Quantum Approximation of Normalized Schatten Norms and Applications to Learning

TL;DR

This paper introduces an efficient quantum sampling method to estimate a normalized Schatten 2-norm based similarity measure for quantum operations, enabling practical applications in quantum circuit learning with size-independent sample complexity.

Contribution

It develops a quantum sampling circuit for estimating the Schatten 2-norm difference between quantum operations with size-independent complexity, linking it to fidelity-based similarity.

Findings

Sample complexity is Poly(1/ε), independent of system size.

Small Schatten 2-norm difference implies high fidelity for processed states.

Application demonstrated in quantum circuit learning tasks.

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 ($\| U_1-U_2 \|_{S_2}$) 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 $\| U_1-U_2 \|_{S_2}$ 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.