Figure 1

Quantum circuit for time-dependent stochastic parameter-shift rule (Stoc.PSR). For every time t, we first prepare a quantum state \(\rho _{0}\) and generate a random number \(s \in [0,t]\). A sequence of gates \(e^{-isH(\boldsymbol{\phi} )}\), \(e^{-it\mu [\partial _{\phi _{j}}H(\boldsymbol{\phi} )]}\), and \(e^{-i(t-s)H(\boldsymbol{\phi} )}\) applies to the circuit and produces \(\rho ^{+}\). We then repeat the scheme while replacing \(e^{-it\mu [\partial _{\phi _{j}}H(\boldsymbol{\phi} )]}\) by \(e^{it\mu [\partial _{\phi _{j}}H(\boldsymbol{\phi} )]}\) and compute \(\rho ^{-}\). The derivative \(\partial _{\phi _{j}}\rho (\boldsymbol{\phi} )\) is obtained via \(\rho ^{+}-\rho ^{-}\). The process is then repeated for all \(\phi _{j}\) to get the QFIM \(Q(\boldsymbol{\phi} )\). Then, we move to the next time \(t'\) and do the same procedure