Reviving Nuclear Magnetic Resonance as a viable approach for quantum computation and Solving simultaneous Pell equations by quantum computational means
This thesis consists of two parts. I describe an approach for a physical realization of a quantum computer by Nuclear Magnetic Resonance (NMR). I propose a new framework for NMR that gives such a physical realization. I construct a quantum mechanical description of NMR from which I can build the fundamental elementary operators for quantum computation. I describe the experiments that build these operators. I give a polynomial time quantum algorithm to solve simultaneous Pell equations as an extension of Hallgren's algorithm for simple Pell equations.
This thesis consists of two parts. I describe an approach for a physical realization of a quantum computer by Nuclear Magnetic Resonance (NMR). I propose a new framework for NMR that gives such a physical realization. I construct a quantum mechanical description of NMR from which I can build the fundamental elementary operators for quantum computation. I describe the experiments that build these operators. I give a polynomial time quantum algorithm to solve simultaneous Pell equations as an extension of Hallgren's algorithm for simple Pell equations.
Cette these contient deux parties. Je decris une approche pour construire une realisation physique d'un ordinateur quantique par Resonance Magnetique Nucleaire (RMN). Je propose un nouveau cadre pour la RMN dans les realisations physiques d'un ordinateur quantique. Je construis une description de la RMN a partir de la mecanique quantique avec laquelle je peux construire les operateurs elementaires essentiels pour le calcul quantique. Je decris les experiences pour construire ces operateurs. Je propose un algorithme quantique en temps polynomial pour resoudre des equations de Pell simultanees comme extension de l'algorithme de Hallgren pour des equations de Pell simples.
Standard CMOS circuits are expected to be no longer able to satisfy nowadays demand for higher performances and lower power consumption. This commits to look for new solutions. Quantum computation, whose fundamental quantity of information is the qubit, is a thoroughly new way of thinking to electronics. There are at least two main reasons for which quantum computing is expected to broaden computation horizons. First, specific algorithms can be engineered, manipulating the probability amplitudes of different states, to exploit the quantum superposition principle and achieve high parallelism. Second, it is expected to be spontaneously suitable to simulate complex quantum physical systems, as large molecules. Quantum computing requires the cooperation of several scientific fields. The leading idea has been to highlight the operations and algorithms a quantum computer must be able to perform and, then, to propose a hardware solution, providing the rigorous physical treatment mandatory for a deep understanding and for an appropriate modelling, something which is hard to find in nowadays literature. Thus, the first chapters address the quantum algorithms, focusing on Grover's search algorithm and deriving, from fundamentals, the corresponding circuit. The latter is then described in Quantum Assembly (QASM), which can be seen as the quantum equivalent of VHDL, and simulated on IBM Q Experience. The following chapters deal with a possible physical implementation of quantum processors: the encoding of qubits on nuclear spins in diamagnetic molecules. When a static magnetic field is applied to a spin-1/2 nucleus, two well-defined energy eigenstates arise, as a result of the Zeeman effect. Each of them can be associated with a qubit eigenstate. Thanks to nuclear magnetic resonance principles, the probability with which one of the two eigenstates is measured can be handled by the superposition of a radio frequency field at the resonance frequency of the nucleus. This can be interpreted as a rotation of the spin, and so of the qubit, about the RF field and enables the execution of single qubit quantum gates, since they are simple rotations of the state vector. The two-qubit CNOT gate is obtained through the interaction of spins via J-coupling. Considering that these gates constitute a universal set, every quantum algorithm can theoretically be run on an NMR processor. Different nuclei have different resonant frequencies, but the addressing of a specific spin can be achieved also in homonuclear molecules, since the resonant frequency depends also on the nuclear environment (nuclear shielding). While unwanted J-couplings can be removed thanks to refocusing, spin decoherence and relaxation phenomena force an upper limit to the allowed timescale. A main advantage of NMR is that this timescale is long, even at room temperature. Conversely, scalability is still an issue. A simple heteronuclear MATLAB model, based on the relations derived from first principles in the thesis, is proposed to prove the feasibility of NMR quantum processors. The input technological parameters, as chemical shielding and J-coupling, can be computed resorting on ORCA, or obtained from experimental data. The model, able to run a universal set of quantum gates and simple algorithms, can be useful to find an optimal operating point as a compromise between molecule physical properties and the quantities which can be controlled by nowadays NMR instrumentation.
Quantum computers can potentially solve real and relevant mathematical and physical problems that are intractable using classical machines. However, the experimental realization of quantum computers represents a significant challenge, because several opposing experimental requirements must be met. A set of coupled quantum bits must be manipulated and measured while coherently retaining entangled quantum states. Yet the manipulation and measurement processes almost inevitably lead to the decay of these fragile states.
This thesis work takes significant steps towards building a practical quantum computer using nuclear spins and liquid state nuclear magnetic resonance (NMR) techniques. I present experimental results for proof of principle of quantum computing in a series of small implementations of quantum algorithms, culminating in the implementation of an adiabatic quantum optimization algorithm. The performance of adiabatic algorithms compared with classical optimization methods is unknown, but small quantum computers could provide crucial insight into answering this question. Furthermore, adiabatic algorithms also shed new light on the usefulness of quantum resources for computational tasks and hence they represent an important class of algorithms.
The second part of this thesis presents several methods developed for improving the control over NMR quantum computing experiments. Even though liquid state NMR quantum computers have well known and accepted scaling limitations, the developed tools are of general use. I will show how several tools are directly transferable to two other implementations of quantum computers; one using optical methods and the other using loops of superconducting material.
This work provides insight into what is needed to build a practical quantum computer, and delivers several tools and techniques that will be useful in future large-scale implementations of quantum computers.
Quantum information processing is the use of inherently quantum mechanical phenomena to perform information processing tasks that cannot be achieved using conventional classical information technologies. One famous example is quantum computing, which would permit calculations to be performed that are beyond the reach of any conceivable conventional computer. Initially it appeared that actually building a quantum computer would be extremely difficult, but in the last few years there has been an explosion of interest in the use of techniques adapted from conventional liquid state nuclear magnetic resonance (NMR) experiments to build small quantum computers. After a brief introduction to quantum computing I will review the current state of the art, describe some of the topics of current interest, and assess the long term contribution of NMR studies to the eventual implementation of practical quantum computers capable of solving real computational problems.
Quantum computation offers the extraordinary promise of solving mathematical and physical problems which are simply beyond the reach of classical computers. However, the experimental realization of quantum computers is extremely challenging, because of the need to initialize, manipulate and measure the state of a set of coupled quantum systems while maintaining fragile quantum coherence. In this thesis work, we have taken significant steps towards the realization of a practical quantum computer: using nuclear spins and magnetic resonance techniques at room temperature, we provided proof of principle of quantum computing in a series of experiments which culminated in the implementation of the simplest instance of Shor’s quantum algorithm for prime factorization (15 = 3 × 5), using a seven-spin molecule. This algorithm achieves an exponential advantage over the best known classical factoring algorithms and its implementation represents a milestone in the experimental exploration of quantum computation. These remarkable successes have been made possible by the synthesis of suitable molecules and the development of many novel techniques for initialization, coherent control and readout of the state of multiple coupled nuclear spins. Furthermore, we devised and implemented a model to simulate both unitary and decoherence processes in these systems, in order to study and quantify the impact of various technological as well as fundamental sources of errors. In summary, this work has given us a much needed practical appreciation of what it takes to build a quantum computer. Furthermore, while liquid NMR quantum computing has well-understood scaling limitations, many of the techniques that originated from this research may find use in other, perhaps more scalable quantum computer implementations.
Quantum computing shows great promise for the solution of many difficult problems, such as the simulation of quantum systems and the factorization of large numbers. While the theory of quantum computing is fairly well understood, it has proved difficult to implement quantum computers in real physical systems. It has recently been shown that nuclear magnetic resonance (NMR) can be used to implement small quantum computers using the spin states of nuclei in carefully chosen small molecules. Here we demonstrate the use of a NMR quantum computer based on the pyrimidine base cytosine, and the implementation of a quantum algorithm to solve Deutsch’s problem (distinguishing between constant and balanced functions). This is the first successful implementation of a quantum algorithm on any physical system.