Weaving Stability from Dissipation: Fixed-Point Engineering for Quantum Information Processing

Dartmouth College, Hanover NH

Investigators

Abstract

Understanding how physical systems approach equilibrium is a central question in statistical physics, and this topic is all the more profound when systems behave according to the laws of quantum mechanics. As a familiar example, consider how a hot cup of tea or a cold glass of water equilibrate to room temperature. One cools down while the other warms up; but in each case individual molecular interactions affect the rate and the fluctuations with which these systems approach equilibrium. A more nuanced example comes from the field of quantum computing, where information can be encoded in the quantum mechanical spin states of individual molecules. One compelling motivation for this is to produce more powerful computers with memory and information processing capabilities far superior to today's classical computers. However, since relaxation towards equilibrium can disorganize the quantum information stored in molecules, equilibration is normally considered an obstacle or a hindrance for quantum computing. The goal of this project is to find ways to utilize the stability of states near equilibrium in order to improve quantum computing protocols. Methods to characterize and influence the stability of multi-particle quantum states will be developed. Techniques to "attract" or nudge any given initial state towards a set of stable equilibrium states that contain genuinely non-classical multi-particle correlations ("entanglement") needed for quantum computing will be explored. This research will address fundamental questions about the conditions under which stabilization to a desired quantum state may be achieved, and it will expand the toolkit and of methods to characterize and control dissipative quantum systems. This project also provides education and training for students working on subjects at the cross-disciplinary boundaries between quantum information science, quantum control theory, applied mathematics, and many-body and statistical physics. Over the last few years, the principal investigator has established rigorous conditions for a general quantum state of interest to be the unique fixed point of a class of continuous-time "frustration-free" Markovian dynamics, subject to realistic locality constraints. Building on these results, this project aims to explore quantum stabilization problems in physical settings that remain largely uncharted as yet -- for instance: (i) Continuous-time Markovian dynamics under more general constraints than imposed by locality, including sensitivity to perturbations and periodically time-varying Floquet-Markov generators; (ii) Constrained discrete-time Markovian dynamics, for which the intriguing possibilities of exact dissipative state preparation and dissipative encoding in finite time arise; (iii) Randomized open-system dynamics, which may shed light on the stabilizability properties of generic (random) target states or subspaces. The methods to be employed will be both analytical and numerical, and will draw on a diverse range of tools from applied mathematics and quantum control theory (including operator algebras, Lyapunov stability techniques, and semi-definite programming), to quantum information theory and many-body physics (notably, entanglement theory, information-preserving structures, and tensor-network techniques). Theoretically, a central theme will be to clarify the extent to which core features from many-body and statistical mechanics -- in particular, the complexity of quantum correlations in the equilibrium state, the lack of frustration, the gapped nature or the commutativity of the underlying "parent" Hamiltonians -- may be brought to bear on the feasibility and efficiency of stabilization tasks for quantum information processing. From a practical standpoint, the goal of these investigations is to improve protocols for dissipative quantum state preparation and quantum information encoding.