  Simulating quantum algorithms with classical resources generally requires
exponential resources. However, heuristic classical approaches are often very
efficient in approximately simulating special circuit structures, for example
with limited entanglement, or based on one-dimensional geometries. Here we
introduce a classical approach to the simulation of general quantum circuits
based on neural-network quantum states (NQS) representations. Considering a set
of universal quantum gates, we derive rules for exactly applying single-qubit
and two-qubit Z rotations to NQS, whereas we provide a learning scheme to
approximate the action of Hadamard gates. Results are shown for the Hadamard
and Fourier transform of entangled initial states for systems sizes and total
circuit depths exceeding what can be currently simulated with state-of-the-art
brute-force techniques. The overall accuracy obtained by the neural-network
states based on Restricted Boltzmann machines is satisfactory, and offers a
classical route to simulating highly-entangled circuits. In the test cases
considered, we find that our classical simulations are comparable to quantum
simulations affected by an incoherent noise level in the hardware of about
$10^{-3}$ per gate.
