Quantum Logic Gates for Coupled Superconducting Phase Qubits

Abstract

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...

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