Due to recent developments and accessible fabrication techniques, superconducting qubits

have become one of the most popular candidates for realizing a large-scale fault-tolerant

quantum computer. Currently, transmon is the most preferred type of superconducting

qubits because it is less sensitivity to noise. To perform operations on the qubits, we need

quantum gates with high fidelity. One of the high-fidelity two-qubit entangling gates used

for superconducting qubits is the Cross-Resonance (CR) gate. In CR gate, two frequency-detuned qubits have a weak coupling and one of them (called control qubit) is driven by a

microwave at the frequency of the other qubit (called target). This induces Rabi oscillations

of the target qubit, whose frequency depends on the state (|0> and |1>) of the control qubit.

This entangles the two qubits, thus providing a natural way to realize CNOT gate. In

this thesis, we study analytically, semi-analytically and numerically the operation of the

Cross-Resonance gate for superconducting qubits to implement the CNOT operation. We

also study various intrinsic errors associated with the CR gate.

Chapter 1 of this thesis gives an introduction. In Chapter 2, we discuss the Hamilitonian of the CR gate. In Chapter 3, we first consider the ideal operation of the CR gate, then derive the next-order analytics, and then develop a semi-analytical approach. Chapter 4 gives a description of our numerical model. Numerical results for the CNOT-equivalent gate duration and compensating single-qubit rotations are discussed in Chapter 5. In Chapter 6, we analyze the error budget for the CNOT-gate intrinsic infidelity. In Chapter 7, we discuss the dependence of infidelity and CNOT duration on various parameters including detuning between control and target qubits, drive frequency, coupling between control and target qubit, smoothness of the pulse ramps, and microwave crosstalk. Chapter 8 presents conclusions.

Quantum computers have the capability to improve the efficiency and speed of many computational tasks. Among different candidates for physical implementation of quantum bits (qubits), superconducting qubits are currently one of the most promising candidates due to their accessible fabrication process and recent rapid developments. Measurement of these qubits is usually done in a circuit quantum electrodynamics (QED) setup, for which experimental and theoretical research is conducted to improve the accuracy and speed of the qubit readout. In this dissertation we study some aspects of the dispersive readout of superconducting qubits, and introduce tools and methods for studying these systems.

In Chapter 2 we show that in presence of neighboring qubits, the system is typically measured in the basis of joint eigenstates of qubits, in contrast to what is expected from the textbook collapse postulate. In such setups, the excitation can switch between the eigenstates, leading to measurement error. In Chapter 3 we study the joint state of the qubit-resonator system during the measurement, and show that the qubit-induced nonlinearity of the resonator squeezes its state, and within the rotating wave approximation (RWA) the system mostly remains in the joint eigenladder that is associated with the qubit's initial state. In Chapter 4 we show that built-in energy resonances in the qubit-resonator Jaynes-Cummings ladder occur at specific resonator populations, and the couplings between these resonant levels are provided by the usually neglected non-RWA terms. Such resonances lead to measurement deterioration by exciting the qubit out of the computational subspace. In Chapter 5 we provide a hybrid phase-space-Fock-space approach for studying the evolution and squeezing of driven nonlinear resonators within Gaussian approximation, which is numerically efficient and sufficiently accurate. In Chapter 6 we study the propagating squeezed field that leaks out of the resonator, and write evolution equations for the correlators of the measured field quadrature. These equations are easy to simulate and can describe the squeezing of the propagating field during the transient, which can be used to optimize the fidelity and speed of the quadrature measurement in the dispersive readout of superconducting qubits.

The potential of superconducting qubits as the medium for a scalable quantum computer has motivated the pursuit of improved interactions within this system. Two challenges for the field of superconducting qubits are measurement fidelity, to accurately determine the state of the qubit, and the efficient transfer of quantum states. In measurement, the current state-of-the-art method employs dispersive readout, by coupling the qubit to a cavity and reading the resulting shift in cavity frequency to infer the qubit's state; however, this is vulnerable to Purcell relaxation, as well as being modeled off a simplified two-level abstraction of the qubit. In state transfer, the existing proposal for moving quantum states is mostly untested against non-idealities that will likely be present in an experiment. In this dissertation, we examine three problems within these two areas.

We first describe a new scheme for fast and high-fidelity dispersive measurement specifically designed to circumvent the Purcell Effect. To do this, the qubit-resonator interaction is turned on only when the resonator is decoupled from the environment; then, after the resonator state has shifted enough to infer the qubit state, the qubit-resonator interaction is turned off before the resonator and environment are recoupled. We also show that the effectiveness of this ``Catch-Disperse-Release'' procedure partly originates from quadrature squeezing of the resonator state induced by the Jaynes-Cummings nonlinearity.

The Catch-Disperse-Release measurement scheme treats the qubit as a two-level system, which is a common simplification used in theoretical works. However, the most promising physical candidate for a superconducting qubit, the transmon, is a multi-level system. In the second work, we examine the effects of including the higher energy levels of the transmon. Specifically, we expand the eigenstate picture developed in the first work to encompass multiple qubit levels, and examine the resulting changes to the system. In particular, we analyze the population of the non-target eigenstates as a result of this expanded model, and provide an analytical form for these deviations from the simpler model in Catch-Disperse-Release (i.e., the dressed state approximation).

Lastly, we assess the robustness of the existing quantum state transfer protocol, testing its performance under typical experimental deviations from the ideal case. We show that the procedure is resilient to almost all non-idealities, except frequency mismatch between the two cavities. We also demonstrate a method to compensate for one such error in frequency-matching.