  The celebrated work of Berezinskii, Kosterlitz and Thouless in the 1970s
revealed exotic phases of matter governed by topological properties of
low-dimensional materials such as thin films of superfluids and
superconductors. Key to this phenomenon is the appearance and interaction of
vortices and antivortices in an angular degree of freedom---typified by the
classical XY model---due to thermal fluctuations. In the 2D Ising model this
angular degree of freedom is absent in the classical case, but with the
addition of a transverse field it can emerge from the interplay between
frustration and quantum fluctuations. Consequently a Kosterlitz-Thouless (KT)
phase transition has been predicted in the quantum system by theory and
simulation. Here we demonstrate a large-scale quantum simulation of this
phenomenon in a network of 1,800 in situ programmable superconducting flux
qubits arranged in a fully-frustrated square-octagonal lattice. Essential to
the critical behavior, we observe the emergence of a complex order parameter
with continuous rotational symmetry, and the onset of quasi-long-range order as
the system approaches a critical temperature. We use a simple but previously
undemonstrated approach to statistical estimation with an annealing-based
quantum processor, performing Monte Carlo sampling in a chain of reverse
quantum annealing protocols. Observations are consistent with classical
simulations across a range of Hamiltonian parameters. We anticipate that our
approach of using a quantum processor as a programmable magnetic lattice will
find widespread use in the simulation and development of exotic materials.
