Triangulated Relativistic Quantum Computation (TRQC) Proof-of-Concept

Supplementary Material: Triangulated Relativistic Quantum Computation (TRQC) Proof-of-Concept Javier Villalba-D´ıez1* and Joaqu´ın Ordieres-Mer´e2 1*Fakult¨at Wirtschaft, Hochschule Heilbronn, Heilbronn, 74081, Germany. 2Escuela T´ecnica Superior de Ingenieros Industriales, Universidad Polit´ecnica de Madrid, Madrid, 28006, Spain. Abstract This Supplementary Material provides the numerical Proof-of-Concept (PoC) accompanying the manuscript, Triangulated Relativistic Quantum Computation: A Curvature-Modulated Unification of Quantum and Relativistic Computing. The PoC confirms the consistency and operational semantics of the TRQC framework by simulating transport across representative graph families and verifying the core theoretical invariants, including causal order independence, remeshing robustness, and the expected O(4N)density-matrix scaling. We present detailed results for transport metrics and address the reviewer’s specific request for peak-over-steps arrival data to mitigate the effects of detector overshoot. 1 Code and Data Availability 1.1 Source Code and Replication The complete Python code for the numerical Proof-of-Concept (PoC) is implemented as a self-contained Jupyter Notebook (proof of concept.ipynb). All simulations rely on the PennyLane quantum programming framework (using the default.mixed device for density-matrix evolution) and standard scientific libraries (numpy,scipy, networkx). The code is publicly available and archived for persistence and reproducibility: •GitHub Repository: https://github.com/Ind50-UPM/2025 TRQC •Archival DOI: 10.5281/zenodo.17625496. 1 All figures, raw timeline data (.npy), and CSV summaries necessary for replication are generated upon execution of the notebook. 2 Design and Methodology of the Proof-of-Concept (PoC) The PoC implements the curvature-guided and spacelike planned evolution described in Section 3 of the main manuscript. 2.1 Parameter Space The simulations aggregate results over multiple folds (random graph realizations or random latent embeddings) across a fixed parameter space to test the framework across varying topologies and sizes (Tab. 1). Table 1 PoC Simulation Parameters and Metrics Parameter/Metric Values Used Purpose Graph Families sphere, geometric2d, ER, scalefree Test closed, boundary, homogeneous, and hub-dominated to... Network Size (N){6,8,10,12}Verify O(4N) scaling and sizedependent transport. Spacelike Rounds (R){2,3}Compare short vs. extended propagation schedules. Folds per (N, R, Family) 5 for N≤10; 3 for N= 12 Stabilize statistical means (adjusted at N= 12 for memory) Final-step Arrival maxdP(final step, d) Conservative metric on fixed early-hop detectors. Peak-over-steps Arrival maxt,d P(step t, d) Captures true transport success (mitigates overshoot). Participation Ratio (PR) 1/Pp2 iMeasures spatial delocalization (wavepacket spreading). 2.2 TRQC Implementation Details •Curvature Estimation: Intrinsic Gaussian curvature (Kv) is derived from vertex angle deficits in latent triangulation, consistent with the discrete Gauss-Bonnet identity (Theorem 3.2, Proposition 3.1). This field is slice-wise normalized and used to define the edge curvature average κe(Eq. 4). •Dynamics: Local evolution is governed by the GKSL generator modulated by curvature (Definition 3.4), ensuring that κemodulates the rates of phase damping (γϕ) and amplitude damping (γamp) while guaranteeing CPTP well-posedness (Proposition 3.4). •Causal Scheduling: Transport across the slice is implemented via a sequence of maximal matchings (spacelike SWAP layers), ensuring order independence and causal factorization within the step (Theorem 3.7). 2 3 Confirmation of Theoretical Invariants The numerical results confirm the mathematical guaranties of the TRQC framework. 3.1 Algebraic Invariance Checks The PoC includes specific diagnostics to test structural stability: •Order Independence (Theorem 3.7): The simulation checks for a non-trivial step where two pairs of commuting SWAPs/channels are applied. Applying the step in reverse order yields a final state difference of ∥∆ρ∥F=0.000e+00, confirming that the global channel is independent of the ordering within the slice when the operators act on disjoint tensor factors. •Remeshing Robustness (Proposition 3.1): Curvature fields computed from a rotated and slightly jittered embedding are compared. The resulting final states ρ differ by a Frobenius norm of ∥ρ0−ρ1∥F=7.514e-05, confirming the robustness of the dynamics to small, physically realistic perturbations in the latent coordinates. •Runtime Scaling: The wall-clock runtime (tseconds) as a function of Nconfirms the expected growth ∝4Ntypical of density-matrix simulations on Nqubits, without unexpected complexity anomalies. 3.2 Curvature-Driven Transport The statistical summary (Tab. 2) illustrates the basic hypothesis that curvature guides transport and delocalization. Table 2 Summary of Best-Performing Simulation Runs by Arrival Mean (± Std) N Rounds Family arrivalmean PRmean tmean(s) 6 3 sphere 0.5719 ±0.3209 1.9864 0.20 8 3 scalefree 0.1596 ±0.3305 1.5457 1.19 10 3 er 0.1657 ±0.3704 1.5218 18.69 12 2 geometric2d 0.2592 ±0.4489 1.6575 390.04 •The highest success in Small N:**sphere** (closed manifold, highly symmetric) with rounds R= 3 achieves the highest overall final-step arrival, consistent with coherent advancement when the path length is short. •Detector Overshoot: At N= 12 with R= 3, the final-step arrival for all families collapses to ≈0.0, despite many runs having PRmean ≈2 (widespread wavepacket). This confirms that the excitation overshoots the conservative earlyhop detector set, a phenomenon visually confirmed by the Peak-over-steps Arrival data logged in the accompanying figures. 3 4 Addressing the Key Suggestion: Peak-over-Steps Arrival The PoC collects the complete history of the probability of detection (logged as arrival timeline.npy) to provide the most informative metric of the capability of transport. 4.1 New Metrics and Conclusion •The final step metric is Conservative: The conservative metric (arrival) is necessary for the initial metric in the paper, but its limitation is exposed at high N/R. •Peak-over-steps Metric: The tracked timeline explicitly captures the **maximum arrival probability over all intermediate spacelike steps** and serves as the key metric to report true transport success. Supplementary figures (e.g., arrival vs steps ...pdf) visually confirm that the peak often occurs far from the final step, validating the need for this alternative metric to truly assess propagation capability in TRQC. 4