Quantum Simulation of Quantum Phase Transitions Using the Convex
  Geometry of Reduced Density Matrices
Transitions of many-particle quantum systems between distinct phases at
absolute-zero temperature, known as quantum phase transitions, require an
exacting treatment of particle correlations. In this work, we present a general
quantum-computing approach to quantum phase transitions that exploits the
geometric structure of reduced density matrices. While typical approaches to
quantum phase transitions examine discontinuities in the order parameters, the
origin of phase transitions -- their order parameters and symmetry breaking --
can be understood geometrically in terms of the set of two-particle reduced
density matrices (2-RDMs). The convex set of 2-RDMs provides a comprehensive
map of the quantum system including its distinct phases as well as the
transitions connecting these phases. Because 2-RDMs can potentially be computed
on quantum computers at non-exponential cost, even when the quantum system is
strongly correlated, they are ideally suited for a quantum-computing approach
to quantum phase transitions. We compute the convex set of 2-RDMs for a
Lipkin-Meshkov-Glick spin model on IBM superconducting-qubit quantum
processors. Even though computations are limited to few-particle models due to
device noise, comparisons with a classically solvable 1000-particle model
reveal that the finite-particle quantum solutions capture the key features of
the phase transitions including the strong correlation and the symmetry
breaking.
Samuel Warren LeeAnn M. Sager-Smith David A. Mazziotti