Methodology for replacing indirect measurements with direct measurements

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 under what conditions 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 method to measure the out-of-time-order correlator based on the presented protocol. Our protocols can significantly reduce the depth of a quantum circuit by making the controlled operation unnecessary, and thus are suitable for quantum-classical hybrid algorithms on near-term quantum computers.