  In the noisy intermediate-scale quantum era, variational quantum algorithms
(VQAs) have emerged as a promising avenue to obtain quantum advantage. However,
the success of VQAs depends on the expressive power of parameterised quantum
circuits, which is constrained by the limited gate number and the presence of
barren plateaus. In this work, we propose and numerically demonstrate a novel
approach for VQAs, utilizing randomised quantum circuits to generate the
variational wavefunction. We parameterize the distribution function of these
random circuits using artificial neural networks and optimize it to find the
solution. This random-circuit approach presents a trade-off between the
expressive power of the variational wavefunction and time cost, in terms of the
sampling cost of quantum circuits. Given a fixed gate number, we can
systematically increase the expressive power by extending the quantum-computing
time. With a sufficiently large permissible time cost, the variational
wavefunction can approximate any quantum state with arbitrary accuracy.
Furthermore, we establish explicit relationships between expressive power, time
cost, and gate number for variational quantum eigensolvers. These results
highlight the promising potential of the random-circuit approach in achieving a
high expressive power in quantum computing.
