Multi-qubit controlled-NOT gates and Greenberger–Horne–Zeilinger state generation using one qubit simultaneously controlling n qubits
Introduction
Since quantum systems have an information processing capability much greater than that of corresponding classical systems, much attention has been paid to quantum computers (QCs), which can solve some problems such as factorizing a large integer via Shor's algorithm [1] and searching for an item from a disordered system via Grover's algorithm [2]. It has been shown that any quantum operation can be decomposed into controlled-NOT (C-NOT) gates between two qubits and rotation on a single qubit [3]. Therefore, the realization of the two-qubit quantum gate plays a key role in the QC. Such gates have been demonstrated experimentally in cavity QED system [4], [5], trapped ions [6], quantum dots system [7], NMR [8], and superconducting qubits [9], [10]. It is well known that an useful QC must allow control of large qubits [11]. However, there are physical limitations on the number of qubits in an QC, and a large-scale quantum computation has not been experimentally achieved yet. Theoretically, some schemes for operating the multi-qubit controlled phase gates have been proposed based on n-controlled qubits acting on one target qubit in cavity QED or ion traps [12], [13], [14], [15]. Based on one qubit simultaneously controlling n target qubits, a novel scheme for achieving multiqubit controlled-phase gates was presented by Yang et al. [16]. Following their idea, we have presented a scheme for implementing n SWAP gates [17]. In this paper, we focus on how to realize n C-NOT gates and GHZ state by using one superconducting flux qubit simultaneously controlling n qubits in circuit cavity QED. For circuit cavity QED, some approach for performing quantum computation and quantum information processing have been proposed (see, e.g., Refs. [18], [19], [20], [21]).
This paper is organized as follows. In the following section, we introduce the model of the flux qubit coupled to a resonator and respective classical microwave pulses, and give the evolution operator of the system. In Section 3, we discuss how to realize n C-NOT gates and GHZ state. A brief discussion is given in Section 4.
Section snippets
Model and unitary evolution
Let us consider a rf-superconducting quantum interference device (rf-SQUID), the rf-SQUID has a configuration formed by two lowest levels ( and ) and an excited level (), as shown in Fig. 1. The Hamiltonian for a rf-SQUID, with junction capacitance C and loop inductance L, can be written in the usual form [22], [23]where is the magnetic flux threading the ring, Q is the total charge on the capacitor, and Q are the conjugate variables of the
Implementation of n C-NOT gates and GHZ state
The goal of this section is to demonstrate how the C-NOT gates and GHZ state can be implemented based on the unitary evolution operator (10).
Discussion
Finally, we give a brief discussion on the experimental issues:
- (i)
The conditions of the large detuning: for SQUIDs interacting with a resonator, the coupling strength is [28]. If we choose , and , a simple calculation shows that , , , and , which satisfy the conditions , , and .
- (ii)
The required operation time for n C-NOT gates or GHZ state generation: in terms of the values of the and , the total
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant no. 11174100, and Key Project of Hunan Provincial Natural Science Foundation under Grant No. 11JJ2003.
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