Quantum querying based on multicontrolled Toffoli gates for causal Feynman loop configurations and directed acyclic graphs

TL;DR

This paper introduces a quantum algorithm leveraging multicontrolled Toffoli gates for querying causal Feynman loop configurations and DAGs, analyzing its efficiency and complexity for higher-loop topologies.

Contribution

The paper presents a novel quantum algorithm based on multicontrolled Toffoli gates for querying complex causal and DAG configurations, including higher-loop topologies.

Findings

Algorithm based on MCX gates is efficient compared to binary clause methods.

Introduces quantum circuit area as a metric for complexity assessment.

Explicit analysis of 3-, 4-, and 5-loop topologies, previously unexplored.

Abstract

Quantum algorithms are a promising framework for unfolding the causal configurations of multiloop Feynman diagrams, which is equivalent to querying the \textit{directed acyclic graph} (DAG) configurations of undirected graphs in graph theory. In this paper, we present a quantum algorithm for querying in both types of applications, using a systematic and sparing logic in the design of an oracle operator. The construction of the quantum oracle is based exclusively on multicontrolled Toffoli (MCX) gates and quantum NOT (Pauli-$X$) gates. The efficiency of the algorithm is evaluated by comparison with a quantum algorithm based on binary clauses. Furthermore, we analyse the impact of traspilation and introduce an appropriate metric to assess the complexity of the algorithm, the \emph{quantum circuit area}. We explicitly analyse three-, four- and five-eloop topologies, which have not previously been explored due to their higher complexity and the current limitations of quantum simulators.