# Quantum Approximation of Normalized Schatten Norms and Applications to Learning

arXiv: https://arxiv.org/abs/2206.11506 · HTML: https://mlsanity.com/paper/2206.11506/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 70 references that are in the map.
- First posted on arXiv 23 Jun 2022.

## Builds on (most cited first, 5 of 5)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [Ansatz-Independent Variational Quantum Classifier](https://mlsanity.com/paper/2102.01759/) (2021)

## Abstract (arXiv)

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 textitefficiently estimated. Given two quantum operations, U1 and U2, represented in their circuit forms, we first develop a quantum sampling circuit to estimate the normalized Schatten 2-norm of their difference (| U1-U2 |S2) with precision ε, using only one clean qubit and one classical random variable. We prove a Poly(frac1ε) 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 | U1-U2 |S2 is sufficiently small (e.g.  ≤ fracε1+√2(1/δ - 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(U1 |psi rangle, U2 |psi rangle)≥ 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.