Diagonal Adaptive Non-local Observables on Quantum Neural Networks

TL;DR

This paper introduces a diagonal form of Adaptive Non-local Observables for quantum neural networks, reducing complexity and measurement costs while maintaining expressive power.

Contribution

It proposes a diagonal ANO method that simplifies measurement design and decreases classical computation without sacrificing the full ANO capabilities.

Findings

Reduces $k$-local observable complexity from $O(4^k)$ to $O(2^k)$

Lowers measurement-side classical computation

Retains the expressive power of full ANO

Abstract

Adaptive Non-local Observables (ANOs) have shown that making quantum observables dynamic can substantially enlarge the function space of Variational Quantum Algorithms, partly shifting hardware demands from circuit synthesis to measurement design. However, this advantage is accompanied by a steep increase in the number of parameters, as well as the classical optimization cost for varying general Hermitian observables. We propose a special form of ANO that significantly reduces this burden by considering only diagonal observables paired with quantum circuits. Mathematically, this is equivalent to the full ANO of a large parameter space since diagonal matrices are canonical representatives of the ANO space modulo unitary similarity. As a result, Diagonal ANO retains the same capability of full ANO while reducing $k$-local observable complexity from $O(4^k)$ to $O(2^k)$ and lowering the corresponding measurement-side classical computation. In this sense, diagonal ANO preserves much of the benefit of full ANO while encompassing conventional VQCs as a special case.