Microwave photons in superconducting circuits is a promising approach for quantum computing due to long coherence times and the easy controllability of superconducting devices. However, as the search of the ultimate design of a quantum computer is still ongoing, various suggestions for the realization of a quantum bit, qubit, exist even in the area of superconducting circuits. In this thesis, we study theoretically quantum computing using itinerant microwave photons as qubits. Basic tools for treating the reflection of wave packets and the noise generated by resistive elements in a circuit are derived starting from a widely used model for superconducting transmission lines. We also present simple quantum devices that can act as quantum gates for microwave qubits, including a novel suggestion for a tunable phase shifter. A tunable phase shifter is a useful component in the study of microwave photons in general as well as in quantum computing. The tunable phase shifter consists of three superconducting quantum interference devices (SQUIDs) that can be treated as tunable inductors in the low-power regime. The results of this thesis can be used both in applications related to quantum computing and in the general study of quantum mechanics in circuits. In particular, we conclude that there exist feasible system parameters, for which it should be possible to demonstrate the tunable phase shifter utilizing available experimental techniques. We also show the principle how the possible nonlinearity of the phase shifter can be used in a two-qubit gate providing interactions between microwave photons.
Microwave photons in superconducting circuits is a promising approach for quantum computing due to long coherence times and the easy controllability of superconducting devices. However, as the search of the ultimate design of a quantum computer is still ongoing, various suggestions for the realization of a quantum bit, qubit, exist even in the area of superconducting circuits. In this thesis, we study theoretically quantum computing using itinerant microwave photons as qubits. Basic tools for treating the reflection of wave packets and the noise generated by resistive elements in a circuit are derived starting from a widely used model for superconducting transmission lines. We also present simple quantum devices that can act as quantum gates for microwave qubits, including a novel suggestion for a tunable phase shifter. A tunable phase shifter is a useful component in the study of microwave photons in general as well as in quantum computing. The tunable phase shifter consists of three superconducting quantum interference devices (SQUIDs) that can be treated as tunable inductors in the low-power regime. The results of this thesis can be used both in applications related to quantum computing and in the general study of quantum mechanics in circuits. In particular, we conclude that there exist feasible system parameters, for which it should be possible to demonstrate the tunable phase shifter utilizing available experimental techniques. We also show the principle how the possible nonlinearity of the phase shifter can be used in a two-qubit gate providing interactions between microwave photons.
Building a scalable quantum computer is one of the greatest challenges of this century. A quantum computer should be able to execute quantum algorithms which promise great speed up in solving certain computational problems. Microwave photons in superconducting circuits offer a promising platform for quantum computing due to their long coherence times, and the ease of their generation and control. Single-qubit quantum gates are implemented with conventional microwave components. However, implementing entangling two-qubit gates for photonic qubits is considerably difficult because photons do not interact with each other naturally. In this thesis, we investigate the effective interactions between photons mediated by superconducting qubits. Especially, the nonlinear effects in the photonic states generated by these interactions are studied. Based on the analysis of the scattering properties of the system, we present a design for a nonlinear phase shifter device, which yields a photon-number-dependent phase shift in the transmitted photons. Furthermore, we show how entangling two-qubit gates can be implemented using the presented nonlinear phase shifter.
We report a generic scheme to implement transmission-type quantum gates for propagating microwave photons, based on a sequence of lumped-element components on transmission lines. By choosing three equidistant superconducting quantum interference devices (SQUIDs) as the components on a single transmission line, we experimentally implement a magnetic-flux-tunable phase shifter and demonstrate that it produces a broad range of phase shifts and full transmission within the experimental uncertainty. Together with previously demonstrated beam splitters, these phase shifters can be utilized to implement arbitrary single-qubit gates. Furthermore, we theoretically show that replacing the SQUIDs by superconducting qubits, the phase shifter can be made strongly nonlinear, thus introducing deterministic photon--photon interactions. These results critically complement the previous demonstrations of on-demand single-photon sources and detectors, and hence pave the way for an all-microwave quantum computer based on propagating photons.
A quantum computer is a promising addition to a classical computer due to increase in performance on certain computational problems. A classical computer computes by manipulating bits, which assume a value either 0 or 1, using logical gates. Similarly, a quantum computation is carried out by manipulating quantum bits, so-called qubits, using quantum gates. The main advantage of qubits is that they can be not only in the states representing 0 and 1, but also in any superposition of these two states. Many different physical realizations for qubits have been proposed. One of the most promising candidates for the hardware of quantum computing are superconducting circuits. Here, the qubit can be represented in number of ways. For example, the two different states can be the direction of current circulating in a superconducting loop, or presence and absence of a photon in a transmission line. In this thesis, we study a tunable phase gate for microwave photons. The gate is implemented by a transmission line interrupted by three superconducting quantum interference devices (SQUIDs), which we model as inductors. We theoretically show that this system fulfills the requirements of a phase gate and that the tunability of the phase shift is frequency dependent. In addition, we consider a non-ideal system by including the effects of the capacitance associated with the SQUIDs. We find that the capacitance has no adverse effects, and in the best case, it may even increase the range of tunability. We also measure the phase shift at frequency of 6.3 GHz and find that the system is well described by the theory. To our knowledge, similar phase gate has not been experimentally studied before.
Quantum optics in superconducting circuits, known as circuit QED, studies the interaction of photons at microwave frequencies with artificial atoms made of Josephson junctions. Although quite young, remarkable progress has been made in this field over the past decade, especially given the interest in building a quantum computer using superconducting circuits. In this thesis based on the appended papers, we look at generation, engineering and detection of microwave photons using superconducting circuits. We do this by taking advantage of the strong coupling, on-chip tunability and huge nonlinearity available in superconducting circuits. First, we present the strong photon-photon interaction measured experimentally, as shown in the giant cross-Kerr effect. In this work, conditional phase shift of about 20 degrees per photon was measured between two coherent fields at single photon level. Given this strong interaction, we propose and analyze a cascaded setup based on the cross-Kerr effect to detect itinerant microwave photons, a long outstanding problem with only recent experimental realizations. We show that a nondestructive detection of microwave photons is possible with few cascaded transmons. The on-chip tunability of Superconducting Quantum Interference Device (SQUID) is exploited to create a tunable superconducting resonator in the next presented experimental work. Finally, we show that by placing the atom at the end of a transmission line, microwave photons can be generated efficiently and on-demand. We also present a setup that can generate the photons in arbitrary wave packets.
We introduce a magnetic-flux-tunable phase shifter for propagating microwave photons, based on three equidistant superconducting quantum interference devices (SQUIDs) on a transmission line. We experimentally implement the phase shifter and demonstrate that it produces a broad range of phase shifts and full transmission within the experimental uncertainty. Together with previously demonstrated beam splitters, this phase shifter can be utilized to implement arbitrary single-qubit gates for qubits based on propagating microwave photons. These results complement previous demonstrations of on-demand single-photon sources and detectors, and hence assist in the pursuit of an all-microwave quantum computer based on propagating photons.
In this thesis, measurements on superconducting electrical circuits that exhibit quantum mechanical effects are presented. Nonlinear superconducting circuits can be regarded as artificial atoms with energy spectra that can be engineered. The artificial atoms can be used as quantum bits for quantum computation. By coupling the artificial atoms to the electromagnetic field inside a transmission line resonator, quantum optical effects can be studied in these electrical circuits.The possibility of using a frequency tunable transmission line resonators for coupling of superconducting quantum bits is investigated. Tunable superconducting resonators consisting of coplanar waveguide structures, terminated to ground in one end via a superconducting quantum interference device (SQUID), are characterized. By applying a magnetic field to the SQUID loop, the boundary condition of the resonator changes leading to a change in resonance frequency.A change of the resonance frequency by more than 250 linewidths is demonstrated. It is also demonstrated that the resonator can be tune by several hundred MHz on a time scale that is much shorter than the photon lifetime of the resonator.Circuits containing both one and two transmon qubits coupled to the resonator are characterized. Quantum mechanical interactions between the resonator and the qubits, in form of vacuum Rabi splitting, are observed. Coherent control of a single qubit coupled to the resonator is demonstrated by performing Rabi oscillations of the qubit. In the two qubit case, coherent interaction between the qubits is observed in terms of a nonlinear scaling of the coupling strength between the resonator and the collective two qubit system.Measurements on a strongly driven single Cooper-pair box, using a quantum capacitance readout, are also presented. A rich structure in the response is well explained in terms of longitudinal dressed states of the single-Cooper pair box.When measuring the response of a single-electron box, a gate dependent excess dissipation is observed. The dissipation is understood by considering the internal dynamics of the single electron box. The excess dissipation is modeled as an effective resistance named the Sisyphus resistance.