Quantum Advantage in Decision Trees: A Weighted Graph and $L_1$ Norm Approach

TL;DR

This paper introduces a weighted graph formulation for single-query quantum decision trees, linking their quantum advantage to the $L_1$ spectral norm and providing heuristics and conditions for exponential quantum advantage.

Contribution

It presents a novel weighted graph approach to analyze quantum decision trees and establishes a necessary condition for quantum advantage based on spectral norm growth.

Findings

High $L_1$ spectral norm is necessary for quantum advantage.

Heuristics for maximizing spectral norm are proposed.

Functions with exponential quantum advantage are demonstrated.

Abstract

The analysis of the computational power of single-query quantum algorithms is important because they must extract maximal information from one oracle call, revealing fundamental limits of quantum advantage and enabling optimal, resource-efficient quantum computation. This paper proposes a formulation of single-query quantum decision trees as weighted graphs. This formulation has the advantage that it facilitates the analysis of the $L_1$ spectral norm of the algorithm output. This advantage is based on the fact that a high $L_1$ spectral norm of the output of a quantum decision tree is a necessary condition to outperform its classical counterpart. We propose heuristics for maximizing the $L_{1}$ spectral norm, show how to combine weighted graphs to generate sequences with strictly increasing norm, and present functions exhibiting exponential quantum advantage. Finally, we establish a necessary condition linking single-query quantum advantage to the asymptotic growth of measurement projector dimensions.