Quantum gradient evaluation through quantum non-demolition measurements

We discuss a quantum non-demolition measurement (QNDM) protocol to estimate the derivatives of a cost function with a quantum computer. The cost function, which is supposed to be classically hard to evaluate, is associated with the average value of a quantum operator. Then a quantum computer is used to efficiently extract information about the function and its derivative by evolving the system with a so-called variational quantum circuit. To this aim, we propose to use a quantum detector that allows us to directly estimate the derivatives of an observable, i.e., the derivative of the cost function. With respect to the standard direct measurement approach, this leads to a reduction of the number of circuit iterations needed to run the variational quantum circuits. The advantage increases if we want to estimate the higher-order derivatives. We also show that the presented approach can lead to a further advantage in terms of the number of total logical gates needed to run the variational quantum circuits. These results make the QNDM a valuable alternative to implementing the variational quantum circuits.