Quantum Approximation of Normalized Schatten Norms and Applications to Learning
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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 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(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. ≤ϵ/1+√(2(1/δ - 1))) then the fidelity of states obtained by processing the same randomly and uniformly picked pure state, |ψ⟩, is as high as needed (F(U_1 |ψ⟩, U_2 |ψ⟩)≥ 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.
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