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.
