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arXiv:2607.02456

A Quantum-Walk Representation of Color-Ordered MHV Scattering Amplitudes

Anirudh Verma, C. M. Chandrashekar

quant-phhep-thSubmitted Jul 2, 2026• Announced Jul 3, 2026• 0 citations
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Abstract

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.

Comments
19 pages, 9 figures

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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.
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.
A graph-theoretical framework for quantum mechanics is developed in which observables and symmetry operators are incorporated into a single complete bi-coloured graph. The proposed Symmetry/Observables Quantum Graph (SOQG) treats Hamiltonians, observables, conserved quantities, and unitary symmetry transformations on equal footing as operators acting on a common Hilbert space. The vertices of the graph represent operators, while the colours of the edges are determined by their commutation relations: commuting operators are connected by maroon edges and non-commuting operators by teal edges. The resulting structure constitutes a finite Ramsey graph that provides a combinatorial representation of operator compatibility. The formalism is applied to several representative quantum systems, including a particle in a central potential, a charged particle in a uniform magnetic field, and a particle in a homogeneous external field. In each case, monochromatic cliques identify maximal compatibility sectors containing both observables and symmetry operators, thereby revealing the interplay between conserved quantities and dynamical symmetries. It is further shown that coloured-graph automorphisms generally differ from the physical symmetry group of the underlying quantum system, leading to the notion of second-order graph symmetry. The SOQG is proved to be invariant under simultaneous unitary transformations of all operators and therefore represents intrinsic properties of the operator algebra rather than a particular Hilbert-space representation. The graph-theoretical structure is analysed using Ramsey theory and the Mantel–Turán extremal theorem, which provide rigorous conditions for the appearance of mutually compatible and mutually incompatible operator subsets. The framework is subsequently generalized to parameter-dependent Symmetry/Observables Quantum Graphs (PSOQGs), whose compatibility structure changes as external parameters modify the underlying commutation relations. For magnetic translation operators, the magnetic flux through the elementary translation cell induces a flux-controlled colour-switching transition between commuting and non-commuting compatibility sectors, accompanied by changes in the graph automorphism group. The proposed framework establishes a direct connection between operator theory, symmetry, graph theory, and extremal combinatorics, providing a new structural perspective on quantum systems.
Relativistic quantum field theory (QFT) is commonly formulated in terms of operators, asymptotic states, and covariant amplitudes, a perspective that tends to obscure the real-time origin of field dynamics and correlations. Here we formulate quantum fields in a real-time Schrödinger-picture framework, in which fields evolve as probability ensembles on the space of field configurations. Within this formulation, the wavefunctional Ψ[φ,t]Ψ[φ,t]Ψ[φ,t] encodes a first-order, causal ensemble dynamics on configuration space. Interactions appear as couplings between configuration-space directions, while propagators arise as derived correlation structures rather than as fundamental postulates. Entanglement, scattering amplitudes, and conformal field theory correlators emerge as distinct projections of the same underlying ensemble evolution, corresponding to equal-time, asymptotic, and symmetry-organized observables. Standard operator, diagrammatic, and path-integral formulations are recovered as computational representations of this single real-time dynamics. This organization makes explicit the distinction between fundamental dynamical structure and representational tools in QFT, and clarifies the scope within which ensemble-averaged correlators account for quantum fluctuations, while also delineating the level at which questions associated with individual realizations and randomness would arise beyond the correlator-based field-theoretic description.
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.
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.
We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2, on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed form, as a function of time and the initial level nnn in the tree, and we find the asymptotic and a full numerical solution for the amplitude. It exhibits a sharp interference peak and a power-law tail, as opposed to the exponentially decaying tail of a broadly peaked distribution of the classical symmetric random walk on a binary tree. The probability peak is orders of magnitude larger than it is for the classical walk (already at small nnn). The quantum walk shows a polynomial algorithmic speedup in nnn over the classical walk, which we conjecture to be of the order 2/32/32/3, based on strong trends in data.