Quantum Approximation of Normalized Schatten Norms and Applications to Learning

Abstract: Efficient measures to determine 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 \textit{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 ($| U_1-U_2 |<em>{S_2}$) with precision $\epsilon$, using only one clean qubit and one classical random variable. We prove a Poly$(\frac{1}{\epsilon})$ 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 ($F$): If $| U_1-U_2 |</em>{S_2}$ is sufficiently small (e.g. $\leq \frac{\epsilon}{1+\sqrt{2(1/\delta - 1)}}$) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state, $|\psi \rangle$, is as high as needed ($$F({U}_1 |\psi \rangle, {U}_2 |\psi \rangle)\geq 1-\epsilon$$) with probability exceeding $1-\delta$. 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.