Continuous-time quantum computing

Quantum computation using continuous-time evolution under a natural hardware Hamiltonian is a promising near- and mid-term direction toward powerful quantum computing hardware. Continuous-time quantum computing (CTQC) encompasses continuous-time quantum walk computing (QW), adiabatic quantum computing (AQC), and quantum annealing (QA), as well as other strategies which contain elements of these three. While much of current quantum computing research focuses on the discrete-time gate model, which has an appealing similarity to the discrete logic of classical computation, the continuous nature of quantum information suggests that continuous-time quantum information processing is worth exploring. A versatile context for CTQC is the transverse Ising model, and this thesis will explore the application of Ising model CTQC to classical optimization problems. Classical optimization problems have industrial and scientific significance, including in logistics, scheduling, medicine, cryptography, hydrology and many other areas. Along with the fact that such problems often have straightforward, natural mappings onto the interactions of readily-available Ising model hardware makes classical optimization a fruitful target for CTQC algorithms. After introducing and explaining the CTQC framework in detail, in this thesis I will, through a combination of numerical, analytical, and experimental work, examine the performance of various forms of CTQC on a number of different optimization problems, and investigate the underlying physical mechanisms by which they operate.Open Acces

Average-case analysis for the MAX-2SAT problem

AbstractWe propose a simple probability model for MAX-2SAT instances for discussing the average-case complexity of the MAX-2SAT problem. Our model is a “planted solution model”, where each instance is generated randomly from a target solution. We show that for a large range of parameters, the planted solution (more precisely, one of the planted solution pairs) is the optimal solution for the generated instance with high probability. We then give a simple linear-time algorithm based on a message passing method, and we prove that it solves the MAX-2SAT problem with high probability for random MAX-2SAT instances under this planted solution model for probability parameters within a certain range

ON CLASSICAL AND QUANTUM CONSTRAINT SATISFACTION PROBLEMS IN THE TRIAL AND ERROR MODEL

Ph.DDOCTOR OF PHILOSOPH

Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT

AbstractThe maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L)·2K/5, where K is the number of clauses and L is their total length. In fact, the running time is only poly(L)·2K2/5, where K2 is the number of clauses containing two literals. This bound implies the bound poly(L)·2L/10. Our results significantly improve previous bounds: poly(L)·2K/2.88 (J. Algorithms 36 (2000) 62–88) and poly(L)·2K/3.44 (implicit in Bansal and Raman (Proceedings of the 10th Annual Conference on Algorithms and Computation, ISAAC’99, Lecture Notes in Computer Science, Vol. 1741, Springer, Berlin, 1999, pp. 247–258.))As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly(M)·2M/3, where M is the number of edges in the graph. This is of special interest for graphs with low vertex degree