Quantum approximation of normalized Schatten norms and applications to learning
Yiyou Chen, Hideyuki Miyahara, Louis‐S. Bouchard, Vwani Roychowdhury · Physical Review A · 2022
Efficient measures to determine the similarity of quantum states, such as the fidelity metric, have been widely studied. In this paper, we address the problem of defining a similarity measure for quantum operations that can be efficiently estimated. Given two quantum operations, ${U}_{1}$ and ${U}_{2}$, represented in their circuit forms, we first develop a quantum sampling circuit to estimate the normalized Schatten 2-norm of their difference ($\ensuremath{\parallel}{U}_{1}\ensuremath{-}{U}_{2}{\ensuremath{\parallel}}_{{S}_{2}}$) with precision $\ensuremath{\epsilon}$, using only one clean qubit and one classical random variable. We prove a $\text{Poly}(\frac{1}{\ensuremath{\epsilon}})$ upper bound on the sample complexity, which is independent of the size of the quantum system. We then show that such a similarity metric is directly related to a functional definition of similarity of unitary operations using the conventional fidelity metric of quantum states ($\mathcal{F}$): If $\ensuremath{\parallel}{U}_{1}\ensuremath{-}{U}_{2}{\ensuremath{\parallel}}_{{S}_{2}}$ is sufficiently small (e.g., $\ensuremath{\le}\frac{\ensuremath{\epsilon}}{1+\sqrt{2(1/\ensuremath{\delta}\ensuremath{-}1)}}$) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state $|\ensuremath{\psi}\ensuremath{\rangle}$ is as high as needed $[\mathcal{F}({U}_{1}|\ensuremath{\psi}\ensuremath{\rangle},{U}_{2}|\ensuremath{\psi}\ensuremath{\rangle})\ensuremath{\ge}1\ensuremath{-}\ensuremath{\epsilon}]$ with probability exceeding $1\ensuremath{-}\ensuremath{\delta}$. We provide example applications of this efficient similarity metric estimation framework to quantum circuit learning tasks, such as finding the square root of a given unitary operation.