  We study theoretically a variety of collective quantum phases occurring in
frustrated saw-tooth chains of Josephson junctions embedded in a
dissipationless transmission line. The basic element of a system, i.e., the
triangular superconducting cell, contains two $0$- and one $\pi$- Josephson
junctions characterized by $E_J$ and $\alpha E_J$ Josephson energies,
accordingly. In the frustrated regime the low energy quantum dynamics of a
single cell is determined by anticlockwise or clockwise flowing persistent
currents (vortex/antivortex). The direct embedding of $\pi$-Josephson junctions
in a transmission line allows to establish a short/long-range interaction
between (anti)vortices of well separated cells. By making use of the
variational approach, we map the superconducting circuit Hamiltonian to an
effective $XX$ spin model with an exchange spin-spin interaction decaying with
the distance $x$ as $x^{-\beta}$, and the local $\hat \sigma_{x,n}$-terms
corresponding to the coherent quantum beats between vortex and antivortex in a
single cell. We obtain that in long arrays as $N \gg \ell_0 \simeq
\sqrt{C/C_0}$, where $C$ and $C_0$ are capacitances of $0$-Josephson junction
and transmission line, accordingly, the amplitude of quantum beats is strongly
suppressed. By means of exact numerical diagonalization, we study the interplay
between the coherent quantum beats and the exchange spin-spin interaction
leading to the appearance of various collective quantum phases such as the
paramagnetic ($P$), compressible superfluid ($CS$) and weakly compressible
superfluid ($w$-$CS$) states.
