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.