How does adiabatic quantum computation fit into quantum automata theory?

Tomoyuki Yamakami

Ari Mizel

Tomoyuki Yamakami

Quantum computation has emerged as a powerful computational medium of our time, having demonstrated the remarkable efficiency in factoring a positive integer and searching databases faster than any currently known classical computing algorithm. Adiabatic evolution of quantum systems have been studied as a potential means that physically realizes quantum computation. Up to now, all the research on adiabatic quantum systems has dealt with polynomial time-bounded computation and little attention has been paid to, for instance, adiabatic quantum systems consuming only constant memory space. Such quantum systems can be modeled in a form similar to quantum finite automata. This exposition dares to ask a bold question of how to make adiabatic quantum computation fit into the rapidly progressing framework of quantum automata theory. As our answer to this eminent but profound question, we first lay out a fundamental platform to carry out adiabatic evolutionary quantum systems (AEQSs) with limited computational resources (in size, energy, spectral gap, etc.) and then establish how to construct such AEQSs by operating suitable families of quantum finite automata. We further explore fundamental structural properties of decision problems (as well as promise problems) solved quickly by the appropriately constructed AEQSs.

M. V. Panduranga Rao

Quantum computing has generated a lot of interested in the past two decades. Research into powerful models of quantum computation has yielded important and elegant results like an efficient algorithm for factoring and a quadratic speed-up for unordered search. At the same time, given the current difficulty in the physical implementation of these general purpose models, considerable effort has also been made in estimating the power of weaker models of quantum computation: models that have a small quantum component. The first part of this thesis is an investigation into the power of interference in quantum computing. While interference in probability amplitudes is a central feature even in powerful models, it is the only extra resource available to quantum finite automata. Of particular interest is interference in automata that have both classical and quantum states (2QCFA) as proposed by Ambainis and Watrous, since it inquires into the power of a classical deterministic finite automaton when augmented with a quantum component of constant size. Our contributions in this part are as follows: • To abstract out the phenomenon of interference in quantum computing, we propose a model called the 2-way Optical Interference Automata (2OIA). The model consists of a 2DFA augmented with a simple optical arrangement. We show different ways of harnessing the power of interference in the form of algorithms on this model to recognize some non-trivial languages. We then go on to show a language recognizable by a Turing machine using O(n2) space but by no 2OIA. • A natural classical model for comparison with 2QCFA is the weighted automaton, since it has the potential to capture interference in sum of path weights. Using the Cortes-Mohri definition of language recognition, we give an efficient simulation of 2QCFAwith algebraic amplitudes by weighted automata over the complex semi ring. • We introduce quantum non-determinism to the Measure-Once 1-way Quantum Finite Automata of Moore and Crutchfield and Kondacs and Watrous and show that even then, the model can recognize only regular languages with bounded error. • We propose a group theoretic generalization of counter automata that allows a notion of counter reversal complexity. To obtain this generalization, we combine concepts from classical counter automata theory with results in 2QCFA. We examine specific instances of this generalization and compare their ii iii powers. We also show an instance recognizing a language that is not recognized by conventional 2-way counter automata. Finally, we show a strict hierarchy among the 1-way versions of the instances Discussed. The second part of the thesis deals with Quantum Adiabatic Optimization algorithms. A common trick for designing faster quantum adiabatic algorithms is to apply the adiabatic condition locally at every instant. However it is often difficult to determine the instantaneous gap between the lowest two eigen values, which is an essential ingredient in the adiabatic condition. We present a simple linear algebraic technique…

Rūsiņš Freivalds

Craig S. Lent, P. Douglas Tougaw, Wolfgang Porod Y

We discuss the fundamental limits of computing using a new paradigm for quantum computation, cellular automata composed of arrays of coulombically coupled quantum dot molecules, which we term quantum cellular automata (QCA). Any logical or arithmetic operation can be performed in this scheme. QCA's provide a valuable concrete example of quantum computation in which a number of fundamental issues come to light. We examine the physics of the computing process in this paradigm. We show to what extent thermodynamic considerations impose limits on the ultimate size of individual QCA arrays. Adiabatic operation of the QCA is examined and the implications for dissipationless computing are explored.>