Recently, quantum metrology with multiplicative Hamiltonians has been
proposed in the variational quantum algorithms, from which the estimation
precision can be adaptively optimized via the variational circuits. For systems
with general Hamiltonians, however, still lack these variational schemes. In
this work, we introduce a quantum-circuit-based approach for studying quantum
metrology with general Hamiltonians. We introduce the stochastic
parameter-shift rule for the derivatives of the evolved quantum state under the
parameterized gates in the circuit, whereby the quantum Fisher information can
be obtained. Here the parameters are those we wish to estimate. We find that
under the family of the parameterized gates, our scheme can be executed in
universal quantum computers. Moreover, in the examples of the magnetic field
estimation, we show the consistency between the results obtained from the
stochastic parameter-shift rule and the exact results while the standard
parameter-shift rule slightly deviates from the exact ones. Our work sheds new
light for studying quantum metrology with general Hamiltonians using quantum
circuit algorithms.Comment: 9 pages, 3 figure