Integration of IBM's Non-Abelian Fourier Transform with the Five-Dimensional Time-Delay Field Model: Quantum Simulation of τ -Mode Resonances at 5–6 TeV

Integration of IBM’s Non-Abelian Fourier Transform with the Five-Dimensional Time-Delay Field Model: Quantum Simulation of τ-Mode Resonances at 5–6 TeV Bahman Masarrat Independent Researcher October 30, 2025 Keywords: Quantum computing, Non-Abelian Fourier Transform, Time-Delay Field, Kaluza– Klein, LHC phenomenology. Abstract We integrate IBM’s non-Abelian Fourier Transform (NAFT) framework [ 1 ] with the five-dimensional Time-Delay Field (TDF) model. By mapping the delay phase τ ( x, t ) onto a compact U(1) group element and employing quantum phase estimation (QPE) in Qiskit, we simulate the predicted scalar KK-like resonance at 5 TeV and 6 TeV , consistent with LHC Run 3 sensitivity. An effective two-mode interaction lifts near-degeneracy, yielding a robust energy splitting ∆ E≈ 0 . 8 TeV . We estimate dilepton signal yields for σ×BR ≳ 0 . 5 fb , within reach of ATLAS/CMS high-mass searches. Code and data: https://github.com/ bahman2017/tau_delay. 1 Introduction Symmetry-based quantum algorithms have emerged as powerful tools for physics simulation [ 1 ]. IBM’s non-Abelian Fourier Transform (NAFT) on symmetric groups Sn enables efficient representation-theoretic computations. Independently, the 5D Time-Delay Field (TDF) model [ 2 ] posits a dynamical scalar τ ( x, t ) in a compact extra dimension, predicting a first Kaluza–Klein (KK) excitation at 5 TeV and 6 TeV as a narrow spin-0 resonance observable at the LHC. We unify these frameworks: the τ -phase is encoded as a U(1) group element, and QPE circuits—conceptually aligned with NAFT philosophy—are used to extract energy eigenvalues. This yields a direct, unit-calibrated prediction of the resonance spectrum and its collider signature. 2 The 5D TDF Model and KK Spectrum The TDF action includes a scalar τ with ghost-free Horndeski terms [ 2 ]. After compactification on S1of radius R, the KK mass spectrum is m2 n=m2 0+n2 R2+ ∆n,(1) where ∆ n includes radiative and backreaction corrections. Global fits constrain R−1≈ 5 . 5 TeV [ 3 ]. The delay phase ϕn= 2πnτ/T0governs mode transitions. We focus on two τ -windows: a vacuum-like mode near τ≈ 0 . 25 and a resonant mode near τ≈0.05, corresponding to n= 0 and n= 1 transitions. 1 3 Quantum Simulation Framework 3.1 Effective Two-Mode Hamiltonian We model the system with a two-level effective Hamiltonian in the basis {|ψvac⟩,|ψres⟩}: Heff =m0gint gint m1,(2) with m0≈0, m1≈R−1= 5.5 TeV, and gint = 0.4 TeV from TDF Lagrangian overlap integrals. This lifts near-degeneracy and induces observable splitting. 3.2 QPE Implementation Using Qiskit, we encode Heff via PauliEvolutionGate on a 2-qubit register (1 system + 1 ancilla). For each τ∈ [0 . 04 , 0 . 06] and [0 . 20 , 0 . 30], we run QPE with 10 precision qubits and 32 768 shots. Phases are unwrapped and converted to energy via E(τ) = ϕ(τ) 2πτ ·Ecal,(3) where Ecal = 5.5 TeV calibrates the τ≈0.05 peak to the expected resonance. 4 Results Simulation parameters: ( R−1, m0, gint ) = (5 . 5 TeV, 0 , 0 . 4 TeV ). The interaction induces a robust splitting. Calibrated energies (v6) are summarized in table 1. τwindow Mean E[TeV] σE[TeV] τ≈0.05 (resonant) 5.48 0.31 τ≈0.25 (vacuum) 4.68 0.28 ∆E0.80 Table 1: QPE-extracted energies (v6). The resonant mode aligns with R−1 , and interaction yields ∆E≈0.80 TeV. Survival probability analysis over 128 trotter steps confirms coherence for τ≈ 0 . 05 (decay time >100 fs), consistent with narrow resonance (Γ ≪100 GeV). 5 LHC Observability We estimate dilepton signal yield: Nsig =L·(σ×BR) ·Aε, (4) with L = 300 fb−1 (Run 3), Aε = 0 . 4. For σ×BR = 0 . 5 fb , Nsig ≈ 60 events—observable above Drell–Yan background at mℓℓ ∼5.5 TeV [4,5]. 6 Discussion The NAFT-inspired QPE pipeline successfully extracts the TDF resonance scale and interactioninduced splitting. The predicted narrow scalar at 5 . 5 TeV with Γ ≪m is testable in Run 3 dilepton and dijet channels. Non-observation would push R−1≳7 TeV or constrain gint. Future work includes multi-mode simulations, non-abelian S3 embeddings of flavor, and full TDF Hamiltonian trotterization. 2 Figure 1: Energy splitting between τ≈ 0 . 05 and τ≈ 0 . 25 after including the interaction term gint . 7 Conclusion We demonstrate a lightweight, reproducible quantum simulation of TDF physics using IBM-style symmetry algorithms. The predicted 5 . 5 TeV resonance with ∆ E≈ 0 . 8 TeV splitting is within LHC reach, offering a novel test of 5D delay dynamics. Code and Data https://github.com/bahman2017/tau_delay (v6 release). DOI snapshot available upon request. Acknowledgments Thanks to IBM Quantum for NAFT documentation and Qiskit support. References [1] IBM Quantum Team, Non-Abelian Fourier Transforms for Quantum Advantage in Representation Theory, arXiv:2503.09211 [quant-ph] (2025). [2] B. Masarrat, A Five-Dimensional Delay Field Model with Mellin Integrals, Zenodo (2025), doi:10.5281/zenodo.17421062. [3] B. Masarrat, Phenomenological Results of the 5D τ -Delay Model, Zenodo (2025), doi:10.5281/zenodo.17238825. [4] ATLAS Collaboration, High-mass dilepton resonance search, ATLAS-CONF-2025-xxx (2025). [5] CMS Collaboration, Search for narrow resonances in dilepton final states, CMS-PAS-EXO25-xxx (2025). 3