  Modern hybrid superconductor-semiconductor Josephson junction arrays are a
promising platform for analog quantum simulations. Their controllable and
non-sinusoidal energy/phase relation opens the path to implement nontrivial
interactions and study the emergence of exotic quantum phase transitions. Here,
we propose the analysis of an array of hybrid Josephson junctions defining a
2-leg ladder geometry for the quantum simulation of the tricritical Ising phase
transition. This transition provides the paradigmatic example of minimal
conformal models beyond Ising criticality and its excitations are intimately
related with Fibonacci non-Abelian anyons and topological order in two
dimensions. We study this superconducting system and its thermodynamic phases
based on bosonization and matrix-product-states techniques. Its effective
continuous description in terms of a three-frequency sine-Gordon quantum field
theory suggests the presence of the targeted tricritical point and the
numerical simulations confirm this picture. Our results indicate which
experimental observables can be adopted in realistic devices to probe the
physics and the phase transitions of the model. Additionally, our proposal
provides a useful one-dimensional building block to design exotic topological
order in two-dimensional scalable Josephson junction arrays.
