Computing prime factors with a Josephson phase qubit quantum processor (1202.5707v1)

Abstract: A quantum processor (QuP) can be used to exploit quantum mechanics to find the prime factors of composite numbers[1]. Compiled versions of Shor's algorithm have been demonstrated on ensemble quantum systems[2] and photonic systems[3-5], however this has yet to be shown using solid state quantum bits (qubits). Two advantages of superconducting qubit architectures are the use of conventional microfabrication techniques, which allow straightforward scaling to large numbers of qubits, and a toolkit of circuit elements that can be used to engineer a variety of qubit types and interactions[6, 7]. Using a number of recent qubit control and hardware advances [7-13], here we demonstrate a nine-quantum-element solid-state QuP and show three experiments to highlight its capabilities. We begin by characterizing the device with spectroscopy. Next, we produces coherent interactions between five qubits and verify bi- and tripartite entanglement via quantum state tomography (QST) [8, 12, 14, 15]. In the final experiment, we run a three-qubit compiled version of Shor's algorithm to factor the number 15, and successfully find the prime factors 48% of the time. 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.

• The paper demonstrates the practical implementation of Shor’s algorithm on a nine-element Josephson phase qubit processor, achieving a 48% success rate in factoring 15.
• The study employs advanced microfabrication techniques to enable high-fidelity entanglement, with reported Bell singlet and three-qubit W-state fidelities of 0.89 and 0.69 respectively.
• The results underscore the potential scalability of superconducting qubit systems by leveraging rapid inter-qubit coupling and coherent multi-qubit operations, despite current coherence time limitations.

Analyzing Prime Factorization via a Josephson Phase Qubit Quantum Processor

This paper presents an empirical paper on the implementation of a quantum algorithm using a solid-state, Josephson phase qubit quantum processor, with a focus on prime factorization via Shor's algorithm. The authors constructed a quantum processor (QuP) composed of a nine-element ensemble including four phase qubits and five coplanar waveguide resonators. These elements were tailored to achieve high-fidelity entanglement necessary for executing quantum circuits.

Initially, the framework demonstrated included the scalability of superconducting qubit systems, exploiting conventional microfabrication techniques, thus showcasing potential for scaling to more complex circuits. The device was capable of performing various operations like high-precision single-qubit gates coupled with swap and controlled-phase gates, facilitated by a bus resonator for multi-qubit entanglement.

The paper stands out as it demonstrates the practical viability of a Josephson phase qubit-based QuP by implementing a compiled version of Shor's algorithm to factor the number 15 with a 48% success rate. The demonstration incorporated rapid entangling capabilities by bringing multiple qubits on resonance, in novel fashion, increasing the effective coupling strength relative to the number of qubits—a characteristic critical for scalable quantum computations.

The authors reported running coherent interactions for up to four qubits with entanglement tomographically verified to include Bell singlet and W-states using Quantum State Tomography (QST). They assert that generating both bi-partite and tri-partite entangled states requires fewer resources than alternative methods, thereby offering a promising approach for more extensive and complex quantum algorithms.

The numerical results, particularly the Bell singlet formed with a fidelity of 0.89 and a three-qubit W-state having a fidelity of 0.69, highlight the device's competence to handle entanglement with significant coherence and accuracy. This is further underlined by the observed $\sqrt{N}$ scaling of coupling strength with number of qubits, as demonstrated during the expeditions on rapid entanglement.

The latter sections detailed the successful execution of Shor's algorithm, where each computational step was rigorously analyzed. Quantum Fourier transformations were achieved, and the results led to successful factoring after classical post-processing 48% of the time. The implications are crucial—demonstrating a pathway to tackle larger composite numbers, a landmark for quantum computation advancements using superconducting technologies.

While limitations in coherence times persist, improvements suggest a trajectory towards larger-scale implementations. Future research is poised to focus on refining coherence and fidelity while exploring more advanced quantum algorithms. Overall, the paper elucidates a pivotal advancement in the field of superconducting qubits, contributing profoundly to the field of quantum computation and solid-state quantum technology development.