Hamiltonian models of classical and quantum stochastic processes

Johannes Bausch

This dissertation investigates the theory of quantum stochastic processes and its applications in quantum many-body physics. The main goal is to analyse complexity-theoretic aspects of both static and dynamic properties of physical systems modelled by quantum stochastic processes. The thesis consists of two parts: the first one addresses the computational complexity of certain quantum and classical divisibility questions, whereas the second one addresses the topic of Hamiltonian complexity theory. In the divisibility part, we discuss the question whether one can efficiently sub-divide a map describing the evolution of a system in a noisy environment, i.e. a CPTP- or stochastic map for quantum and classical processes, respectively, and we prove that taking the nth root of a CPTP or stochastic map is an NP-complete problem. Furthermore, we show that answering the question whether one can divide up a random variable $X$ into a sum of $n$ iid random variables $Y_i$, i.e. $X=\sum_{i=1}^n Y_i$, is poly-time computable; relaxing the iid condition renders the problem NP-hard. In the local Hamiltonian part, we study computation embedded into the ground state of a many-body quantum system, going beyond "history state" constructions with a linear clock. We first develop a series of mathematical techniques which allow us to study the energy spectrum of the resulting Hamiltonian, and extend classical string rewriting to the quantum setting. This allows us to construct the most physically-realistic QMAEXP-complete instances for the LOCAL HAMILTONIAN problem (i.e. the question of estimating the ground state energy of a quantum many-body system) known to date, both in one- and three dimensions. Furthermore, we study weighted versions of linear history state constructions, allowing us to obtain tight lower and upper bounds on the promise gap of the LOCAL HAMILTONIAN problem in various cases. We finally study a classical embedding of a Busy Beaver Turing Machine into a low-dimensional lattice spin model, which allows us to dictate a transition from a purely classical phase to a Toric Code phase at arbitrarily large and potentially even uncomputable system sizes.

Gian‐Carlo Rota

Sergio A Albeverio, Giulio Casati, D. Merlini

Xiaoyang Wang, Yinchenguang Lyu, Changyu Yao +1 more

We propose to use quantum computers to simulate infection spreading in networks. We first show the analogy between the infection distribution and spin-lattice configurations with Ising-type interactions. Then, since the spreading process can be modeled as a classical Markovian process, we show that the spreading process can be simulated using the evolution of a quantum thermal dynamic model with a parameterized Hamiltonian. In particular, we analytically and numerically analyze the evolution behavior of the Hamiltonian, and prove that the evolution simulates a classical Markovian process, which describes the well-known epidemiological stochastic susceptible and infectious (SI) model. A practical method to determine the parameters of the thermal dynamic Hamiltonian from epidemiological inputs is exhibited. As an example, we simulate the infection spreading process of the SARS-Cov-2 variant Omicron in a small-world network.

Hans Maassen

A brief description is given of dilation theory in connection with systems theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Abel Klein