Date of Award
Campus Access Dissertation
Doctor of Philosophy (PhD)
A comprehensive understanding of chaos in quantum systems remains elusive. In this dissertation, we explore the concept of chaos, starting from the known definitions and postulated conditions for chaos in quantum systems, and then apply a quantum information theoretical approach to them. It is not difficult to observe that the essential properties of chaos are encapsulated in its eigenvalues and eigenvectors; however, we do not measure eigenvalues and eigenvectors, but rather observe chaos through specific probes. This dissertation introduces a novel tool, "Isospectral Twirling", designed to understand the interplay between what determines chaos and what probes it.
This tool is then utilized on probes pertinent to chaos and information theory, for instance, the Out-of-Time-Order Correlator (OTOC), Loschmidt Echo, Entanglement Entropy, and Coherence. To facilitate a comparison between chaotic and integrable systems, we employ random matrix theory and three spectral distributions: the Gaussian Unitary Ensemble (GUE) representing chaos, and the Gaussian Diagonal Ensemble (GDE) and Poisson distributions describing integrable systems. The newly introduced tool proves effective in differentiating chaotic quantum dynamics from non-chaotic ones.
Further, this dissertation explores the influence of chaos on two fundamental notions: the simulability and learnability of a quantum system. From the perspective of simulability, it is widely accepted that a circuit of n qubits composed of Clifford gates can be simulated on a classical computer by an algorithm scaling as poly(n)exp(k). We demonstrate that for a quantum circuit to simulate quantum chaotic features, k=O(n) is necessary, implying that quantum chaos cannot be simulated on a classical computer.
Regarding learnability, our efforts are concentrated on the relatively simpler task of unscrambling, which is related to the notion of scrambling—the destruction of local correlations and the dispersion of information throughout the system. Remarkably, even in non-classical simulability contexts, it is still possible to construct a Clifford decoder. We demonstrate this by developing an algorithm capable of accomplishing this task. We find that the notable properties of scramblers can be efficiently encoded in a Clifford decoder, as long as the system is not entirely chaotic.
In conclusion, we introduce a new measure for the quantum property of non-stabilizerness, often referred to as "magic," by considering the Rényi entropy of the probability distribution associated with a pure quantum state. This state is defined by the square of the expectation value of Pauli strings. We argue that this is a competent measure of non-stabilizerness from a resource theory perspective, providing bounds with other known measures. The stabilizer Rényi entropy offers the benefit of easy computation as it forgoes the need for a minimization procedure. An experimental protocol for measurement by randomized measurements is presented. We propose that the non-stabilizerness is closely linked to out-of-time-order correlation functions and that maximal levels of non-stabilizerness are essential for quantum chaos.
Oliviero, Salvatore F. E., "Information Theoretic Approach to Quantum Chaos" (2023). Graduate Doctoral Dissertations. 862.