  Many experimental proposals for noisy intermediate scale quantum devices
involve training a parameterized quantum circuit with a classical optimization
loop. Such hybrid quantum-classical algorithms are popular for applications in
quantum simulation, optimization, and machine learning. Due to its simplicity
and hardware efficiency, random circuits are often proposed as initial guesses
for exploring the space of quantum states. We show that the exponential
dimension of Hilbert space and the gradient estimation complexity make this
choice unsuitable for hybrid quantum-classical algorithms run on more than a
few qubits. Specifically, we show that for a wide class of reasonable
parameterized quantum circuits, the probability that the gradient along any
reasonable direction is non-zero to some fixed precision is exponentially small
as a function of the number of qubits. We argue that this is related to the
2-design characteristic of random circuits, and that solutions to this problem
must be studied.
