  Operator size growth describes the scrambling of operators in quantum
dynamics and stands out as an essential physical concept for characterizing
quantum chaos. Important as it is, a scheme for direct measuring operator size
on a quantum computer is still absent. Here, we propose a quantum algorithm for
direct measuring the operator size and its distribution based on Bell
measurement. The algorithm is verified with spin chains and meanwhile, the
effects of Trotterization error and quantum noise are analyzed. It is revealed
that saturation of operator size growth can be due to quantum chaos itself or
be a consequence of quantum noises, which make a distinction between quantum
integrable and chaotic systems difficulty on noisy quantum processors.
Nevertheless, it is found that the error mitigation will effectively reduce the
influence of noise, so as to restore the distinguishability of quantum chaotic
systems. Our work provides a feasible protocol for investigating quantum chaos
on noisy quantum computers by measuring operator size growth.
