  We systematically map low-bond-dimension PEPs tensor networks to quantum
circuits. By measuring and reusing qubits, we demonstrate that a simulation of
an $N \times M$ square-lattice PEPs network, for arbitrary $M$, of bond
dimension $2$ can be performed using $N+2$ qubits. We employ this approach to
calculate the values of a long-range loop observable in the topological Wen
plaquette model by mapping a $3\times 3$ PEPs tensor network to a 5-qubit
quantum circuit and executing it on the Honeywell System Model H1-1 trapped-ion
device. We find that, for this system size, the noisy observable values are
sufficient for diagnosing topological vs. trivial order, as the Wen model is
perturbed by a magnetic field term in the Hamiltonian. We provide an overview
of the experimental procedure and its results. We then explain in greater
detail our method for mapping 2D tensor networks to quantum circuits and its
scaling properties. Our results serve as a proof-of-concept of the utility of
the measure-and-reuse approach for simulating large two-dimensional quantum
systems on small quantum devices.
