Phase transitions and gaps in quantum random energy models

Quantum Phase Transitions in Random Systems

Phase transitions and random quantum satisfiability

Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. We study random ensembles of the QMA1-complete quantum satisfiability (QSAT) problem introduced by Bravyi [1]. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show ...

Quantum energy gaps and first-order mean-field transitions

We study first-order mean-field quantum phase transitions, in particular the exponentially fast closure of the energy gap with the system size. We consider exactly solvable ferromagnetic models and discuss their relation to the Grover problem to which they reduce in a particular limit. We compute the coefficient in the exponential closure of the gap using an instantonic approach. We also discus...

Phase transitions and random matrices

Phase transitions generically occur in random matrix models as the parameters in the joint probability distribution of the random variables are varied. They affect all main features of the theory and the interpretation of statistical models. In this paper a brief review of phase transitions in invariant ensembles is provided, with some comments to the singular values decomposition in complex no...

Phase Transitions in Random Structures

These notes provides a perspective on phase transitions in random discrete structures informed by recent ideas and techniques from statistical physics. The lectures are based on the publications [11, 20, 22].