Quantum Information Processing By NMR : Quantum State Discrimination, Hadamard Spectroscopy, Liouville Space Search, Use Of Geometric Phase For Gates And Algorithms

Abstract

The progess in NMRQIP can be outlined in to four parts.1) Implementation of theoretical protocols on small number of qubits. 2) Demonstration of QIP on various NMR systems. 3) Designing and implementing the algorithms for mixed initial states. 4) Developing the techniques for coherent and decoherent control on higher number(up to 15) of qubits. This thesis contains some efforts in the direction of first three points. Quantum-state discrimination has important applications in the context of quantum communication and quantum cryptography. One of the characteristic features of quantum mechanics is that it is impossible to devise a measurement that can distinguish nonorthogonal states perfectly. However, one can distinguish them with a finite probability by an appropriate measurement strategy. In Chapter 2, we describe the implementation of a theoretical protocol of programmable quantum-state discriminator, on a two-qubit NMR System. The projective measurement is simulated by adding two experiments. This device does the unambiguous discrimination of a pair of states of the data qubit that are symmetrically located about a fixed state. The device is used to discriminate both linearly polarized states and eillipitically polarized states. The maximum probability of successful discrimination is achieved by suitably preparing the ancilla quubit. The last step of any QIP protocol is the readout. In NMR-QIP the readout is done by using density matrix tomography. It was first proposed by Ernst and co-workers that a two-dimensional method can be used to correlate input and output states. This method uses an extra (aniclla) qubit, whose transitions indicate the quantum states of the remaining qubits. The 2D spectrum of ancilla qubit represent the input and output states along F1 and F2 dimensions respectively. However the 2D method requires several t1 increments to achieve the required spectral width and resolution in the indirect dimension, hence leads to large experimental time. In chapter 3, the conventional 2D NMRQIP method is speeded-up by using Hadamard spectroscopy. The Hadamard method is used to implement various two-, three-qubit gates and qutrit gates. We also use Hadamard spectroscopy for information storage under spatial encoding and to implement a parallel search algorithm. Various slices of water sample can be spatially encoded by using a multi-frequency pulse under the field gradient. Thus the information of each slice is projected to the frequency space. Each slice represents a classical bit, where excitation and no excitation corresponds to the binary values 0 and 1 respectively. However one has to do the experiment for each binary information, by synthesizing a suitable multi-frequency pulse. In this work we show that by recording the data obtained by various Hadamard encoded multi-frequency pulses, one can suitably decode it to obtain any birnary information, without doing further experiments. Geometric phases depend only on the geometry of the path executed in the projective Hilbert space, and are therefore resilient to certain types of errors. This leads to the possibility of an intrinsically fault-tolerant quantum computation. In liquid state NMRQIP. Controlled phase shift gates are achieved by using qubit selective pulses and J evolutions, and also by using geometir phases. In order to achieve higher number of qubits in NMR, one explores dipolar couplings which are larger in magnitude, yielding strongly coupled spectra. In such systems since the Hamiltonian consists of terms, it is difficult to apply qubit selective pulses. However such systems have been used for NMRQIP by considering 2n eigen states as basis states of an n-qubit system. In chapter 4, it is shown that non-adiabatic geometric phases can be used to implement controlled phase shift gates in strongly dipolar coupled systems. A detailed theoretical explanation of non-adiabatic geometric phases in NMR is given, by using single transition operators. Using such controlled phase shift gates, the implementation of Deutsch-Jozsa and parity algorithms are demonstrated. Search algorithms play an important role in the filed of information processing. Grovers quantum search algorithm achieves polynomial speed-up over the classical search algorithm. Bruschweiler proposed a Liouville space search algorithm which achieve polymonial speed-up. This algorithm requires a weakly coupled system with a mixed initial state. In chapter 5 we modified the Bruschweiler’s algorithm, so that it can be implemented on a weakly as well as strongly coupled system. The experiments are performed on a strongly dipolar coupled four-qubit system. The experiments from four spin-1/2 nuclei of a molecule oriented in a liquid crystal matrix. Chapter 6 describes the implementation of controlled phase shift gates on a quadrupolar spin-7/2 nucleus, using non-adiabatic geometric phases. The eight energy levels of spin-7/2 nucleus, form a three qubit system. A general procedure is given, for implementing a controlled phase shift gate on a system consisting of any number of energy levels. Finally Collin’s version of three-qubit DJ algorithm using multi-frequency pulses, is implemented in the spin-7/2 system.