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arXiv:2507.14252 · SciPost Physics

A quantum algorithm for the n-gluon MHV scattering amplitude

Erik Bashore, Stefano Moretti, Timea Vitos

Published Apr 16, 2026• 0 citations
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Abstract

We propose a quantum algorithm for computing the n n -gluon maximally helicity violating (MHV) tree-level scattering amplitude. We revisit a newly proposed method for unitarisation of non-unitary operations and present how this implementation can be used to create quantum gates responsible for the color and kinematic factors of the gluon scattering amplitude. As a proof-of-concept, we detail the full conceptual algorithm that yields the squared amplitude and implement the corresponding building blocks on simulated noiseless quantum circuits for n=4 n = 4 to analyze its performance. The algorithm is found to perform well with parameter optimizations, suggesting it to be a good candidate for implementing on quantum computers also for higher multiplicities.

DOI
10.21468/scipostphys.20.4.114

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We propose a quantum algorithm for computing the n n -gluon maximally helicity violating (MHV) tree-level scattering amplitude. We revisit a newly proposed method for unitarisation of non-unitary operations and present how this implementation can be used to create quantum gates responsible for the color and kinematic factors of the gluon scattering amplitude. As a proof-of-concept, we detail the full conceptual algorithm that yields the squared amplitude and implement the corresponding building blocks on simulated noiseless quantum circuits for n=4 n = 4 to analyze its performance. The algorithm is found to perform well with parameter optimizations, suggesting it to be a good candidate for implementing on quantum computers also for higher multiplicities.
We introduce a graph-theoretic framework for representing color-ordered maximally helicity violating (MHV) scattering amplitudes in quantum chromodynamics using coined quantum walks on permutation trees. Each root-to-terminal path corresponds to a distinct color ordering of the external gluons, while local transition amplitudes are assigned according to the spinor-product structure of the Parke--Taylor amplitudes. The walk evolves in coherent superpositions over permutation sectors, giving a dynamical picture of the underlying combinatorics. A quantum-channel formulation based on Kraus operators is also introduced to describe sector-resolved contributions, while a weighted collection operator coherently combines the terminal sectors at a common reference node. A quantum Fourier transform on the coin space is then employed to combine the encoded contributions into the corresponding color-decomposed amplitude. Together, these constructions establish a unified graph-based framework connecting permutation trees, quantum walks, and open quantum systems providing a framework for quantum algorithms to simulate scattering processes in quantum field theory. As an example, numerical results for low-point gluon amplitudes demonstrate that the proposed representation faithfully captures the characteristic Parke--Taylor structure and is consistent with analytical results.
Scattering processes in high-energy physics are inherently quantum mechanical, yet are typically analyzed at the level of final states, where entanglement appears as a property of the outcome rather than a consequence of the underlying dynamics. We reformulate scattering at the level of the process itself by representing helicity transition matrices as quantum circuits. Once the kinematic configuration and scattering channel are fixed, the problem reduces to a finite-dimensional quantum map, making a circuit description natural. Within this framework, an example of the process e+e−→μ+μ−e^+e^-\to μ^+μ^-e+e−→μ+μ− is shown, which decomposes into unitary and non-unitary components, corresponding to coherent mixing and postselection effects. This representation reorganizes the amplitude into distinct operational elements, providing a perspective in which collider processes can be viewed as constrained quantum circuits and their entanglement structure can be understood in terms of the underlying circuit dynamics, opening the door to analyzing their properties using the language of quantum information.
The efficient simulation of accurate collider data at modern collider experiments will be crucial for the detection of Beyond the Standard Model signatures in the search for new physics. Simulation of collision processes at hadron colliders like the LHC are carried out by Monte Carlo event generators which simulate the evolution from an initial hard scattering event, downwards in energy scales, towards the formation of long lived particles. However, as we enter a new era of higher luminosity experiments, accurate simulations of these pseudo-event generated data will become further computationally intensive with the vast amounts of data to generate. With the rapid development of quantum computing hardware, quantum computation offers itself as an alternative computing paradigm that may be harnessed to provide a natural framework to model several physical processes in high-energy physics, where the inherent quantum features of the device may be exploited to provide speedups or enhance current simulations. In this thesis, we present general and extendable quantum algorithms for two crucial parts of event generation: the calculation of matrix elements for the hard interaction and the QCD parton shower stage. First, a novel algorithm is proposed for the calculation of helicity amplitudes by outlining a proposal of constructing helicity spinors directly on a quantum circuit and manipulating the spinors to compute helicity amplitudes. This was used to calculate multiple helicity amplitudes for simple tree level scattering processes simultaneously as a proof-of-principle demonstration. The second algorithm outlines a proposal for a Monte Carlo-inspired parton shower algorithm which was used to simulate two shower steps of a simplified QCD model. The final algorithm extends the quantum parton shower algorithm onto a quantum walk framework which demonstrates significant scaling improvements, simulating more realistic shower depths. These algorithms utilise the quantum computers’ ability to remain in a quantum state throughout the computation and represents a first step towards a quantum computing algorithm describing full collision events at the LHC.
Magic, also known as non-stabilizerness, measures the usefulness of a quantum state for quantum computation. While magic is defined relative to a choice of computational basis, in some physical settings the available data determine this basis only up to local phase conventions. In this paper, we generalize the notion of magic and formulate it in a phase-independent manner, and hence define a generalized stabilizer Rényi entropy. As a case study, we consider higher-multiplicity tree-level gluon scattering, interpreting the outgoing helicities as qubits. In this setting, the helicity data naturally determine a local basis for each qubit but leave a phase ambiguity. For 3→23\to 23→2 scattering, we find that the final-state phase-independent magic is generically larger than the maximum attainable in 2→22\to 22→2 scattering. For 2→32 \to 32→3 scattering, we find a nonzero minimal value approached in the soft limit. Moreover, when the three outgoing momenta become symmetric, the magic approaches a local minimum only a few percent above the soft-limit value. In all cases considered, the color dependence cancels from the phase-independent stabilizer Rényi entropy.
Quantum computers are expected to give major speed-ups for the simulation of quantum systems. In this work, we present quantum gates that simulate the colour part of the interactions of quarks and gluons in perturbative quantum chromodynamics (QCD). As a first application, we implement these circuits on a simulated noiseless quantum computer and use them to calculate colour factors for various examples of Feynman diagrams. This work constitutes a first key step towards a quantum simulation of generic scattering processes in perturbative QCD.
Precision measurements at electron-positron colliders provide stringent tests of the Standard Model and powerful probes of possible higher-dimensional interactions. We present a hybrid quantum-classical framework for computing leading-order helicity amplitudes for e+e−→ℓ+ℓ−e^+e^-\to \ell^+\ell^-e+e−→ℓ+ℓ− scattering on gate-based quantum hardware and using the resulting cross sections to constrain both Standard Model couplings and effective field theory operators. In our approach, external kinematics are encoded into single-qubit Weyl spinors, and full helicity amplitudes are reconstructed by coherently combining diagrammatic contributions within a single quantum circuit. Classical post-processing yields physical amplitudes and differential cross sections that can be directly compared with collider data. As a proof of concept, we compute unpolarised angular distributions and perform binned likelihood fits to precision electron-positron measurements. The extracted bounds are statistically consistent with Standard Model expectations, demonstrating that quantum-assisted amplitude evaluation can interface directly with phenomenological analyses and experimental data. This work establishes a concrete pathway toward applying quantum computing to precision collider physics and effective field theory studies.