  In quantum computing, the indirect measurement of unitary operators such as
the Hadamard test plays a significant role in many algorithms. However, in
certain cases, the indirect measurement can be reduced to the direct
measurement, where a quantum state is destructively measured. Here we
investigate in what cases such a replacement is possible and develop a general
methodology for trading an indirect measurement with sequential direct
measurements. The results can be applied to construct quantum circuits to
evaluate the analytical gradient, metric tensor, Hessian, and even higher order
derivatives of a parametrized quantum state. Also, we propose a new method to
measure the out-of-time-order correlator based on the presented protocol. Our
protocols can reduce the depth of the quantum circuit significantly by making
the controlled operation unnecessary and hence are suitable for
quantum-classical hybrid algorithms on near-term quantum computers.
