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

Sebastian Alberto Grillo, Bernardo Daniel Dávalos, Rodney Fabian Franco Torres, Franklin de Lima Marquezino, Edgar López Pezoa

TL;DR

The paper addresses the question of when single-query quantum algorithms can outperform classical counterparts by modeling them as weighted dynamical graphs (WDG) and analyzing the $L_1$ spectral norm of the output. It introduces a degree-2 polynomial–WDG correspondence and develops an optimization framework to maximize the $L_1$ norm, including Kronecker-product-based graph compositions that yield exponentially growing norms. A concrete exponential-advantage example and a necessary condition tied to the growth of measurement projector dimensions are provided, highlighting measurement as a fundamental resource. The framework offers a combinatorial, tractable path to design resource-efficient quantum procedures and delineates clear directions for extending the approach to broader classes of problems and circuits.

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.

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

TL;DR

The paper addresses the question of when single-query quantum algorithms can outperform classical counterparts by modeling them as weighted dynamical graphs (WDG) and analyzing the spectral norm of the output. It introduces a degree-2 polynomial–WDG correspondence and develops an optimization framework to maximize the norm, including Kronecker-product-based graph compositions that yield exponentially growing norms. A concrete exponential-advantage example and a necessary condition tied to the growth of measurement projector dimensions are provided, highlighting measurement as a fundamental resource. The framework offers a combinatorial, tractable path to design resource-efficient quantum procedures and delineates clear directions for extending the approach to broader classes of problems and circuits.

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 spectral norm of the algorithm output. This advantage is based on the fact that a high 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 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.
Paper Structure (6 sections, 13 theorems, 49 equations, 3 figures, 1 table)

This paper contains 6 sections, 13 theorems, 49 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomials of degree at most two as weighted graphs
  4. The $L_{1}$ spectral norm maximization problem.
  5. $L_1$ spectral norm and measurement
  6. Conclusion

Key Result

Theorem 2

Given a QQM algorithm, we also introduce the following notation: If the QQM algorithm applies $t$ queries over $x=x_0x_1...x_n$, then:

Figures (3)

  • Figure 1: Weighted graph for Equation \ref{['ej1']}.
  • Figure 2: Two weighted graph.
  • Figure 3: Scheme of the proof for certificate complexity.

Theorems & Definitions (33)

  • Definition 1: Complete Set of Orthogonal Projectors
  • Theorem 2
  • Definition 3
  • Theorem 4
  • Definition 5: Certificate complexity
  • Theorem 6
  • Theorem 7
  • proof
  • Definition 8
  • Theorem 9
  • ...and 23 more