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Triangulated relativistic quantum computation: a curvature-modulated unification of quantum and relativistic computing
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DOI：10.1007/s40509-025-00376-5.png)
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Abstract
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We introduce Triangulated Relativistic Quantum Computation (TRQC), a mathematically consistent framework that integrates relativistic causal constraints with quantum channel dynamics by promoting intrinsic curvature to a first-class control parameter. Spacetime is modeled as an oriented simplicial complex endowed with a time labeling that induces a causal partial order on events. Quantum degrees of freedom are finite-dimensional and attached to vertices, while local evolution along edges is given by completely positive trace-preserving (CPTP) maps generated by a curvature-modulated Lindbladian. Curvature on spacelike slices is estimated from vertex angle deficits of a latent triangulation-intrinsic to the induced piecewise-Euclidean metric of the chosen embedding; this estimator is O(d)-invariant under global orthogonal transformations of the latent embedding, with an explicit per-slice scale convention. We prove: (i) gauge invariance of the angle-deficit and curvature density; (ii) well-posedness and norm-continuity of curvature-modulated CPTP semigroups; (iii) causal factorization and no-signaling across spacelike-separated subcomputations via order-independence within slices; (iv) triangulation invariance under commuting-locality with preserved per-cell generators, and triangulation-independence in a Lie-Trotter refinement limit; (v) a discrete Gauss-Bonnet identity on closed slices; and (vi) quantum speed limits and Lindbladian perturbation bounds with explicit curvature dependence. In the flat limit, TRQC reduces to standard quantum circuits; in an entanglement-breaking limit, it reduces to classical relativistic computation. We outline algorithms for curvature evaluation on moving meshes, causal scheduling, and remeshing-robust Trotterization, and we sketch applications to relativistic quantum networking, analog simulation on curved/hyperbolic lattices, geometry-aware error correction, and transport on curved or fractal nanostructures. Beyond offering new theoretical guarantees, TRQC provides a practical semantics for designing and simulating quantum information processing in nontrivial geometries and time-dilated settings.
Keywords:
Relativistic quantum computation
Discrete Gaussian curvature
Lindblad semigroups
Causal triangulations
Triangulation invariance
No signaling
Quantum speed limits
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