Implementation of the Four-Bit Deutsch-Jozsa Algorithm with Josephson Charge Qubits

Abstract

PACS: 03.67.Lx; 85.25.Cp We show that the Deutsch-Jozsa algorithm for up to four qubits can be realized with a setup of Josephson charge qubits. While existing proposals for the implementation of the algorithm (for up to three qubits) are based on certain classifications of the oracles, this approach becomes increasingly cumbersome for higher qubit numbers. Here we present a method to implement all balanced functions for the four-qubit algorithm by a fixed sequence of operations. The free parameters which define the implemented function are the rotation angles of 15 one-qubit operations. IntroductionDuring the past few years, implementations for quantum bits in many different areas of physics have been proposed. That is, potentially there are many physical systems which are suitable to realize one-qubit and two-qubit operations. Due to the universality of quantum computation this is, in principle, sufficient to realize any arbitrary N-bit quantum operation [1].At the present technological level, however, the feasibility of quantum computation still crucially depends on the details of a given physical implementation such as the available Hamiltonians (and therefore, in particular, the available two-bit operations), the maximal decoherence time, the scalability, etc. Therefore, the test whether or not a certain hardware setup can really be used as a quantum computer is the realization of a complete algorithm.Here we investigate the implementation of the Deutsch-Jozsa algorithm in a system of Josephson charge qubits [2]. Superconducting nanocircuits are promising candidates for the realization of a quantum computer. Recently, considerable progress has been achieved in the experimental design and control of the quantum dynamics of such systems [3][4][5][6][7].The Deutsch problem was the first example for which an algorithm on a quantum computer can outperform the solution on a classical computer [8]. It has been proposed by Deutsch in 1985 and since then has undergone several refinements by Deutsch, Jozsa, and others [9][10][11]. The implementation of the Deutsch-Jozsa algorithm has been realized for up to three qubits in liquid state NMR [12]; proposals exist to implement the three-bit algorithm also for Josephson charge qubits [13] and for molecular systems [14].Systematic implementations for three qubits start from certain classifications of the oracles. Since this approach becomes rather cumbersome for four and more qubits, a universal strategy for implementing the oracles becomes desirable, i.e. the treatment of

Efficient quantum circuits for diagonal unitaries without ancillas

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“…These authors showed that an n-qubit diagonal unitaryê ( ) if x can be implemented exactly using a circuit with − 2 1 n z-axis rotation operators with rotation angles proportional to the discrete Walsh transform coefficients of f (x). The circuit depth 3 was found to be − + 2 3 n 1 [12]. In this analysis f (x) was a real-valued function of an n-bit string, taking 2 n discrete values.…”

“…In particular, we are interested in the important case where f(x) is a real-valued function of a continuous variable x, rather than an n-bit string. We show that this can be accomplished using circuits of a similar type to those in [11,12]. By identifying elementary operators whose eigenvalues in the x-basis are Walsh functions, we show that an arbitrary diagonal unitary may be approximated efficiently on a finite register by using a Walsh-Fourier series approximation for f(x).…”

Quantum Face Recognition Protocol with Ghost Imaging

“…Quantum machine learning algorithms employing quantum features such as superposition and entanglement [7][8][9][10][11][12][13][14][15] promise enhancements in terms of the computing resources and the speed compared to the classical counterparts. Several experimental researches have been done to implement these algorithms [16][17][18][19][20][21]. In this article, we present a quantum algorithm for face recognition as one of the potential applications of quantum algorithms in machine learning.…”

Quantum computation with classical light: Implementation of the Deutsch–Jozsa algorithm

“…However, due to the aforementioned problems, the maximum number of qubits used in such implementations has not yet exceeded a few. For example, the greatest number of qubits used for the Deutsch-Jozsa Algorithm was four [23]. Thus there has not yet been a computational problem solved on a quantum computer which was inaccessible for classical computers (Turing machines).…”