Based on a network graph analysis of the underlying circuit, a quantum theory of arbitrary superconducting charge qubits is derived. Describing the dissipative elements of the circuit with a Caldeira-Leggett model, we calculate the decoherence and leakage rates of a charge qubit. The analysis includes decoherence due to a dissipative circuit element such as a voltage source or the quasiparticle resistances of the Josephson junctions in the circuit. The theory presented here is dual to the quantum circuit theory for superconducting flux qubits. In contrast to spin-boson models, the full Hilbert space structure of the qubit and its coupling to the dissipative environment are taken into account. Moreover, both self- and mutual inductances of the circuit are fully included.
Based on a network graph analysis of the underlying circuit, a quantum theory of arbitrary superconducting charge qubits is derived. Describing the dissipative elements of the circuit with a Caldeira-Leggett model, we calculate the decoherence and leakage rates of a charge qubit. The analysis includes decoherence due to a dissipative circuit element such as a voltage source or the quasiparticle resistances of the Josephson junctions in the circuit. The theory presented here is dual to the quantum circuit theory for superconducting flux qubits. In contrast to spin-boson models, the full Hilbert space structure of the qubit and its coupling to the dissipative environment are taken into account. Moreover, both self- and mutual inductances of the circuit are fully included.
We present a multilevel quantum theory of decoherence for a general circuit realization of a superconducting qubit. Using electrical network graph theory, we derive a Hamiltonian for the circuit. The dissipative circuit elements (external impedances, shunt resistors) are described using the Caldeira-Leggett model. The master equation for the superconducting phases in the Born-Markov approximation is derived and brought into the Bloch-Redfield form in order to describe multilevel dissipative quantum dynamics of the circuit. The model takes into account leakage effects, i.e., transitions from the allowed qubit states to higher excited states of the system. As a special case, we truncate the Hilbert space and derive a two-level (Bloch) theory with characteristic relaxation (T1) and decoherence (T2) times. We apply our theory to the class of superconducting flux qubits; however, the formalism can be applied for both superconducting flux and charge qubits.
In the Born–Markov approximation, decoherence property of superconducting quantum circuit with a flux qubit is investigated. Using Kirchhoff's laws of classical circuit and quantum theory, we derive a Hamiltonian of the circuit. Then, combining with the Bloch–Redfield equation and in the two-level approximation, the energy relaxation time and the decoherence time of superconducting qubits is studied. Compared to the spin-boson model, this method not only investigates a decoherence being caused by the dissipative environment, but also the decoherence being generated by the dissipative elements in a superconducting electronic circuit. Hence, it is good for studying the decoherence of superconducting qubits.
Quantum computers utilize qubits to store and process quantum information. In superconducting quantum computers, qubits are implemented as quantum superconducting resonant circuits. The circuits are operated only at the two energy states, which form the computational basis for the qubit. To suppress leakage to uncomputational states, superconducting qubits are designed to be anharmonic oscillators, which is achieved using one or more Josephson junctions, a nonlinear superconducting element. One of the main challenges in developing quantum computers is minimizing the decoherence caused by environmental noise. Decoherence is characterized by two coherence times, T1 for depolarization processes and T2 for dephasing. This thesis reviews and investigates the decoherence properties of superconducting qubits. The main goal of the thesis is to analyze the tradeoff between anharmonicity and dephasing in a qubit unimon. Recently developed unimon incorporates a single Josephson junction shunted by a linear inductor and a capacitor. Unimon is tunable by external magnetic flux, and at the half flux quantum bias, the Josephson energy is partially canceled by the inductive energy, allowing unimon to have relatively high anharmonicity while remaining fully protected against low-frequency charge noise. In addition, at the sweet spot with respect to the magnetic flux, unimon becomes immune to first-order perturbations in the flux. The sweet spot, however, is relatively narrow, making unimon susceptible to dephasing through the quadratic coupling to the flux noise. In the first chapter of this thesis, we present a comprehensive look into the basic theory of superconducting qubits, starting with two-state quantum systems, followed by superconductivity and superconducting circuit elements, and finally combining these two by introducing circuit quantum electrodynamics (cQED), a framework for building superconducting qubits. We follow with a theoretical discussion of decoherence in two-state quantum systems, described by the Bloch-Redfield formalism. We continue the discussion by estimating decoherence using perturbation theory, with special care put into the dephasing due to the low-frequency 1/f noise. Finally, we review the theoretical model of unimon, which is used in the numerical analysis. As a main result of this thesis, we suggest a design parameter regime for unimon, which gives the best ratio between anharmonicity and T2.
A complete analysis of the decoherence properties of a Josephson junction qubit is presented. The qubit is of the flux type and consists of two large loops forming a gradiometer, one small loop, and three Josephson junctions. We develop a complete, quantitative description of the inductances and capacitances of the circuit. Including two stray capacitances makes the quantum mechanical modeling of the system five dimensional. To make the required calculation tractable, we devise a general Born-Oppenheimer approximation which reduces the effective dimensionality in the calculation to one. Contributions to relaxation (T1) and dephasing $({T}_{\ensuremath{\phi}})$ arising from two different control circuits, one coupled to the small loop and one coupled to a large loop, is computed. We explore T1 and ${T}_{\ensuremath{\phi}}$ along an optimal line in the space of applied fluxes; along this ``S line'' we see significant and rapidly varying contributions to the decoherence parameters, primarily from the circuit coupling to the large loop.
We calculate for the current-biased Josephson junction the decoherence of the qubit state from noise and dissipation. The effect of dissipation can be entirely accounted for through a semiclassical noise model that appropriately includes the effect of zero-point and thermal fluctuations from dissipation. The magnitude and frequency dependence of this dissipation can be fully evaluated with this model to obtain design constraints for small decoherence. We also calculate decoherence from spin echo and Rabi control sequences and show they are much less sensitive to low-frequency noise than for a Ramsey sequence. We predict small decoherence rates from 1/f noise of charge, critical current, and flux based on noise measurements in prior experiments. Our results indicate this system is a good candidate for a solid-state quantum computer.
We describe in this paper how the nonlinear Josephson inductance is the crucial circuit element for all Josephson qubits. We discuss the three types of qubit circuits, and show how these circuits use this nonlinearity in unique manners. We give a brief derivation of the BCS theory, highlighting the appearance of the macroscopic phase parameter. The Josephson equations are derived using standard first and second order perturbation theory that describe quasiparticle and Cooper-pair tunneling. An exact calculation of the Josephson effect then follows using the quasiparticle bound-state theory, and then expand upon this theory to describe quasiparticle excitations as transitions from the ground to excited bound states from nonadiabatic changes in the bias. Although quasiparticle current is typically calculated only for a constant DC voltage, the advantage to this approach is seen where we qualitatively describe quasiparticle tunneling with AC voltage excitations, as appropriate for the qubit state. This section describes how the Josephson qubit is typically insensitive to quasiparticle damping, even to the extent that a phase qubit can be constructed from microbridge junctions.