The third chapter examines the capabilities of liquid-state NMR systems for quantum computing. It begins by grounding the reader in the basics of spin dynamics and NMR spectroscopy, followed by a discussion on the encoding of qubits into the spin states of the nucleus of atoms inside molecules. The narrative progresses to describe the implementation of single-qubit gates via external magnetic fields, weaving in key concepts such as the rotating-wave approximation, the Rabi cycle, and pulse shaping. The technique for orchestrating two-qubit gates, leveraging the intrinsic couplings between the spins of nuclei of atoms within a molecule, is subsequently detailed. Additionally, the chapter explains the process of detecting qubits’ states through the collective nuclear magnetization of the NMR sample and outlines the steps for qubit initialization. Attention then shifts to the types of noise that affect NMR quantum computers, shedding light on decoherence and the critical T1 and T2 times. The chapter wraps up by providing a synopsis, evaluating the strengths and weaknesses of liquid-state NMR for quantum applications, and a note on the role of entanglement in quantum computing.

The final chapter details some methods for evaluating the performance of quantum computers. It begins by delineating the essential features of quantum benchmarks and organizes them into a three-tiered framework. Initially, it discusses early-stage benchmarks that provide a detailed analysis of basic operations on a few qubits, emphasizing fidelity tests and tomography. Then, it progresses to intermediate-stage benchmarks that provide a more generalized appraisal of gate quality, circuit depth, and length. Concluding the benchmarking spectrum, later-stage benchmarks are introduced, aimed at evaluating the overall reliability and efficiency of quantum computers operating with a large number of qubits (e.g. 1000 or more).

The chapter contains a brief summary of what transformations are and how we understand symmetry.

We explain why we write the book, its purpose, and intended audience.

Problems with calculations of Berry properties in real and reciprocal spaces and physical characteristics involving manifestations of Berry properties are included.

Problems considering identical particles in the context of addition of angular momenta, perturbation theory, chemistry, and many-body physics are included.

Problems on properties of the Dirac equation and emphasis on nontrivial topology and specific materials (such as graphene) are included.

Problems involving calculations of various properties associated with the density operator and entropies and their relations to more general situations in physics are included.

This chapter begins the final section of the book, which presents both review and new results of original research on decoherence and measurement theory. In this chapter, it is shown that normal quantum mechanics can lead to irreversible behavior in an open system, in contrast to the expectation of the Poincaré theorem that predicts repeating, cyclical behavior for all closed systems. The quantum Boltzmann equation, which implies the famous H-theorem that underlies all statistical mechanics, is derived.

This chapter presents the surprising mathematical result that classical systems can indeed have entanglement. However, the degree to which they can be entangled is strictly limited, while quantum systems have no limit to their amount of entanglement.

This chapter surveys some of the ways in which the Copenhagen interpretation of quantum mechanics has led to a various views of the world with spiritual and moral implications; the perspective of this chapter is that most of these views are not demanded by the actual theory and experiments of quantum mechanics.