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arXiv:2404.03544 · Physical review. D/Physical review. D.

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

Selomit Ramírez-Uribe, Andrés E. Rentería-Olivo, Germán Rodrigo A

Published Jun 25, 2026• 1 citation
arXivHTMLPDF

Abstract

Quantum algorithms are a promising framework for unfolding the causal configurations of multiloop Feynman diagrams, which is equivalent to querying the 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 analyze the impact of traspilation and introduce an appropriate metric to assess the complexity of the algorithm, the quantum circuit area. We explicitly analyze three-, four- and five-eloop topologies, which have not previously been explored due to their higher complexity and the current limitations of quantum simulators.

DOI
10.1103/b6jg-p8mr

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Abstract Quantum algorithms provide a promising framework in high-energy physics, in particular, for unraveling the causal configurations of multiloop Feynman diagrams by identifying Feynman propagators with qubits, a challenge analogous to querying directed acyclic graphs in graph theory. In this paper, we present the Minimum Clique-optimised quantum Algorithm (MCA), an automated quantum algorithm designed to efficiently query the causal structures within the Loop-Tree Duality. The MCA quantum algorithm is optimised by exploiting graph theory techniques, specifically, by analogy with the Minimum Clique Partition problem. The evaluation of the MCA quantum algorithm is exhibited by analysing the transpiled quantum circuit depth and quantum circuit area.
Abstract We present a novel benchmark application of a quantum algorithm to Feynman loop integrals. The two on-shell states of a Feynman propagator are identified with the two states of a qubit and a quantum algorithm is used to unfold the causal singular configurations of multiloop Feynman diagrams. To identify such configurations, we exploit Grover’s algorithm for querying multiple solutions over unstructured datasets, which presents a quadratic speed-up over classical algorithms when the number of solutions is much smaller than the number of possible configurations. A suitable modification is introduced to deal with topologies in which the number of causal states to be identified is nearly half of the total number of states. The output of the quantum algorithm in IBM Quantum and QUTE Testbed simulators is used to bootstrap the causal representation in the loop-tree duality of representative multiloop topologies. The algorithm may also find application and interest in graph theory to solve problems involving directed acyclic graphs.
The first application of a quantum algorithm to Feynman loop integrals is reviewed. The connection between quantum computing and perturbative quantum field theory is feasible due to fact that the two on-shell states of a Feynman propagator are naturally encoded in a qubit. The particular problem to be addressed is the identification of the causal singular configurations of multiloop Feynman diagrams. The identification of such configurations is carried out through the implementation of a modified Grover's quantum algorithm for querying multiple solutions over unstructured datasets.
The first application of a quantum algorithm to Feynman loop integrals is reviewed. The connection between quantum computing and perturbative quantum field theory is feasible due to fact that the two on-shell states of a Feynman propagator are naturally encoded in a qubit. The particular problem to be addressed is the identification of the causal singular configurations of multiloop Feynman diagrams. The identification of such configurations is carried out through the implementation of a modified Grover's quantum algorithm for querying multiple solutions over unstructured datasets.
A proof-of-concept application of a quantum algorithm to multiloop Feynman integrals in the Loop-Tree Duality (LTD) framework is applied to a representative four-loop topology. Causality obtained through the LTD formalism, is a suitable problem to address with quantum computers given the straightforward possibility to encode the two on shell states of a propagator on the two states of a qubit. A modification of Grover's quantum search algorithm is developed for querying multiple solutions over the unstructured set of solutions generated after the application of the LTD. The quantum algorithm is successfully implemented on IBM Quantum and QUTE simulators.
In this thesis, we study concepts in quantum computing using graphical languages, specifically using the ZX-calculus. The core of the research revolves around (graphical) stabilizer decompositions. The first major focus is on the decomposition of non-stabilizer states created from star edges. We discuss previous results and then present novel decompositions that yield a theoretical improvement. The second major focus is on weighting algorithms, applied to the special class of multi-control Toffoli gate dense quantum circuits. The representation of the corresponding gates is based on star edges. The applicability of known methods, such as CNOT-grouping, traditionally used for other classes, is examined in the context of this specific class. We then present a novel weighting algorithm that attempts to determine the best vertex to decompose. A refined version is implemented to simulate a known class of quantum querying algorithms, which is used to search for causal configurations of multiloop Feynman diagrams. For this case, as well as for a generalized benchmark consisting of randomly generated quantum circuits, we demonstrate occasional improvements in the final number of terms against traditional methods. These results are discussed by considering different simplification strategies. This thesis also provides a brief but broad outline of the important preliminaries.