaa@3BC0@

|ψ 3 =± z x ( 1 ) x.z+f( x ) |z 2 N [ |0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiiFai abeI8a5LqbaoaaaiaabaWcdaWgaaadbaWaaSbaaeaacaaIZaaabeaa aeqaaaGccaGLQmcajugibiabg2da9iabgglaXMqbaoaaqafabaWaaa buaeaadaWcaaqaamaabmaabaqcLbsacqGHsislcaaIXaaajuaGcaGL OaGaayzkaaWaaWbaaeqabaWcdaahaaqcfayabeaajugWaiaadIhaca GGUaGaamOEaiabgUcaRiaadAgalmaabmaajuaGbaqcLbmacaWG4baa juaGcaGLOaGaayzkaaaaaaaajugibiaacYhajuaGdaaacaqaaKqzGe GaamOEaaqcfaOaayPkJaaabaqcLbsacaaIYaqcfa4aaWbaaeqabaqc LbmacaWGobaaaaaaaKqbagaajugWaiaadIhaaKqbagqajugibiabgg HiLdaajuaGbaqcLbmacaWG6baajuaGbeqcLbsacqGHris5aKqbaoaa dmaakeaajuaGdaWcaaGcbaqcLbsacaGG8bqcfa4aaaGaaOqaaKqzGe GaaGimaaGccaGLQmcajugibiabgkHiTiaacYhajuaGdaaacaGcbaqc LbsacaaIXaaakiaawQYiaaqaaKqbaoaakaaakeaajugibiaaikdaaS qabaaaaaGccaGLBbGaayzxaaaaaa@7387@                                   (11)

Thus,

|ψ 3 =± z x ( 1 ) x.z+g( a ).x |z 2 N [ |0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiiFai abeI8a5LqbaoaaaiaabaWaaSbaaWqaaKqbaoaaBaaameaalmaaBaaa meaajugWaiaaiodaaWqabaaabeaaaeqaaaGccaGLQmcajugibiabg2 da9iabgglaXMqbaoaaqafabaWaaabuaeaadaWcaaqaaSWaaeWaaKqb agaajugWaiabgkHiTiaaigdaaKqbakaawIcacaGLPaaalmaaCaaaju aGbeqaaSWaaWbaaKqbagqabaqcLbmacaWG4bGaaiOlaiaadQhacqGH RaWkcaWGNbWcdaqadaqcfayaaKqzadGaamyyaaqcfaOaayjkaiaawM caaKqzadGaaiOlaiaadIhaaaaaaiaacYhajuaGdaaacaqaaKqzGeGa amOEaaqcfaOaayPkJaaabaqcLbsacaaIYaqcfa4aaWbaaeqabaqcLb macaWGobaaaaaaaKqbagaajugWaiaadIhaaKqbagqajugibiabggHi LdaajuaGbaqcLbmacaWG6baajuaGbeqcLbsacqGHris5aKqbaoaadm aakeaajuaGdaWcaaGcbaqcLbsacaGG8bqcfa4aaaGaaOqaaKqzGeGa aGimaaGccaGLQmcajugibiabgkHiTiaacYhajuaGdaaacaGcbaqcLb sacaaIXaaakiaawQYiaaqaaKqbaoaakaaakeaajugibiaaikdaaSqa baaaaaGccaGLBbGaayzxaaaaaa@7991@                                       (12)

x ( 1 ) x.z +g( a ).x = 2 N δ g ( a ),z. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaqafabaWaae WaaeaajugibiabgkHiTiaaigdaaOGaayjkaiaawMcaaaWcbaqcLbsa caWG4baaleqajugibiabggHiLdqcfa4aaWbaaeqabaGaamiEaiaac6 cacaWG6baaaOWaaWbaaSqabeaacqGHRaWkcaWGNbWaaeWaaeaacaWG HbaacaGLOaGaayzkaaGaaiOlaiaadIhaaaqcLbsacqGH9aqpcaaIYa qcfa4aaWbaaeqabaGaamOtaaaajugibiabes7aKLqbaoaaBaaabaGa am4zaaqabaGcdaWgaaWcbaWaaeWaaeaacaWGHbaacaGLOaGaayzkaa GaaiilaiaadQhacaGGUaaabeaaaaa@566A@                                                                     (13)   

Thus,

|ψ 3 =± z x ( 1 ) x.z+g( a ).x |z 2 N [ |0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaiiFai abeI8a5LqbaoaaaiaabaWaaSbaaWqaaKqbaoaaBaaameaalmaaBaaa meaajugWaiaaiodaaWqabaaabeaaaeqaaaGccaGLQmcajugibiabg2 da9iabgglaXMqbaoaaqafabaWaaabuaeaadaWcaaqaamaabmaabaqc LbsacqGHsislcaaIXaaajuaGcaGLOaGaayzkaaWcdaahaaqcfayabe aalmaaCaaajuaGbeqaaKqzadGaamiEaiaac6cacaWG6bGaey4kaSIa am4zaSWaaeWaaKqbagaajugWaiaadggaaKqbakaawIcacaGLPaaaju gWaiaac6cacaWG4baaaaaacaGG8bqcfa4aaaGaaeaajugibiaadQha aKqbakaawQYiaaqaaKqzGeGaaGOmaSWaaWbaaKqbagqabaqcLbmaca WGobaaaaaaaKqbagaajugWaiaadIhaaKqbagqajugibiabggHiLdaa juaGbaqcLbmacaWG6baajuaGbeqcLbsacqGHris5aKqbaoaadmaake aajuaGdaWcaaGcbaqcLbsacaGG8bqcfa4aaaGaaOqaaKqzGeGaaGim aaGccaGLQmcajugibiabgkHiTiaacYhajuaGdaaacaGcbaqcLbsaca aIXaaakiaawQYiaaqaaKqbaoaakaaakeaajugibiaaikdaaSqabaaa aaGccaGLBbGaayzxaaaaaa@7864@                                                             

=± z 2Nδ g( a ), z |z 2 N [ |0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeyySaeBcfa4aaabuaeaadaWcaaqaaSWaaWbaaKqbagqabaqcLbma caaIYaGaamOtaiabes7aKbaacaWGNbWcdaqadaqcfayaaKqzadGaam yyaaqcfaOaayjkaiaawMcaaKqzadGaaiilaiaadQhajuaGdaahaaqa beaajugWaiaacYhalmaaaiaajuaGbaqcLbmacaWG6baajuaGcaGLQm caaaaabaqcLbmacaaIYaqcfa4aaWbaaeqabaqcLbmacaWGobaaaaaa aKqbagaajugibiaadQhaaKqbagqajugibiabggHiLdqcfa4aamWaaO qaaKqbaoaalaaakeaajugibiaacYhajuaGdaaacaGcbaqcLbsacaaI WaaakiaawQYiaKqzGeGaeyOeI0IaaiiFaKqbaoaaaiaakeaajugibi aaigdaaOGaayPkJaaabaqcfa4aaOaaaOqaaKqzGeGaaGOmaaWcbeaa aaaakiaawUfacaGLDbaaaaa@67A4@

=±|g ( a )[ |0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeyySaeRaaiiFaiaadEgajuaGdaaacaqaamaabmaabaqcLbsacaWG HbaajuaGcaGLOaGaayzkaaaacaGLQmcadaWadaGcbaqcfa4aaSaaaO qaaKqzGeGaaiiFaKqbaoaaaiaakeaajugibiaaicdaaOGaayPkJaqc LbsacqGHsislcaGG8bqcfa4aaaGaaOqaaKqzGeGaaGymaaGccaGLQm caaeaajuaGdaGcaaGcbaqcLbsacaaIYaaaleqaaaaaaOGaay5waiaa w2faaaaa@4EC3@

=±|g ( a 1 )g( a 2 )...g( a N )[ |0|1 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyypa0 JaeyySaeRaaiiFaiaadEgajuaGdaaacaqaamaabmaabaqcLbsacaWG HbWcdaWgaaadbaGaaGymaaqabaaajuaGcaGLOaGaayzkaaqcLbsaca WGNbqcfa4aaeWaaeaajugibiaadggalmaaBaaameaacaaIYaaabeaa aKqbakaawIcacaGLPaaajugibiaac6cacaGGUaGaaiOlaiaacEgaju aGdaqadaqaaKqzGeGaamyyaSWaaSbaaWqaaiaad6eaaeqaaaqcfaOa ayjkaiaawMcaaaGaayPkJaWaamWaaOqaaKqbaoaalaaakeaajugibi aacYhajuaGdaaacaGcbaqcLbsacaaIWaaakiaawQYiaKqzGeGaeyOe I0IaaiiFaKqbaoaaaiaakeaajugibiaaigdaaOGaayPkJaaabaqcfa 4aaOaaaOqaaKqzGeGaaGOmaaWcbeaaaaaakiaawUfacaGLDbaaaaa@5EF4@                                                                    (14)

from which

g( a 1 ),g( a 2 )...g ( a N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaO WaaeWaaeaajugibiaadggakmaaBaaaleaajugWaiaaigdaaSqabaaa kiaawIcacaGLPaaajugibiaacYcacaWGNbGcdaqadaqaaKqzGeGaam yyaKqbaoaaBaaabaGaaGOmaaqabaaakiaawIcacaGLPaaajugibiaa c6cacaGGUaGaaiOlaiaacEgakmaaaiaabaWaaeWaaeaajugibiaadg gajuaGdaWgaaqaaiaad6eaaeqaaaGccaGLOaGaayzkaaaacaGLQmca aaa@4EA8@                                                                                   (15)

can be obtained. That is to say, if we measure g( a 1 ),g( a 2 )...g ( a N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaO WaaeWaaeaajugibiaadggajuaGdaWgaaqaaiaaigdaaeqaaaGccaGL OaGaayzkaaqcLbsacaGGSaGaam4zaOWaaeWaaeaajugibiaadggaju aGdaWgaaqaaiaaikdaaeqaaaGccaGLOaGaayzkaaqcLbsacaGGUaGa aiOlaiaac6cacaWGNbGcdaaacaqaamaabmaabaqcLbsacaWGHbGcda WgaaWcbaqcLbmacaWGobaaleqaaaGccaGLOaGaayzkaaaacaGLQmca aaa@4EA9@ then we can retrieve the following values

g( a 1 ),g( a 2 ),g( a 3 ),...g( a N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaO WaaeWaaeaajugibiaadggajuaGdaWgaaqaaiaaigdaaeqaaaGccaGL OaGaayzkaaqcLbsacaGGSaGaai4zaOWaaeWaaeaajugibiaadggaju aGdaWgaaqaaiaaikdaaeqaaaGccaGLOaGaayzkaaqcLbsacaGGSaGa ai4zaOWaaeWaaeaacaWGHbqcfa4aaSbaaeaacaaIZaaabeaaaOGaay jkaiaawMcaaKqzGeGaaiilaiaac6cacaGGUaGaaiOlaiaadEgakmaa bmaabaqcLbsacaWGHbGcdaWgaaWcbaqcLbmacaWGobaaleqaaaGcca GLOaGaayzkaaaaaa@5486@                                                                                (16)

Using a single query. All we have to do is to perform one quantum measurement. The speed to determine N values improves by a factor of N as compared to the classical counterpart. Notice that we receive the Bernstein-Vazirani algorithm when g:aiai MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zai aacQdacaWGHbGaamyAaiabgkziUkaadggacaWGPbaaaa@4008@  

In the present work we have presented a generalization of the Bernstein-Vazirani algorithm. The advantage that is promised over its classical counterpart is O( N ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4taO WaaeWaaeaajugibiaad6eaaOGaayjkaiaawMcaaaaa@3C9C@ , much like in the vein of quantum parallelism, where the N coefficients of a pure state ψN MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=xjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiYdK NcdaaacaqaaKqzadGaamOtaaGccaGLQmcaaaa@3D97@  can be known in one single quantum query.

Acknowledgement

J. Batle acknowledges fruitful discussions with J. Rossell´o, Maria del Mar Batle and Regina Batle.

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