Grover Algorithm Applied to Four Qubits System

Transcription

1 Computer and Inormation Science Vol., No. ; May Grover Algorithm Applied to Four Qubits System Z. Sakhi (Corresponding author) Laboratory o Inormation Technology and Modelisation, and Laboratory o Inormation processing Ben M sik Faculty o Sciences, Hassan II-mohamedia University PO box 79, Casablanca, Marocco Tel: zb.sakhi@yahoo.r A. Tragha Laboratory o Inormation Technology and Modelisation Ben M sik Faculty o Sciences, Hassan II-mohamedia University PO box 79, Casablanca, Marocco Tel: atragha@yahoo.r R. Kabil Laboratory o Inormation processing Ben M sik Faculty o Sciences, Hassan II-mohamedia University PO box 79, Casablanca, Marocco Tel: relkabil@yahoo.r M. Bennai Laboratory o Condensed Matter Physics Ben M sik Faculty o Sciences, Hassan II-mohamedia University PO box 79, Casablanca, Marocco Tel: mdbennai@yahoo.r Received: January 8, Accepted: February, doi:.9/cis.vnp Abstract We present some applications o quantum algorithm on many qubits system. We ocus in particular on Grover algorithm which we apply to our qubits case. We give deinition o some quantum gates used in the context o Grover algorithm as Hadamard gate. We consider specially the case o our qubits system and show that Grover algorithm allows as obtaining a maximal probability to get the result. Keywords: Qubits, Grover algorithm, Probability. Introduction Quantum inormation processing has emerged recently as a new inormation science combining physic, inormatics and electronic. The mean idea behind quantum inormation computing is the use the undamental laws o quantum physics in inormation processing. Feynman (R. P. Feynman, 98) in early 98 years, has proposed to use the power o quantum mechanics to simulate quantum phenomena. Thus, the inormation can be encoded in a superposition o states o photons, atoms, or ions which were deined as qubits. On the other hand, the increasing miniaturization o electronic circuits will be certainly limited by quantum eects at nanometer scale. This observation was irst pointed out by Moore (D. Deutsch, 98). Recently, a renewed interest in the subject was seen, since the seminal work o Shor (P. W. Shor, 99) and Grover (L. K. Grover, 997). This make possible to solve many problems which can t be reach in classical computing. In this perspective, Grover algorithm has showed an increasing capacity to solve many problems and Published by Canadian Center o Science and Education

2 Computer and Inormation Science Vol., No. ; May was applied in many context: quantum cryptography (Li-Yi-Hsu, ), charge Josephson junction qubits system (Xiao-Hu Zheng, Ping Dong, Zheng-Yuan Xue, Zhuo-Liang Cao, 7). Note that Josephson junction qubits are one o the strongest candidates or quantum computing physical systems. In practice, many diicult are in order to realize a concrete and a real physical quantum inormation system, especially in the case o many qubits circuits introducing various coupling scheme. Thus, the study o many qubits algorithms has become very important. We signal that our qubits case is very special and entanglement o our qubits systems was studied by many authors and in dierent context(f. Verstraete, J. Dehaene, B. De Moor, H. Verschelde, ). In the present work, we are interested on Grover algorithm which we apply in the case o our qubits system. We begin, in the next section, by recalling some basic quantum gates and circuits which are undamental in deining quantum algorithms. Some eatures o Grover algorithm are presented in section. The last two sections are devoted to our work and to a conclusion. We show specially that Grover algorithm allows as to obtain a maximal probability to get the result.. Quantum Circuits and logic Gates. One qubit gate We begin by presenting some interesting examples o one-qubit and two-qubits gates. Recall that a qubit is any state which is a linear combination o states and :. This state is a vector in a two dimensional complex vector space. In order to construct an adequate quantum algorithm, one has to introduce a quantum logical gates similar to the classical ones. The most known quantum gates are: Hadamard and CNOT gates. The irst one which is used in the context o Grover algorithm, is a one qubit gate. This gate is very important because it allows as constructing a superposed state rom individual qubits. In matrix representation, the Hadamard gate, is a one-qubit rotation, mapping the qubit-basis states and to two superposition states with equal weight o the computational basis states and. This corresponds to the transormation matrix given by: H in the {, } basis. Many quantum algorithms use the Hadamard transorm as an initial step, since it maps n qubits initialized with to a superposition o all n orthogonal states in the, basis with equal weight. This is a general eature, valid also or two or more qubits.. Two qubits gate: Controlled note In the two qubits A and B case, the Hilbert space is a tensor product o the one qubit ones: The corresponding basis is H AB ; ; ; H A H, where:,,, Another important -qubits quantum gate is the CNOT gate, which is the quantum generalization o the classical gate described earlier. It has two input qubits, the control and the target qubit, respectively. The target qubit is lipped only i the control qubit is set to. On the other hand, the output o the CNOT gate can be entangled while the input is non-entangled. For more details on quantum circuits see (Yang Liu, Gui Lu Long and Yang Sun, 8). The Controlled NOT gate (or CNOT) is a quantum gate that is an essential component in the construction o a various quantum algorithms. The matrix representation o this gate is B 6 ISSN E-ISSN

3 Computer and Inormation Science Vol., No. ; May CNOT=. It was shown by Barenco et al. (A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, and H. Weinurter, 99), that the CNOT gate and any one qubit gates can be considered as an universal set or quantum computing. In other word, any unitary transormation or an n qubits system can be decomposed in one qubit gate and a CNOT gate. Note that there exist also many other quantum gates, but the above cited ones are the most used in the realization o quantum circuits and quantum algorithms.. Grover algorithm In this section, we recall the essential eature or Grover algorithm. Suppose we have an unstructured database with N elements which are numbered rom to N-, and the elements are not ordered. Classically, we would test each element at a time, until we get the one searched or. In Grover algorithm, only O( N ) trials are needed (M. A. Nielsen, I. L. Chuang, ). Grover s algorithm has two registers: n qubits in the irst and one qubit in the second. The irst step is to create a superposition o all n computational basis states. This is achieved by initializing the irst register in the state and apply the operator H n. Then we deine a unction which recognizes the solution as: :,..., N,, (k) = i k is the searched element, = otherwise. Thus we can resume the Grover algorithm as consisting o the ollowing steps: ) Consider an initial state: n, ) We apply the Hadamard gate on the irst n qubits to get a uniorm superposition o all possible arguments, ) Apply the oracle. Note that the inormation on are included in the (n+) th qubit, ) Apply against the Hadamard gate, ) Do an observation. Note that the above algorithm can be illustrated in the ooling diagram: Figure In this igure, note that we have a succession o a one Grover iteration (G) and the states o the irst register correspond to the irst iteration. In the next section, we apply this algorithm to the case o our qubits system.. Four qubits system case Consider now the particular case o our qubits system or which we apply the above procedure. We begin by aecting to the irst and the second register our qubits and one qubit respectively. Then we initialize the irst register to be in state and the second register in the state. Apply now the Hadamard operator to the irst register. We obtain: N H i N i Thus i 6 i On the other hand, we apply Hadamard gate to the second register to get entangled ollowing state H Now, to apply the Grover operator G, one must to calculate the iteration number given by [] Published by Canadian Center o Science and Education 7

4 Computer and Inormation Science Vol., No. ; May k round N. In our case, where N=6, we have k =. Note that the calculation is done in the ollowing working basis,,,,,,,,,,,,,,, I we assume that the unknown element is, thus one can deduce i 6 i i 6 Where: i i i The next steep o Grover algorithm consist o applying the oracle U, as: i U i i U U U Introduce i i i Now, i we apply the inversion operator about the mean, one can easily obtain i 6 i i The second iteration (oracle) in Grover algorithm leads to U 6 8 ISSN E-ISSN

5 Computer and Inormation Science Vol., No. ; May Inversion operator about the mean leads to By applying the th iteration, one can obtain easily i 6 i i I i 6 i i U Apply again the inversion operator about the mean, on get I 6 i i i 6 Finally, to test Grover algorithm, we calculate the probability to ind the state. We ind: P 6 Thus, the probability o getting the result is around 96%.. Conclusion In this paper, we have presented an introduction to quantum algorithm in relation to many qubits system. We have in particular, considered the Grover algorithm which we have applied to the special case o our qubits and calculated the Grover probability in this case. In a perspective, one must use the same procedure to analyze a realistic case o physical system like Josephson junction qubits. In this context, it will be interesting to consider various coupling types o qubits eect on Grover algorithm. Further more general and concrete realization o Grover algorithm must be realized on a real quantum system such as many bodies Josephson qubits. Reerences A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, and H. Weinurter. (99). Elementary gates or quantum computation, Phys. Rev. A., D. Deutsch. (98). Int. J. theor. Phys. (98). F. Verstraete, J. Dehaene, B. De Moor, H. Verschelde. (). Four qubits can be entangled in nine dierent ways, Phys. Rev. A, 6 (). L. K. Grover. (997). Phys. Rev. Lett. 79 (997). Li-Yi-Hsu. (). Quantum secret-sharing protocol based on Grover's algorithm, Phys. Rev. A. 68 () 6. M. A. Nielsen, I. L. Chuang. (). Quantum Computation and Quantum Inormation, (Cambridge University Press, Cambridge, ). P. W. Shor. (99). Polynomial-time algorithms or prime actorization and discrete logarithms on a quantum computer, in Proceedings o the th Annual Symposium on the Foundations o Computer Science, edited by S. Goldwasser, page, Los Alamitos, CA, (99), IEEE Computer Society..96 Published by Canadian Center o Science and Education 9

6 Computer and Inormation Science Vol., No. ; May R. P. Feynman. (98). Simulating physics with computers, Int. J. Theor. Phys. (98) Xiao-Hu Zheng, Ping Dong, Zheng-Yuan Xue, Zhuo-Liang Cao. (7). Implementation o Grover search algorithm with Josephson charge qubits, Physica C. (7) Yang Liu, Gui Lu Long and Yang Sun. (8). Analytic one-bit and CNOT gate constructions o general n-qubit controlled gates, International Journal o Quantum Inormation (IJQI). Volume: 6, Issue: (8) 7-6. Figure. ISSN E-ISSN