Based on a quantum analysis of two capacitively coupled current-biased Josephson junctions, we propose two fundamental two-qubit quantum logic gates. Each of these gates, when supplemented by single-qubit operations, is sufficient for universal quantum computation. Numerical solutions of the time-dependent Schrödinger equation demonstrate that these operations can be performed with good fidelity. The current-biased Josephson junction is an easily fabricated device with great promise as a scalable solid-state qubit [1], as demonstrated by the recent observations of Rabi oscillations [2,3]. This phase qubit is controlled through manipulation of the bias currents and application of microwave pulses resonant with the energy level splitting [2].In this Letter we analyze the quantum dynamics of two coupled phase qubits. (The classical dynamics of this system has also been studied recently [4]). We identify two quantum logic gates that, together with single-qubit operations, provide all necessary ingredients for a universal quantum computer. We perform full dynamical simulations of these gates through numerical integration of the time-dependent Schrödinger equation. These two-qubit operations may be experimentally probed with the methods already used to observe single junction Rabi oscillations [2,3]. Such experiments are of fundamental importance: the successful demonstration of macroscopic quantum entanglement holds profound implications for the universal validity of quantum mechanics [5]. Important progress toward this goal are the temporal oscillations of coupled charge qubits [6] and spectroscopic measurements [7] on the system considered here. Finally, our methods are applicable to the other promising superconducting proposals based on charge, flux, and hybrid realizations [8].Figure 1(a) shows the circuit diagram of our coupled qubits. Each junction has characteristic capacitance C J and critical current I c , and they are coupled by capacitance C C . The two degrees of freedom of this system are the phase differences γ 1 and γ 2 , with dynamics governed by the Hamiltonian [9]Here we have employed the charging and Josephson energies E C = e 2 /2C J and E J = I c /2e, the normalized bias currents J 1 = I 1 /I c , J 2 = I 2 /I c , and the dimensionless coupling parameter ζ = C C /(C C + C J ). This coupling scheme has been recently analyzed [9, 10, 11] and results in a system with easily tuned energy levels and adjustable effective coupling. While ζ is typically fixed by fabrication, the energy levels and the effective coupling of the associated eigenstates are under experimental control through J 1 and J 2 . As shown below, the two junctions are decoupled for J 1 and J 2 sufficiently different, but if J 1 and J 2 are related in certain ways, the junctions are maximally coupled. To illustrate this method of control, we define a reference bias current J 0 and consider the variation of J 1 and J 2 through a detuning parameter ǫ:Quantum logic gates are implemented by varying ǫ with time as shown in Fig. 1(b). This ra...

We present spectroscopic evidence for the creation of entangled macroscopic quantum states in two current-biased Josephson-junction qubits coupled by a capacitor. The individual junction bias currents are used to control the interaction between the qubits by tuning the energy level spacings of the junctions in and out of resonance with each other. Microwave spectroscopy in the 4 to 6 gigahertzrange at 20 millikelvin reveals energy levels that agree well with theoretical results for entangled states. The single qubits are spatially separate, and the entangled states extend over the 0.7-millimeter distance between the two qubits.

Recently, quantized versions of random walks have been explored as effective elements for quantum algorithms. In the simplest case of one dimension, the theory has remained divided into the discretetime quantum walk and the continuous-time quantum walk. Though the properties of these two walks have shown similarities, it has remained an open problem to find the exact relation between the two. The precise connection of these two processes, both quantally and classically, is presented. Extension to higher dimensions is also discussed. where ψ(n, t) is a complex amplitude at the (continuous) time t and (discrete) lattice position n.The discrete-time quantum walk (DTQW), introduced by Aharonov et al. [4] and independently by Meyer [5], is a discrete unitary mapping such aswhere ψ R (n, τ ) and ψ L (n, τ ) are complex amplitudes at the (discrete) time τ and (discrete) lattice position n, and the labels R and L indicate an additional degree of freedom, often taken as the state of a coin which tells the walker (located at position n) which way to step. This discrete dynamics has a rich mathematical structure that is quite foreign to the CTQW, and has been the subject of extensive theoretical investigation. In particular, there has been significant extension of the DTQW to include decoherence [6], quantum chaotic [7] and quasiperiodic effects [8]. In addition, certain continuum limits have been used to connect the DTQW to more familiar wavelike propagation [9,10,11]. Despite this large body of work, the relation of these two quantum walks remains an open problem.This problem is truly fundamental for quantum computation, for at least two reasons. First, it is quite unnatural to have two distinct ways to quantize classical diffusion. Determining whether quantum mechanics speeds up a classical process is difficult enough, but even moreso if there is no unique quantization. For the processes considered here, the coin degree of freedom appears unnecessary, and indeed there is a perfectly reasonable discretetime quantum process that can be implemented without a coin [12]-this will be discussed below. Second, the spreading properties of the two quantum walks are quite similar [11]. From an initially localized state, both evolutions generate a probability distribution that is nearly constant save for two peaks at ±ct (here c = 2γ for (1) and c = cos θ for (2)), decaying to zero thereafter (see Figure 1). Both have standard deviations of position that grow linearly in time, quadratically faster than classical diffusion. These similarities suggest that, besides the fact that both are unitary quantum processes, there should be some underlying connection between the two walks. Nevertheless, the precise relationship has remained elusive.

We present a method to synthesize an arbitrary quantum state of two superconducting resonators. This state-synthesis algorithm utilizes a coherent interaction of each resonator with a tunable artificial atom to create entangled quantum superpositions of photon number (Fock) states in the resonators. We theoretically analyze this approach, showing that it can efficiently synthesize NOON states, with large photon numbers, using existing technology.

The present state-of-the-art in cooling mechanical resonators is a version of "sideband" cooling. Here we present a method that uses the same configuration as sideband cooling -coupling the resonator to be cooled to a second microwave (or optical) auxiliary resonator -but will cool significantly colder. This is achieved by varying the strength of the coupling between the two resonators over a time on the order of the period of the mechanical resonator. As part of our analysis, we also obtain a method for fast, high-fidelity quantum information-transfer between resonators.PACS numbers: 85.85.+j,42.50.Dv,85.25.Cp, There is presently a great deal of interest in cooling high-frequency micro-and nano-mechanical oscillators to their ground states. This interest is due to the need to prepare resonators in states with high purity to exploit their quantum behavior in future technologies [1,2]. The key measure of a cooling scheme is the cooling factor, which we will denote by f cool . The cooling factor is the ratio of the average number of phonons in the resonator at the ambient temperature, n T , to the average number of phonons achieved by the cooling method, which we will denote by n cool . The present state-of-the-art for cooling mechanical resonators is sideband cooling, which was originally developed in the context of cooling trapped ions [3][4][5]. This method is a powerful and practical technique, able to achieve large cooling factors, and these have been demonstrated in the laboratory [6][7][8][9][10][11][12][13][14][15].In the context of mechanical resonators, sideband cooling involves coupling the resonator to be cooled (from now on the "target") to a microwave or optical resonator (the "auxiliary") whose frequency is sufficiently high that it sits in its ground state at the ambient temperature. The resonators are coupled together by a linear interaction, and one that is straightforward to implement experimentally. In particular, if we denote the annihilation operators for the target and auxiliary resonator by a and b, respectively, then the full Hamiltonian of the two resonators iswhere x a = a + a † and x b = b + b † are the position operators of the respective resonators. The coupling is modulated at the difference frequency between the resonators, ν = Ω − ω. This converts the high frequency of the auxiliary resonator so that the two resonators are effectively on-resonance, and thus exchange energy at the coupling rate g. With this frequency conversion, the auxiliary constitutes a source of essentially zero entropy (and thus zero temperature) for the target resonator [16].When the rate of the coupling, g, is significantly smaller than the frequency ω of the target resonator (so that one is within the rotating-wave approximation (RWA)-see, e.g. [17]), then the linear coupling between the resonators is merely excitation (phonon/photon) exchange between the two. If the auxiliary is now damped sufficiently rapidly, then the excitation exchange, combined with the relatively fast damping of the auxiliary at effec...