arXiv subjects

Edward Gillman

Publications and source records attributed to Edward Gillman.

12 recordsLinked to original sources

arXiv2026

Rare Event Analysis of Large Language Models

Being probabilistic models, during inference large language models (LLMs) display rare events: behaviour that is far from typical but highly significant. By definition all rare events are hard to see, but the enormous scale of LLM usage means that events completely unobserved during development are likely to become prominent in deployment. Here we present an end-to-end framework for the systematic analysis of rare events in LLMs. We provide a practical implementation spanning theory, efficient generation strategies, probability estimation and error analysis, which we illustrate with concrete examples. We outline extensions and applications to other models and contexts, highlighting the generality of the concepts and techniques presented here.

cs.LG↗

arXiv2022

Combining Reinforcement Learning and Tensor Networks, with an Application to Dynamical Large Deviations

We present a framework to integrate tensor network (TN) methods with reinforcement learning (RL) for solving dynamical optimisation tasks. We consider the RL actor-critic method, a model-free approach for solving RL problems, and introduce TNs as the approximators for its policy and value functions. Our "actor-critic with tensor networks" (ACTeN) method is especially well suited to problems with large and factorisable state and action spaces. As an illustration of the applicability of ACTeN we solve the exponentially hard task of sampling rare trajectories in two paradigmatic stochastic models, the East model of glasses and the asymmetric simple exclusion process (ASEP), the latter being particularly challenging to other methods due to the absence of detailed balance. With substantial potential for further integration with the vast array of existing RL methods, the approach introduced here is promising both for applications in physics and to multi-agent RL problems more generally.

cond-mat.stat-mech↗

arXiv2022

Using (1 + 1)D Quantum Cellular Automata for Exploring Collective Effects in Large Scale Quantum Neural Networks

Central to the field of quantum machine learning is the design of quantum perceptrons and neural network architectures. A key question in this regard is the impact of quantum effects on the way in which such models process information. Here, we approach this question by establishing a connection between $(1+1)D$ quantum cellular automata, which implement a discrete nonequilibrium quantum many-body dynamics through the successive application of local quantum gates, and recurrent quantum neural networks, which process information by feeding it through perceptrons interconnecting adjacent layers. This relation allows the processing of information in quantum neural networks to be studied in terms of the properties of their equivalent cellular automaton dynamics. We exploit this by constructing a class of quantum gates (perceptrons) that allow for the introduction of quantum effects, such as those associated with a coherent Hamiltonian evolution, and establish a rigorous link to continuous-time Lindblad dynamics. We further analyse the universal properties of a specific quantum cellular automaton, and identify a change of critical behavior when quantum effects are varied, demonstrating that they can indeed affect the collective dynamical behavior underlying the processing of information in large-scale neural networks.

quant-ph↗

arXiv2022

Asynchronism and nonequilibrium phase transitions in $(1+1)$D quantum cellular automata

Probabilistic cellular automata provide a simple framework for the exploration of classical nonequilibrium processes. Recently, quantum cellular automata have been proposed that rely on the propagation of a one-dimensional quantum state along a fictitious discrete time dimension via the sequential application of quantum gates. The resulting $(1+1)$-dimensional space-time structure makes these automata special cases of feed-forward quantum neural networks. Here we show how asynchronism -- introduced via non-commuting gates -- impacts on the collective nonequilibrium behavior of quantum cellular automata. We illustrate this through a simple model, whose synchronous version implements a contact process and features a nonequilibrium phase transition in the directed percolation universality class. Non-commuting quantum gates lead to an "asynchronism transition", i.e. a sudden qualitative change in the phase transition behavior once a certain degree of asynchronicity is surpassed. Our results show how quantum effects may lead to abrupt changes of non-equilibrium dynamics, which may be relevant for understanding the role of quantum correlations in neural networks.

quant-ph↗

arXiv2021

Quantum and classical temporal correlations in $(1 + 1)D$ Quantum Cellular Automata

We employ $(1 + 1)$-dimensional quantum cellular automata to study the evolution of entanglement and coherence near criticality in quantum systems that display non-equilibrium steady-state phase transitions. This construction permits direct access to the entire space-time structure of the underlying non-equilibrium dynamics. It contains the full ensemble of classical trajectories and also allows for the analysis of unconventional correlations, such as entanglement in the time direction between the "present" and the "past". Close to criticality, the dynamics of these correlations - which we quantify through the second-order Renyi entropy - displays power-law behavior on its approach to stationarity. Our analysis is based on quantum generalizations of classical non-equilibrium systems: the Domany-Kinzel cellular automaton and the Bagnoli-Boccara-Rechtman model, for which we provide estimates for the critical exponents related to the classical and quantum components of the entropy. Our study shows that $(1 + 1)$-dimensional quantum cellular automata permit an intriguing perspective on the nature of classical and quantum correlations in out-of-equilibrium systems.

quant-ph↗

arXiv2020

Numerical Simulation of Critical Quantum Dynamics without Finite Size Effects

Classical $(1+1)D$ cellular automata, as for instance Domany-Kinzel cellular automata, are paradigmatic systems for the study of non-equilibrium phenomena. Such systems evolve in discrete time-steps, and are thus free of time-discretisation errors. Moreover, information about critical phenomena can be obtained by simulating the evolution of an initial seed that, at any finite time, has support only on a finite light-cone. This allows for essentially numerically exact simulations, free of finite-size errors or boundary effects. Here, we show how similar advantages can be gained in the quantum regime: The many-body critical dynamics occurring in $(1+1)D$ quantum cellular automata with an absorbing state can be studied directly on an infinite lattice when starting from seed initial conditions. This can be achieved efficiently by simulating the dynamics of an associated one-dimensional, non-unitary quantum cellular automaton using tensor networks. We apply our method to a model introduced recently and find accurate values for universal exponents, suggesting that this approach can be a powerful tool for precisely classifying non-equilibrium universal physics in quantum systems.

quant-ph↗

arXiv2020

Non-equilibrium phase transitions in $(1+1)$-dimensional quantum cellular automata with controllable quantum correlations

Motivated by recent progress in the experimental development of quantum simulators based on Rydberg atoms, we introduce and investigate the dynamics of a class of $(1+1)$-dimensional quantum cellular automata. These non-equilibrium many-body models, which are quantum generalisations of the Domany-Kinzel cellular automaton, possess two key features: they display stationary behaviour and non-equilibrium phase transitions despite being isolated systems. Moreover, they permit the controlled introduction of local quantum correlations, which allows for the impact of quantumness on the dynamics and phase transition to be assessed. We show that projected entangled pair state tensor networks permit a natural and efficient representation of the cellular automaton. Here, the degree of quantumness and complexity of the dynamics is reflected in the difficulty of contracting the tensor network.

quant-ph↗

arXiv2020

A Tensor Network Approach to Finite Markov Decision Processes

Tensor network (TN) techniques - often used in the context of quantum many-body physics - have shown promise as a tool for tackling machine learning (ML) problems. The application of TNs to ML, however, has mostly focused on supervised and unsupervised learning. Yet, with their direct connection to hidden Markov chains, TNs are also naturally suited to Markov decision processes (MDPs) which provide the foundation for reinforcement learning (RL). Here we introduce a general TN formulation of finite, episodic and discrete MDPs. We show how this formulation allows us to exploit algorithms developed for TNs for policy optimisation, the key aim of RL. As an application we consider the issue - formulated as an RL problem - of finding a stochastic evolution that satisfies specific dynamical conditions, using the simple example of random walk excursions as an illustration.

cond-mat.stat-mech↗

arXiv2019

Numerical Simulation of Critical Dissipative Non-Equilibrium Quantum Systems with an Absorbing State

The simulation of out-of-equilibrium dissipative quantum many body systems is a problem of fundamental interest to a number of fields in physics, ranging from condensed matter to cosmology. For unitary systems, tensor network methods have proved successful and extending these to open systems is a natural avenue for study. In particular, an important question concerns the possibility of approximating the critical dynamics of non-equilibrium systems with tensor networks. Here, we investigate this by performing numerical simulations of a paradigmatic quantum non-equilibrium system with an absorbing state: the quantum contact process. We consider the application of matrix product states and the time-evolving block decimation algorithm to simulate the time-evolution of the quantum contact process at criticality. In the Lindblad formalism, we find that the Heisenberg picture can be used to improve the accuracy of simulations over the Schrödinger approach, which can be understood by considering the evolution of operator-space entanglement. Furthermore, we also consider a quantum trajectories approach, which we find can reproduce the expected universal behaviour of key observables for a significantly longer time than direct simulation of the average state. These improved results provide further evidence that the universality class of the quantum contact process is not directed percolation, which is the class of the classical contact process.

quant-ph↗

arXiv2019

Quantum contact process

The contact process is a paradigmatic classical stochastic system displaying critical behavior even in one dimension. It features a non-equilibrium phase transition into an absorbing state that has been widely investigated and shown to belong to the directed percolation universality class. When the same process is considered in a quantum setting much less is known. So far mainly semi-classical studies have been conducted and the nature of the transition in low dimensions is still a matter of debate. Also from a numerical point of view, from which the system may look fairly simple --- especially in one dimension --- results are lacking. In particular the presence of the absorbing state poses a substantial challenge which appears to affect the reliability of algorithms targeting directly the steady-state. Here we perform real-time numerical simulations of the open dynamics of the quantum contact process and shed light on the existence and on the nature of an absorbing state phase transition in one dimension. We find evidence for the transition being continuous and provide first estimates for the critical exponents. Beyond the conceptual interest, the simplicity of the quantum contact process makes it an ideal benchmark problem for scrutinizing numerical methods for open quantum non-equilibrium systems.

cond-mat.stat-mech↗

arXiv2018

The Kibble Zurek Mechanism of Topological Defect Formation in Quantum Field Theory with Matrix Product States

The Kibble Zurek mechanism in a relativistic $ϕ^{4}$ scalar field theory in $D = (1 + 1)$ is studied using uniform matrix product states. The equal time two point function in momentum space $G_{2}(k)$ is approximated as the system is driven through a quantum phase transition at a variety of different quench rates $τ_{Q}$. We focus on looking for signatures of topological defect formation in the system and demonstrate the consistency of the picture that the two point function $G_{2}(k)$ displays two characteristic scales, the defect density $n$ and the kink width $d_{K}$. Consequently, $G_{2}(k)$ provides a clear signature for the formation of defects and a well defined measure of the defect density in the system. These results provide a benchmark for the use of tensor networks as powerful non-perturbative non-equilibrium methods for relativistic quantum field theory, providing a promising technique for the future study of high energy physics and cosmology.

quant-ph↗

arXiv2017

Topological Defects in Quantum Field Theory with Matrix Product States

Topological defects (kinks) in a relativistic $ϕ^{4}$ scalar field theory in $D=(1+1)$ are studied using the matrix product state tensor network. The one kink state is approximated as a matrix product state and the kink mass is calculated. The approach used is quite general and can be applied to a variety of theories and tensor networks. Additionally, the contribution of kink-antikink excitations to the ground state is examined and a general method to estimate the scalar mass from equal time ground state observables is provided. The scalar and kink mass are compared at strong coupling and behave as expected from universality arguments. This suggests that the matrix product state can adequately capture the physics of defect-antidefect excitations and thus provides a promising technique to study challenging non-equilibrium physics such as the Kibble-Zurek mechanism of defect formation.

quant-ph↗