  Multicritical Ising models and their perturbations are paradigmatic models of
statistical mechanics. In two space-time dimensions, these models provide a
fertile testbed for investigation of numerous non-perturbative problems in
strongly-interacting quantum field theories. In this work, analog
superconducting quantum electronic circuit simulators are described for the
realization of these multicritical Ising models. The latter arise as
perturbations of the quantum sine-Gordon model with $p$-fold degenerate minima,
$p =2, 3,4,\ldots$. The corresponding quantum circuits are constructed with
Josephson junctions with $\cos(n\phi + \delta_n)$ potential with $1\leq n\leq
p$ and $\delta_n\in[-\pi,\pi]$. The simplest case, $p = 2$, corresponds to the
quantum Ising model and can be realized using conventional Josephson junctions
and the so-called $0-\pi$ qubits. The lattice models for the Ising and
tricritical Ising models are analyzed numerically using the density matrix
renormalization group technique. Evidence for the multicritical phenomena are
obtained from computation of entanglement entropy of a subsystem and
correlation functions of relevant lattice operators. The proposed quantum
circuits provide a systematic approach for controlled numerical and
experimental investigation of a wide-range of non-perturbative phenomena
occurring in low-dimensional quantum field theories.
