  Hamiltonian moments in Fourier space - expectation values of the unitary
evolution operator under a Hamiltonian at different times - provide a
convenient framework to understand quantum systems. They offer insights into
the energy distribution, higher-order dynamics, response functions, correlation
information and physical properties. This paper focuses on the computation of
Fourier moments within the context of a nuclear effective field theory on
superconducting quantum hardware. The study integrates echo verification and
noise renormalization into Hadamard tests using control reversal gates. These
techniques, combined with purification and error suppression methods,
effectively address quantum hardware decoherence. The analysis, conducted using
noise models, reveals a significant reduction in noise strength by two orders
of magnitude. Moreover, quantum circuits involving up to 266 CNOT gates over
five qubits demonstrate high accuracy under these methodologies when run on IBM
superconducting quantum devices.
