Experimental realization of Shor's quantum factoring algorithm using nuclear magnetic resonance

This paper surveys our recent research on quantum information processing by nuclear magnetic resonance (NMR) spectroscopy. We begin with a geometric introduction to the NMR of an ensemble of indistinguishable spins, and then show how this geometric interpretation is contained within an algebra of multispin product operators. This algebra is used throughout the rest of the paper to demonstrate that it provides a facile framework within which to study quantum information proces...

In this chapter we review the contributions of Nuclear Magnetic Resonance to the study of quantum correlations, including its capabilities to prepare initial states, generate unitary transformations, and characterize the final state. These are the three main demands to implement quantum information processing in a physical system, which NMR offers, nearly to perfection, though for a small number of qubits. Our main discussion will concern liquid samples at room temperature.

Lectures on quantum computing. Contents: Algorithms. Quantum circuits. Quantum Fourier transform. Elements of number theory. Modular exponentiation. Shor`s algorithm for finding the order. Computational complexity of Schor`s algorithm. Factoring integers. NP-complete problems.

This thesis addresses the problems of initialization and separability in liquid state NMR based quantum information processors. We prepare pure quantum states lying above the entanglement threshold. Our pure state quantum computer derives its purity from the highly polarized nuclear spin states in the para-hydrogen molecule. The thesis begins with a critique of conventional NMR based quantum information processing outlining the major strengths and weaknesses of the technology...

Quantum mechanics provides spectacular new information processing abilities (Bennett 1995, Preskill 1998). One of the most unexpected is a procedure called quantum teleportation (Bennett et al 1993) that allows the quantum state of a system to be transported from one location to another, without moving through the intervening space. Partial implementations of teleportation (Bouwmeester et al 1997, Boschi et al 1998) over macroscopic distances have been achieved using optical ...

This thesis focuses on the experimental creation and detection of a variety of quantum correlations using nuclear magnetic resonance hardware. Quantum entanglement, being most common and counter-intuitive, is one of the main type considered in this thesis. Quantum correlations play a major role in achieving, the much talked, computational speedup. Creation and detection of such correlations experimentally is a major thrust area in experimental quantum information processing f...

We describe an array of quantum gates implementing Shor's algorithm for prime factorization in a quantum computer. The array includes a circuit for modular exponentiation with several subcomponents (such as controlled multipliers, adders, etc) which are described in terms of elementary Toffoli gates. We present a simple analysis of the impact of losses and decoherence on the performance of this quantum factoring circuit. For that purpose, we simulate a quantum computer which ...

While a bit is the fundamental unit of binary classical information, a qubit is the fundamental unit of quantum information. In quantum information processing (QIP), it is customary to call the qubits under study as system qubits, and the additional qubits as ancillary qubits. In this thesis, I describe various schemes to exploit the ancillary qubits to efficiently perform many QIP tasks and their experimental demonstrations in nuclear magnetic resonance (NMR) systems. Partic...

Very recently, Monz, et al. [arXiv:1507.08852] have reported the demonstration of factoring 15 using a scalable Shor algorithm with an ion-trap quantum computer. In this note, we remark that the report is somewhat misleading because there are three flaws in the proposed circuit diagram of Shor algorithm. We also remark that the principles behind the demonstration have not been explained properly, including its correctness and complexity.

We consider how to optimize memory use and computation time in operating a quantum computer. In particular, we estimate the number of memory qubits and the number of operations required to perform factorization, using the algorithm suggested by Shor. A $K$-bit number can be factored in time of order $K^3$ using a machine capable of storing $5K+1$ qubits. Evaluation of the modular exponential function (the bottleneck of Shor's algorithm) could be achieved with about $72 K^3$ e...