  A key open question in quantum computing is whether quantum algorithms can
potentially offer a significant advantage over classical algorithms for tasks
of practical interest. Understanding the limits of classical computing in
simulating quantum systems is an important component of addressing this
question. We introduce a method to simulate layered quantum circuits consisting
of parametrized gates, an architecture behind many variational quantum
algorithms suitable for near-term quantum computers. A neural-network
parametrization of the many-qubit wave function is used, focusing on states
relevant for the Quantum Approximate Optimization Algorithm (QAOA). For the
largest circuits simulated, we reach 54 qubits at 4 QAOA layers, approximately
implementing 324 RZZ gates and 216 RX gates without requiring large-scale
computational resources. For larger systems, our approach can be used to
provide accurate QAOA simulations at previously unexplored parameter values and
to benchmark the next generation of experiments in the Noisy Intermediate-Scale
Quantum (NISQ) era.
