A quantum computer is an information processing machine, much like an ordinary classical computer, but its function is based on quantum mechanical principles. To be able to construct such a machine would be a fantastic accomplishment---to have total control over a quantum system is a dream for both physicists and science-fiction enthusiasts. The basic information unit in a quantum computer is the quantum bit, or qubit for short. A quantum computer consists of many coupled qubits. To get a single qubit to work properly, would be a major step towards building this machine. Here we study two different qubit ideas. The central element in both setups is the superconducting tunnel junction---the Josephson junction. By connecting the Josephson junctions to standard electronics in a clever way, a qubit can be realised. With these constructions it is in principle very easy to manipulate and read out the quantum probabilities, by varying voltages and currents in time. However, this ease of manipulation has a cost: strong interactions with uncontrolled degrees of freedom of the environment transfer information from the qubit. For superconducting qubits this decoherence is typically very fast. There are ways to deal with the decoherence. One way is to tune the circuit parameters so that the decoherence becomes minimal. Another way is to engineer the qubits so fast so that the effect of decoherence becomes small. In this thesis, we will apply both these strategies. Specifically, the measurement speed of the second qubit we study, turns out to be very sensitive to the topology of the phase space of the detector variables.
A quantum computer is an information processing machine, much like an ordinary classical computer, but its function is based on quantum mechanical principles. To be able to construct such a machine would be a fantastic accomplishment---to have total control over a quantum system is a dream for both physicists and science-fiction enthusiasts. The basic information unit in a quantum computer is the quantum bit, or qubit for short. A quantum computer consists of many coupled qubits. To get a single qubit to work properly, would be a major step towards building this machine. Here we study two different qubit ideas. The central element in both setups is the superconducting tunnel junction---the Josephson junction. By connecting the Josephson junctions to standard electronics in a clever way, a qubit can be realised. With these constructions it is in principle very easy to manipulate and read out the quantum probabilities, by varying voltages and currents in time. However, this ease of manipulation has a cost: strong interactions with uncontrolled degrees of freedom of the environment transfer information from the qubit. For superconducting qubits this decoherence is typically very fast. There are ways to deal with the decoherence. One way is to tune the circuit parameters so that the decoherence becomes minimal. Another way is to engineer the qubits so fast so that the effect of decoherence becomes small. In this thesis, we will apply both these strategies. Specifically, the measurement speed of the second qubit we study, turns out to be very sensitive to the topology of the phase space of the detector variables.
Few technological ideas in recent memory have stirred as much excitement—or skepticism—as quantum computing. The promise is genuine: for certain computational tasks, a quantum machine can do in seconds what would take a classical supercomputer longer than the age of the universe. At the heart of realising that promise is the qubit, the quantum analogue of a classical bit, and among the many physical systems that can host a qubit, superconducting circuits have quietly become the frontrunner. They can be fabricated in the same cleanrooms used to make conventional chips, controlled with ordinary microwave electronics, and scaled up in ways that other approaches cannot yet match. This thesis traces the full arc from first principles to working code. We start with the quantum mechanics a reader needs—superposition, entanglement, measurement—and build from therethrough the physics of superconductivity to the engineering of real qubit circuits. A key character in that story is the Josephson junction: a thin sandwich of metal and insulator whose quirky nonlinear behaviour is what makes the whole enterprise of superconducting qubits possible. A major focus of this work is the surface code, a topological quantum error-correcting code widely regarded as the most promising approach for achieving fault-tolerant quantum computation. We provide a rigorous analysis of the surface code’s lattice structure, stabilizer formalism, syndrome extraction, and decoding algorithms—in particular the minimum-weight perfect matching (MWPM) algorithm. A full Python/Qiskit implementation of a distance-3 surface code is presented, including qubit initialization, syndrome measurement circuits, error simulation, and decoding logic. We close by looking ahead. Modular processors, 3D circuit integration, and the long road to fault-tolerant machines are all on the horizon, and we try to give an honest assessment of where things stand and what the real obstacles are. Throughout, the goal has been to bridge theory and practice—to be precise enough to be useful, but grounded enough to stay connected to what is actually happening in laboratories today.
The theory of superconducting phase qubits---also known as current-biased Josephson junctions---is presented. In the first part of this thesis, I introduce quantum computation, quantum simulation, and their deep connection with symplectic integration. I then consider the fundamental many-body theory of superconductivity and Josephson junctions and show how the quantum dynamics of a single macroscopic degree of freedom, the gauge invariant phase difference, emerges. A complete study of the Hilbert space structure of such a variable is performed for the current-biased junction. The resulting resonance structure is studied in detail, using various formalisms including the WKB approximation, instanton methods, the complex scaling transformation, basis set stabilization, numerical integration, and dynamical simulation using Lie algebraic wave-packet propagation. The second part of this thesis explores how the current-biased junction can be used as an element of a quantum computer---a quantum bit (qubit). Single qubit operations are studied, followed by the presentation of the theory of coupled qubit devices. My key result is the design and optimization of quantum logic gates with high fidelity (F ~ 0.9999) for capacitively coupled phase qubits with short gate times (~ 10 ns). Finally, I examine an advanced qubit-coupling scheme, a resonant coupling method utilizing a harmonic oscillator as the auxiliary degree of freedom. The models and methods presented here have been developed in direct collaboration with an experimental program. These experiments are the first to show spectroscopic evidence for entanglement between two and three macroscopic degrees of freedom in a superconducting circuit.
Quantum-state engineering, i.e., active control over the coherent dynamics of suitable quantum-mechanical systems, has become a fascinating prospect of modern physics. With concepts developed in atomic and molecular physics and in the context of NMR, the field has been stimulated further by the perspectives of quantum computation and communication. Low-capacitance Josephson tunneling junctions offer a promising way to realize quantum bits (qubits) for quantum information processing. The article reviews the properties of these devices and the practical and fundamental obstacles to their use. Two kinds of device have been proposed, based on either charge or phase (flux) degrees of freedom. Single- and two-qubit quantum manipulations can be controlled by gate voltages in one case and by magnetic fields in the other case. Both kinds of device can be fabricated with present technology. In flux qubit devices, an important milestone, the observation of superpositions of different flux states in the system eigenstates, has been achieved. The Josephson charge qubit has even demonstrated coherent superpositions of states readable in the time domain. There are two major problems that must be solved before these devices can be used for quantum information processing. One must have a long phase coherence time, which requires that external sources of dephasing be minimized. The review discusses relevant parameters and provides estimates of the decoherence time. Another problem is in the readout of the final state of the system. This issue is illustrated with a possible realization by a single-electron transistor capacitively coupled to the Josephson device, but general properties of measuring devices are also discussed. Finally, the review describes how the basic physical manipulations on an ideal device can be combined to perform useful operations.
Quantum computing is a novel paradigm for information processing that promises computational speedup for certain computational tasks while not being driven by Moore's law. Superconducting circuits based on Josephson junctions are one of the leading approaches for implementing the quantum bits, or qubits, in a quantum processor. Most superconducting qubit architectures involve circuit quantum electrodynamics, where the qubits are analogous to artificial atoms that are coupled to superconducting linear resonant cavities. This scheme allows for the readout of individual qubits, the manipulation of single-qubit states, and the generation of entanglement between qubits. The performance of current state-of-the-art systems is limited by decoherence of the qubits and associated gate errors. Nonetheless, experiments have been performed that already demonstrate a significant computational speedup compared to classical processing. Future implementations of fault-tolerant quantum computers will require the development of quantum error correction. Initial tests of elements of error correction in superconducting qubit arrays are underway.
Among the potential physical implementations of quantum bits (qubits) solid state devices appear most promising for large scale applications and integration in electronic circuits. Devices based on low-capacitance Josephson junctions exploit the coherence of the superconducting state, combined with the possibility to control individual charges by Coulomb blockade effects. The logical states in these systems differ by one Cooper-pair charge. Single- and two-bit operations can be performed by applying a sequence of gate voltages. The phase coherence time is sufficiently long to allow a series of these steps. If, in addition, the strength of the Josephson coupling is controlled (e.g. by currents in SQUID loops) the system is ideal for quantum computation. Recent experiments by Nakamura et al. show the feasibility of this approach. In addition to the manipulations of qubits the resulting quantum state has to be read out. This quantum measurement process can be accomplished by coupling a single-electron transistor (SET) capacitively to the qubit. We investigate the process by analyzing the time-evolution of the density matrix of the coupled system. The measurement process is characterized by three time scales: the dephasing time, which is fast when a dissipative current flows in the SET, a longer ‘measurement time’ needed to read out the signal, and a third, even longer ‘mixing time’, characterizing measurement induced transitions which destroy the information about the initial quantum state. The noise spectrum of the current in the SET displays these time scales and allows their direct observation.
Superconducting qubits are fabricated "loss-free" electrical circuits on a chip with size features of tens of nanometers. If cooled to cryogenic temperatures below -273 °C they behave as quantum elements, similar to atoms and molecules. Such a qubit can be manipulated by fast-oscillating magnetic fields and its resulting quantum state can be measured using a proper magnetic detector. These qubits could be the basic elements for a quantum processor. To that order, multiple qubits have to controllably interact, and to that purpose very high quality resonators are fabricated on the same chip. Effectively the interaction results from the phase-coherent exchange of a single photon between qubit and resonator. In this way it is highly analogous to the so-called cavity QED in atomic physics where single atoms couple to a single trapped photon, all inside a cavity resonator. In this thesis, a superconducting flux qubit has been coupled to an LC resonator and coherent exchange of a single photon between qubit and resonator has been achieved. A striking option in these superconducting circuits, not attainable in "common" atomic cavity-QED, is that the interaction between qubit and photon can be made very large, up to the limit that one no longer can talk about the qubit and resonator as such. This limit requires a fully different description and it leads to novel effects, e.g. in the energy spectrum of the qubit-resonator composite. The Bloch-Siegert spectral shift is experimentally demonstrated in this thesis. As a small but critical technical aside a new type of control line filter has been developed, which allows strong suppression of excess electrical noise that forms a threat for qubit coherence. Having two modes of operation, it shorts all low-frequency noise if control is idle while being transparent if control is active, where short or open is determined by the control line signal level itself.