Sampling combinatorial energy landscapes by classical and quantum computation

Abstract

Combinatorial and, by extension, non-convex optimization problems are among the most difficult to solve computationally due to the inadequacy of local search methods alone to find global optima. The notion of an energy landscape provides a unified language to describe the origin of combinatorial complexity across a wide variety of fields including physics, materials science, and artificial intelligence. This thesis utilizes probabilistic techniques for efficiently sampling energy landscapes that rely on classical high-performance computers and near-term quantum computers. Such sampling techniques not only provide effective heuristics for solving combinatorial optimization problems, but also enable an explanation of physical phenomena that rely on features of the energy landscape other than the global optimum. Six use cases are considered that are of relevance to the design of future clean energy systems: i) a predictive theory of materials polymorphism based on the partitioning of the continuous energy landscape into structurally equivalent regions, ii) a theory of the structure of functional glasses based on thermally averaging structurally inequivalent regions, iii) solution of the minimum dominating set variant of the optimal phasor measurement unit placement problem on the power grid by quantum-annealing over a discrete, combinatorial solution landscape, iv) a reformulation and solution of the Markov decision process formalism underlying optimal control via quantum annealing, v) a prescription for the variational preparation of fractional quantum Hall states on a digital quantum computer that accommodates the size of the Hilbert space on quantum hardware while shifting the burden of variational minimization to classical, global optimization heuristics, and vi) the simulation of quantum cellular automata on a digital quantum computer and establishment thereby as primitives that could help design noise-resilient classically-parameterized unitary circuits such as those used in quantum machine learning. Through these domain examples, the outlook established herein is that efficient methods for exploring combinatorial energy landscapes, combined with resilience-minded near-term quantum algorithm construction and intelligent division of labor between quantum and classical resources, provide a promising path towards solving some of the most challenging problems in renewable energy science.