  A quadrillion dimensional Hilbert space hosted by a quantum processor with
over 50 physical qubits has been expected to be powerful enough to perform
computational tasks ranging from simulations of many-body physics to complex
financial modeling. Despite few examples and demonstrations, it is still not
clear how we can utilize such a large Hilbert space as a computational
resource; in particular, how a simple and small quantum system could solve
non-trivial computational tasks. In this paper, we show a simple Ising model
capable of performing such non-trivial computational tasks in a quantum neural
network model. An Ising spin chain as small as ten qubits can solve a practical
image classification task with high accuracy. To evaluate the mechanism of its
computation, we examine how the symmetries of the Hamiltonian would affect its
computational power. We show how the interplay between complexity and
integrability/symmetries of the quantum system dictates the performance as
quantum neural network.
