Quantum walks as thermalizations, with application to fullerene graphs
arXiv:2304.01572 · doi:10.1016/j.physa.2024.129823
Abstract
We consider to what extent quantum walks can constitute models of thermalization, analogously to how classical random walks can be models for classical thermalization. In a quantum walk over a graph, a walker moves in a superposition of node positions via a unitary time evolution. We show a quantum walk can be interpreted as an equilibration of a kind investigated in the literature on thermalization in unitarily evolving quantum systems. This connection implies that recent results concerning the equilibration of observables can be applied to analyse the node position statistics of quantum walks. We illustrate this in the case of a family of graphs known as fullerenes. We find that a bound from Short et al., implying that certain expectation values will at most times be close to their time-averaged value, applies tightly to the node position probabilities. Nevertheless, the node position statistics do not thermalize in the standard sense. In particular, quantum walks over fullerene graphs constitute a counter-example to the hypothesis that subsystems equilibrate to the Gibbs state. We also exploit the bridge created to show how quantum walks can be used to probe the universality of the eigenstate thermalisation hypothesis (ETH) relation. We find that whilst in C60 with a single walker, the ETH relation does not hold for node position projectors, it does hold for the average position, enforced by a symmetry of the Hamiltonian. The findings suggest a unified study of quantum walks and quantum self-thermalizations is natural and feasible.
arXiv comment13 pages, 7 figures
References in corpus (21)
- From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
- Quantum Computation and Decision Trees
- The foundations of statistical mechanics from entanglement: Individual states vs. averages
- Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems
- Quantum walks: a comprehensive review
- Canonical Typicality
- Quantum mechanical evolution towards thermal equilibrium
- Eigenstate Thermalization Hypothesis
- On the relationship between continuous- and discrete-time quantum walk
- Quantum equilibration in finite time
- House of Graphs: a database of interesting graphs
- Open quantum system dynamics and the mean force Gibbs state
- Canonical thermalization
- Distribution of local entropy in the Hilbert space of bi-partite quantum systems: Origin of Jaynes' principle
- Experimental Implementation of Quantum Walks on IBM Quantum Computers
- Predicting quantum advantage by quantum walk with convolutional neural networks
- Entanglement Typicality
- From Schrödinger's Equation to the Quantum Search Algorithm
- Unifying typical entanglement and coin tossing: on randomization in probabilistic theories
- Eigenstate Thermalisation on Average
- Quantum walk mixing is faster than classical on periodic lattices