During the last ten years, superconducting circuits have passed from being interesting physical devices to becoming contenders for near-future useful and scalable quantum information processing (QIP). Advanced quantum simulation experiments have been shown with up to nine qubits, while a demonstration of quantum supremacy with fifty qubits is anticipated in just a few years. Quantum supremacy means that the quantum system can no longer be simulated by the most powerful classical supercomputers. Integrated classical-quantum computing systems are already emerging that can be used for software development and experimentation, even via web interfaces. Therefore, the time is ripe for describing some of the recent development of superconducting devices, systems and applications. As such, the discussion of superconducting qubits and circuits is limited to devices that are proven useful for current or near future applications. Consequently, the centre of interest is the practical applications of QIP, such as computation and simulation in Physics and Chemistry.

The formation of bound states at surfaces of materials with an energy gap in the bulk electron spectrum is a well known physical phenomenon. At superconductor surfaces, quasiparticles with energies inside the superconducting gap may be trapped in bound states in quantum wells, formed by total reflection against the vacuum and total Andreev reflection against the superconductor. Since an electron reflects as a hole and sends a Cooper pair into the superconductor, the surface states give rise to resonant transport of quasiparticle and Cooper pair currents, and may be observed in tunnelling spectra. In superconducting junctions these surface states may hybridize and form bound Andreev states, trapped between the superconducting electrodes. In d-wave superconductors, the order parameter changes sign under 90 • rotation and, as a consequence, Andreev reflection may lead to the formation of zero energy quasiparticle bound states, midgap states (MGS). The formation of MGS is a robust feature of d-wave superconductivity and provides a unified framework for many important effects which will be reviewed: large Josephson current, low-temperature anomaly of the critical Josephson current, π -junction behaviour, 0 → π junction crossover with temperature, zero-bias conductance peaks, paramagnetic currents, time reversal symmetry breaking, spontaneous interface currents, and resonance features in subgap currents. Taken together these effects, when observed in experiments, provide proof for d-wave superconductivity in the cuprates.

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We study the radio-frequency single-electron transistor (rf-SET) as a readout device for charge qubits. We measure the charge sensitivity of an rf-SET to be 6.3microe/sqrt[Hz] and evaluate the backaction of the rf-SET on a single Cooper-pair box. This allows us to compare the needed measurement time with the mixing time of the qubit imposed by the measurement. We find that the mixing time can be substantially longer than the measurement time, which would allow readout of the state of the qubit in a single shot measurement.

Recent demonstrations of macroscopic quantum coherence in Josephson junction based electronic circuits have opened an entirely new dimension for research and applications in the established field of Josephson electronics. In this article we discuss basic Josephson circuits for qubit applications, methods of quantum description of these circuits, and circuit solutions for qubit couplings. Principles of manipulation and readout of superconducting qubits are reviewed and illustrated with recent experiments using various qubit types.

We theoretically investigate selective coupling of superconducting charge qubits mediated by a superconducting stripline cavity with a tunable resonance frequency. The frequency control is provided by a flux biased dc-SQUID attached to the cavity. Selective entanglement of the qubit states is achieved by sweeping the cavity frequency through the qubit-cavity resonances. The circuit is able to accommodate several qubits and allows to keep the qubits at their optimal points with respect to decoherence during the whole operation. We derive an effective quantum Hamiltonian for the basic, two-qubit-cavity system, and analyze appropriate circuit parameters. We present a protocol for performing Bell inequality measurements, and discuss a composite pulse sequence generating a universal control-phase gate.Coherent coupling of superconducting qubits has been experimentally demonstrated for all major qubit types (charge 1,2 , flux 3,4,5 , and phase 6,7 qubits) using permanent direct qubit-qubit coupling, capacitive or inductive. A major challenge is to implement a tunable coupling of qubits required for any useful gate operation. Numerous suggestions in this direction have been discussed in recent literature together with related quantum gate protocols (for a review see, e.g. Ref. 8).There are two conceptually different approaches to the tunable coupling. The first approach is to employ direct coupling schemes using Josephson junctions in the non-resonant regime as passive controllable elements, either capacitive, 9 or inductive. 10,11,12,13,14 The second approach, which we adopt in this paper, suggests qubit coupling via a dynamic intermediate element, e.g., LCoscillator or Josephson junction, which becomes entangled with a qubit during a two-qubit operation. In this scheme, the entanglement is achieved by tuning the qubit and the mediator in resonance, and then transferring the entanglement to another qubit by tuning the mediator and the second qubit in the resonance. Such coupling method has been first suggested 15 and experimentally tested 16 for the ion trap qubits. For superconducting qubits, qubit-oscillator entanglement has been demonstrated experimentally for a charge qubit coupled to a microwave stripline cavity, 17 and a flux qubit coupled to a SQUID oscillator;18,19 the gate protocols based on controllable qubit-oscillator coupling have been theoretically discussed in Refs. 20,21.The experimental setup with the qubit coupling to a distributed oscillator -stripline cavity 17,22 possesses potential for scalability -several qubits can be coupled to the cavity. In this paper we investigate the possibility to use this setup for implementation of tunable qubit-qubit coupling and simple gate operations. Tunable qubitcavity coupling is achieved by varying the cavity frequency by controlling magnetic flux through a dc-SQUID attached to the cavity (see Fig. 1). An advantage of this method is the possibility to keep the qubits at the optimal points with respect to decoherence during the whole two-qubit operation. The qubi...