Dynamics and NMR Implementation of Controlled-NOT Gates for Quantum Computing

NMR experiments implementing two-bit controlled-NOT logic gates on alanine (JAB = 35.1 Hz) and 2,3dibromothiophene (JAB = 5.6 Hz) were performed. Spectra were collected at a variety of tip angles (angles between the spin and the axis of magnetization) by applying a selective RF pulse of constant power and variable duration. From this collection of spectra, the effective Hamiltonian of the spin system was derived and found to contain an internal Hamiltonian. In a spin system with weak coupling, the internal Hamiltonian contains spin-spin coupling terms. The effective Hamiltonian gives a more complete description than the currently used transition Hamiltonian. Understanding the dynamics of a spin system not only furthers the field of NMR but has application in the subject of quantum computing. NMR pulse sequences for four-, eightand 16-spin controlled-NOT logic gates were developed. A pattern is evident and the pulse sequence for any number of spins can be derived. Disregarding the differences in the spin-spin coupling constants of different spin systems, these results suggest that the total time to implement a controlled-NOT logic gate in NMR does not increase exponentially with the number of spins in the system. Thesis Supervisor: Title: David Cory Associate Professor of Nuclear Engineering The computational speed of the classic computer is limited by the size of the computer's components and the speed at which they can be driven. Eventually, computer components will be so small that quantum effects cannot be ignored. A computer that relies on quantum coherence may permit a future generation of computers to have capabilities that far outreach their classic predecessors, which adhere to the laws of classical physics. However, a quantum computer of a size greater than six-bits has yet to be built and the physical realization of such a quantum computer is still many years away. These machines will perform reversible, unitary computations on a system including quantum superpositions. The reality of being able to create an efficient, user-friendly two state system with coherence times that are long enough to do multiple computations is a formidable goal. Quantum computing systems are being researched. The ion trap method uses the quantized energy levels of an atom as the states of the bit [Cirac95]. Computations are performed using lasers to impart detectable energy and momentum to the system. An extension of the ion trapping idea uses high-Q optical cavities to transfer information between ions and the field modes. If realized, this will be the first example of quantum information transfer [Davidovich94]. Another family of paradigms uses a coupled system of spin = % particles. Spin up and spin down provide an easy analogue of the classical binary states. Nuclear spins are preferred over electron spins because the smaller magnetic moment of the proton means a weaker dipolar magnetic interaction and a much longer time period of coherence. Because standard techniques have already been developed and spin systems can be put into pseudo-pure states that evolve under a well-studied Hamiltonian, NMR (nuclear magnetic resonance) has been a fairly successful method of implementing unitary, reversible quantum computations [Cory96], [Cory97], [Gershenfeld97], [Lloyd93]. In NMR quantum computing, each molecule acts as a separate quantum computer and the number of resolvable spins in the molecule determines the number of bits in the computer. The nuclear spins are quantized in a magnetic field and a computation is performed on the initial spin state by applying a series of RF (radio-frequency) pulses resonant at the frequency of the desired transition. Because each NMR sample has on the order of 1023 molecules, a computation is massively redundant; each measurement is a measurement of an ensemble of states. While quantum computation relies on the deterministic, measurement of a random state, measurement of an ensemble provides a stochastic average of a large number of random states. [Cory97]. The study of quantum computing, the science of computation using components small enough that their dynamics are governed by the principles of quantum mechanics, is already many years old. In the 1980's, Feynman introduced the idea of using systems governed by quantum mechanics to explore complex quantum phenomena. This exploration of quantumbased systems led to the idea that other problems could be solved by adhering the rules of quantum mechanics. The possibilities of solving problems once deemed unsolvable or solving problems with more efficient algorithms, inspired many. Shor proved that using quantum algorithms, computers would be able to factor large numbers in polynomial time as opposed to the exponential time it takes classical computers [Shor94]. Grover developed a quantum algorithm that searches through a database with N items in an average of /N tries as opposed to the N/2 average associated with classical computer algorithms [Grover96]. Deutsch, DiVincenzo and Barenco have transformed the idea of classical universal gates into quantum complete and quantum universal gates. With the suggestion that quantum algorithms are sometimes more efficient and in some cases capable of solving computational "hard" problems, the excitement behind the field of quantum computing continues to grow. People are talking about the eventual necessity of quantum computers to safeguard encrypted data. Physicists are developing quantum techniques with the hope that a system based on quantum principles will provide insight into quantum phenomena. While many people have embraced the idea of quantum computing, the immediate gratification is coming from its secondary results. Implementation of quantum computing ideas, in addition to further advancing the somewhat elusive goal of building a quantum computer, is advancing quantum information theory, laser applications, ion trapping methods, methods of error correction, the understanding of complex quantum phenomena, and the field of NMR. This paper focuses on the use of NMR to study ideas and algorithms of quantum computing. Following Chapter One's introductions to NMR and quantum computing logic gates, Chapter Two describes experiments implementing a two-spin controlled-NOT gate with the PoundOverhauser technique. The chapter contains a discussion of the Hamiltonian associated with the state transition in a two-spin quantum controlled-NOT gate and argues that the standard picture of the transition Hamiltonian is incomplete. Chapter Three discusses multiple-spin controlled-NOT gates and the NMR pulse sequences needed to implement them. The final chapter discusses the impact of this work, future experiments and the NMR paradigm of quantum computing. Though problems exist, research being done to implement today's quantum computing ideas with NMR will be applicable to any paradigm of quantum computing, advance new theories of quantum computing and inspire new ideas in the field of NMR. Chapter One. Introduction. Quantum Computing: An Application of NMR. Basics of NMR. NMR (nuclear magnetic resonance) is a method of spectroscopy first developed in the 1940's. NMR is most commonly used in chemical and structural analysis and also medical imaging. The sample's nuclear spins are quantized in a magnetic field of several Tesla. The state is governed by the Zeeman Hamiltonian, Hzeeman = (-y h Bo) Iz (1) RF pulses are used to manipulate the state of the spin. Pulses are designed to induce either all of the transitions allowed by quantum mechanics or a selective set of them. The RF Hamiltonian in a stationary frame of reference is, HRF = RF (Ix ). (2) The component of the spin magnetization that is put into the transverse plane precesses about the axis of magnetization. The rate of this precession is governed by T2, the spin-spin relaxation. Because of this precession, it is useful to use the HRF in a rotating frame, HRF-rotating e tme e(iwRF Z + ABtime) * HRF * e(-i WRF * JAB*time) (3) Following the RF pulse, the T1 (spin-lattice) relaxation dictates the return of the spin state to thermal equilibrium. The relaxation results in a measurable FID (free induction decay) that maps the transfer of magnetization from the transverse plane to the magnetization axis (z-axis). The dynamics of a single nuclear spin in a magnetic field are completely described by the Bloch equations. Coupled spins require a density matrix approach. In addition to the Zeeman and RF Hamiltonians, there is an internal Hamiltonian, which for our purposes consists of the chemical shift and scalar coupling, Hnternal = A W IB + 2JAB . (4) Atomic electron clouds affect the nuclei and result in a chemical shift. Furthermore, the spin of an atom can be coupled to the spin of other atoms in the molecule. The degree of coupling between two spins is denoted by the coupling constant, JAB. In the liquid phase, because the magnetic dipolar interaction averages to zero, a spin's resonant frequency is unaffected by the spin in atoms of other molecules. Spin-spin coupling and a chemical shift result in a Fourier transformed spectra like the one shown in Figure One. The splitting in the peaks represents the different transitions shown in the energy level diagram of Figure Two [Cory97]. Figure One. A Fourier transform of the FID collected after applying a non-selective 90' pulse to the equilibrium state of a two-spin system. From left to right, the peaks represent the population of spins undergoing the transitions (IT to TT), (44 to Tf), (T to ft), and (4I to IT),. In other words, the left doublet represents the transitions of the B spin coupled to the A spin while the right doublet represents the transitions of the A spin coupled to the B spin. Each doublet is centered on the resonant frequency of the spin (wA or w.) and the width of the splitting (due to spin-spin coupling) is equal to JAB. Frequency (and energy) increases from left to right.