Quantum Approximation of Normalized Schatten Norms and Applications to Learning
This provides an efficient similarity measure for quantum operations, aiding quantum learning tasks, though it is incremental as it builds on existing fidelity and norm concepts.
The paper tackles the problem of efficiently estimating similarity between quantum operations by developing a quantum sampling circuit to approximate the normalized Schatten 2-norm with precision ε, achieving a sample complexity polynomial in 1/ε independent of system size. It shows this metric relates to state fidelity, enabling applications like quantum circuit learning, such as finding unitary square roots.
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 ($\| U_1-U_2 \|_{S_2}$) with precision $ε$, using only one clean qubit and one classical random variable. We prove a Poly$(\frac{1}ε)$ 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 $\| U_1-U_2 \|_{S_2}$ is sufficiently small (e.g. $ \leq \fracε{1+\sqrt{2(1/δ- 1)}}$) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state, $|ψ\rangle$, is as high as needed ($F({U}_1 |ψ\rangle, {U}_2 |ψ\rangle)\geq 1-ε$) with probability exceeding $1-δ$. 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.