Josephson phase-quantum-bits, ("qubits"), together with superconducting resonators, comprise the essential quantum elements in a state-of-the-art quantum processor (QuP). A QuP can be used to exploit quantum mechanics to find the prime factors of composite numbers by running Shor's algorithm[57].

In this thesis, I describe the first solid-state demonstration of a compiled version of Shor's algorithm. To meet this challenge, I designed a QuP so that I could map the problem of factoring the number N = 15 onto a quantum circuit that is compatible with our technological capabilities. The QuP is composed of nine quantum elements: four phase qubits and five superconducting coplanar waveguide (CPW) resonators. Using this device, I ran a three-qubit complied version of Shor's algorithm and successfully found the prime factors 48% of the time (compared to the ideal success rate of 50 %). In addition, the QuP produced coherent interactions between five quantum elements, and bi- and tripartite entanglement, which was verified via quantum state tomography (QST).

Scaling up to nine quantum elements and performing these experiments represent key milestones to realizing a quantum computer. Continued improvements in the superconducting qubit coherence times and more complex circuits should provide the resources necessary to factor larger composite numbers and run more intricate quantum algorithms in the near future.