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 ($\Vert U_1-U_2{\Vert }_{S_2}$) 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 $\Vert U_1-U_2{\Vert }_{S_2}$ is sufficiently small (e.g. $\le \frac{\epsilon }{1+\sqrt{2(1/\delta -1)}}$) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state, $|\psi ⟩$, is as high as needed ($F(U_1|\psi ⟩,U_2|\psi ⟩)\ge 1-\epsilon$) 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.

25 pages, 4 figures, 6 tables, 1 algorithm

References in corpus (16)

• Quantum algorithm for solving linear systems of equations
• A Quantum Approximate Optimization Algorithm
• The Quantum Chernoff Bound
• Quantum state discrimination and its applications
• Fidelity of quantum operations
• All entangled states are useful for channel discrimination
• Distinguishing Arbitrary Multipartite Basis Unambiguously Using Local Operations and Classical Communication
• Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition
• Distinguishability of Quantum States by Separable Operations
• Quantum state discrimination: a geometric approach
• Quantum Neural Network Classifiers: A Tutorial
• Unambiguous discrimination among oracle operators
• Exact Quantum Search by Parallel Unitary Discrimination Schemes
• Distinguishability, classical information of quantum operations
• Optimal Perfect Distinguishability between Unitaries and Quantum Operations
• Ancilla-Assisted Discrimination of Quantum Gates

Cited by in corpus (2)

• Cross-Platform Verification in Quantum Networks
• Quantum Wasserstein distance between unitary operations