Triangulated relativistic quantum computation: a curvature-modulated unification of quantum and relativistic computing

INSTITUTE FOR QUANTUM STUDIES CHAPMAN UNIVERSITY Quantum Stud.: Math. Found. (2026) 13:2 https://doi.org/10.1007/s40509-025-00376-5 REGULAR PAPER Triangulated relativistic quantum computation: a curvaturemodulated unification of quantum and relativistic computing Javier Villalba-Díez ·Joaquín Ordieres-Meré Received: 9 October 2025 / Accepted: 7 December 2025 © The Author(s) 2025 Abstract 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 nosignaling 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. J. Villalba-Díez Fakultät Wirtschaft, Hochschule Heilbronn, Max-Planck-Str.39, 74081 Heilbronn, Baden-Württemberg, Germany J. Villalba-Díez (B) Department of Mechanical Engineering, Universidad de La Rioja, Edificio Departamental, c/ San José de Calasanz, 31, 26004 Logroño, La Rioja, Spain e-mail: javier[email protected] J. Ordieres-Meré Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, C/ José Gutierrez Abascal, 2, 28006 Madrid, Madrid, Spain 0123456789().: V,-vol 123 2 Page 2 of 21 J. Villalba-Díez, J. Ordieres-Meré Keywords Relativistic quantum computation ·Discrete Gaussian curvature ·Lindblad semigroups ·Causal triangulations ·Triangulation invariance ·No signaling ·Quantum speed limits 1 Introduction Quantum information science has matured into a discipline where abstract mathematical structure and practical engineering demands coevolve [1–3]. On the one hand, the standard circuit and Hamiltonian models provide the operational backbone of quantum computation; in circuit-to-Hamiltonian constructions, a dedicated clock register enforces a global ordering of gates, with local-clock variants also studied [4]. On the other hand, the physical carriers of quantum information—photons, trapped ions, superconducting circuits, cold atoms, and spin defects— are increasingly deployed in settings where geometry and relativity cannot be ignored: optical links between moving satellites [5], quantum repeaters on aircraft and high-altitude platforms [6], photonic and excitonic devices patterned on curved membranes [7], and networked processors whose clocks tick at different proper rates [8,9]. Throughout this work, the local Hilbert spaces attached to nodes are finite-dimensional (Assumption 1); infinite-dimensional channels are treated only under fixed finite truncations. These trends expose a conceptual gap. Today’s standard models handle unitary gates, noise, and scheduling, but they do not natively encode how intrinsic curvature, holonomy, and causal structure shape the channels by which quantum states evolve and information flows. This paper addresses that gap by developing Triangulated Relativistic Quantum Computation (TRQC), a framework that provides a curvature-modulated causal channel semantics compatible with relativistic constraints [10] and elevates intrinsic curvature to a first-class control parameter. The core idea is direct: represent spacetime as a causally oriented simplicial complex and compute intrinsic Gaussian curvature on spacelike slices via vertex angle deficits. Use that curvature to modulate both coherent and incoherent components of the local generators, Hamiltonians and Lindblad operators [11], while enforcing completely positive, trace-preserving (CPTP) [12] evolution and no-signaling across spacelike regions. This approach yields a mathematically controlled calculus with discrete analogs of foundational geometric identities (Gauss–Bonnet) [13] and with invariances that are essential for robust modeling on moving or re-meshed geometries. The motivation is twofold, conceptual and practical. Conceptually, quantum computation and communication are constrained by relativity [14]: signals propagate within light cones; operations may be only partially ordered; clocks are local; and, in gravitational fields or accelerating frames, proper time differs across nodes. Practically, emerging architectures operate on the move—satellite-based quantum key distribution (QKD) [15], drone-mounted quantum sensors [16], and inter-node links subject to Doppler and gravitational shifts [17]. In condensed-matter and photonic platforms, curved or hyperbolic embeddings are no longer esoteric [18]: patterned waveguides, curved nanomembranes [19], and synthetic gauge fields [20] realize effective curvature and holonomy that alter dispersion, interference, and localization. A framework that treats curvature and causal order as fundamental, rather than as afterthoughts, is thus timely and, we argue, necessary. Despite progress across several adjacent areas, a unified, channel-level treatment has been missing. Continuum models of quantum particles constrained to curved surfaces capture geometric potentials and curvature-induced bound states, but they do not supply a CPTP, discretized semantics that composes cleanly under partial orders or remeshing. Graph-theoretic curvatures offer valuable surrogates when smooth geometry is absent, yet they are rarely coupled to physically motivated Lindblad noise while preserving complete-positivity. Numerical solvers on curved domains can approximate transport and spectra, but they lack causal factorization guarantees and triangulation invariance. Most importantly, a few approaches treat curvature as a control field that modulates not only phases and couplings, but also noise rates—a key ingredient for realistic performance envelopes in devices where geometry impacts dephasing, loss, and cross-talk. TRQC fills this gap with five design pillars. First, causal triangulations: spacetime is represented by an oriented simplicial complex with a time labeling that induces a partial order on events. Computations proceed by composing local channels along edges, respecting this partial order. Second, intrinsic curvature from angle deficits: curvature on a spacelike slice is computed from vertex angle deficits on a latent triangulation, using areas that are invariant under 123 Triangulated relativistic quantum computation Page 3 of 21 2 global orthogonal transformations of the latent embedding. Crucially, triangles themselves are flat in Euclidean space; curvature is encoded in the deficit around a vertex, ensuring that the estimator is intrinsic and gaugeinvariant. Third, curvature-modulated Lindbladians: local generators depend on a scalar curvature value averaged over the relevant neighborhood. Both Hamiltonian terms and dissipators receive curvature-dependent corrections or rates, guaranteeing CPTP evolution for all curvature values through the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) structure [21]. Fourth, causal factorization and no-signaling: channels applied on spacelike-separated regions commute and tensor-factorize, making the global map independent of within-slice orderings and forbidding signaling outside the light cone. Fifth, algebraic invariance under commuting refinements: under commuting-locality conditions, and in a Lie–Trotter refinement limit, retriangulations that preserve the per-cell generators leave the global channel unchanged. This is an algebraic (non-topological) invariance that enables remeshing without altering the channel semantics. The mathematical content supporting these pillars is developed and proved within the paper. We establish O(d)-invariance of the curvature estimator (with an explicit scale convention) under global orthogonal transforms of the latent embedding; we prove well-posedness and norm-continuity of curvature–modulated semigroups; we formalize causal factorization, monoidality, and no-signaling for spacelike-separated subcomputations; we show triangulation invariance under commuting/locality hypotheses and convergence to a triangulation-independent limit under refinement; we present a discrete Gauss–Bonnet identity validating the curvature construction on closed slices, and we derive quantum speed limits with explicit curvature dependence in the unitary case, together with a Lipschitztype perturbation bound for general Lindbladians. Two limiting regimes anchor the framework: in the flat limit (zero curvature), TRQC reduces to standard quantum circuits; in an entanglement-breaking limit with commuting maps on disjoint factors, it reduces to a classical relativistic computation governed by a causal partial order. The practical implications are immediate. In relativistic quantum networking, TRQC provides a scheduling semantics aligned with causal order and proper-time clocks, while geometry-aware noise models inform routing, synchronization, and key rates. In analog quantum simulation and topological photonics, curvature and holonomy enter both the coherent and dissipative parts of the dynamics, offering programmable control of phase interference and transport on curved or hyperbolic lattices. For error correction on curved codes, curvature modulates syndrome graph expansion and can be injected into decoders and noise models. For quantum transport on curved or fractal nanostructures, curvature-driven dephasing extends beyond purely Hamiltonian treatments, enabling predictions for mobility edges and anomalous diffusion that depend on geometric features. The central research question is: Can we construct a rigorously defined, curvature-aware, causally composable, and triangulation-invariant channel calculus for quantum information processing in relativistic and curved settings? Our hypothesis is that vertex angle-deficit curvature on spacelike triangulations, used to modulate Lindbladian generators, yields (a) CPTP, well-posed semigroups continuous in curvature; (b) causal factorization and no-signaling across spacelike regions; (c) invariance of the global channel to within-slice orderings and to mesh refinements under commuting/locality or Trotter conditions; and (d) physically meaningful complexity bounds and speed limits that expose explicit curvature dependence. This paper makes the following contributions: •A curvature-modulated Lindbladian calculus on causal triangulations, with proofs of O(d)-invariance (with an explicit scale convention), well-posedness, and continuity. •Theorems establishing causal factorization, order-independence, and no-signaling for spacelike-separated subcomputations. •Algebraic invariance under commuting refinements when per-cell generators are preserved, and a triangulationindependent limit under Lie–Trotter refinement. •A discrete Gauss–Bonnet identity on closed slices, validating the intrinsic curvature estimator. •Curvature-dependent quantum speed limits (unitary case) and a contractive Duhamel-type perturbation bound for general Lindbladians. •Recoveries of standard limits: flat-space quantum circuits and classical relativistic computation in the entanglement-breaking regime. 123 2 Page 4 of 21 J. Villalba-Díez, J. Ordieres-Meré The remainder of the manuscript is organized to balance formal development and practical guidance. Section 2 surveys the mathematical and physical context that motivates TRQC and clarifies the precise gap it fills. Section 3 presents the full mathematical framework: causal triangulations and slice Hilbert spaces; the latent, orthogonally gauged curvature estimator; curvature–modulated generators; and the proofs of our main structural theorems, followed by implementation details for curvature evaluation, causal Trotterization, and remeshing. Section4interprets the theoretical results in light of realistic architectures and identifies strengths, limitations, and open questions. Section5synthesizes the main insights and outlines concrete paths for extending TRQC to adaptive control, graphcurvature surrogates on fractals, and hardware-aware compilers that treat geometry and proper time as optimization variables. Assumption 1 (Standing scope: finite dimension and bounded generators) Throughout the paper, all local Hilbert spaces Hvare finite-dimensional, and all superoperators acting on B(Hv)are bounded. Consequently, GKSL generators generate uniformly continuous semigroups on the trace class, CPTP maps are contractions in the trace norm, and the diamond norm is well defined and finite. Infinite-dimensional channels (e.g., bosonic modes) are out of scope unless an explicit finite-dimensional truncation is fixed throughout; our continuity and Lipschitz bounds rely on boundedness. As a guiding philosophy, the framework is designed not only to be mathematically sound but also to feel natural to practitioners: curvature and causal order are built in from the outset; complete-positivity and no-signaling are guaranteed by construction; and triangulation invariance makes mesh updates and mobility safe. In our experience, such alignment between theory and practice is essential for progress whenever quantum devices leave the laboratory bench and enter the richer, curved, and time-dilated world in which they will ultimately operate. 2 Background and related work The operational constraints of relativity force any physically faithful model of distributed quantum information processing to abandon the fiction of a single global clock. Instead, events are partially ordered by causal reachability, and local clocks advance according to proper time. In such settings, computation is naturally described by causal networks in which nodes execute local transformations while respecting light-cone constraints. Classical formulations have long embraced partial orders for asynchronous systems [22]; in the quantum regime, the corresponding objects are CPTP maps [23] arranged along causal links and composed according to a triangulated cobordism from an initial spacelike antichain to a final one [24]. The TRQC viewpoint instantiates this semantics on an oriented simplicial complex (T,t), where tinduces the causal partial order on vertices and edges [25,26]. Spacelike layers are maximal antichains; within each layer, transformations commute on disjoint tensor factors, yielding order-independence [27]. This categorical (monoidal) structure of channel composition is a structural backbone of TRQC and generalizes conventional circuit models to relativistic schedules [28]. Open quantum dynamics on finite-dimensional systems is generated, in the Markovian regime, by GKSL operators [29]. The associated semigroups etLare CPTP for all t≥0[30], are stable under composition [12], and admit Trotter-type factorization when split into summands [31]. These properties make GKSL dynamics the natural substrate for a channel-level calculus that must be both physically consistent (positivity, trace preservation, and no-signaling under spacelike separation) and composable across a partial order. TRQC adopts GKSL generators at the level of edges in the triangulation, with the distinctive twist that the Hamiltonian and the rates are modulated by intrinsic curvature computed on spacelike slices [32]. The Lie–Trotter and Duhamel expansions provide quantitative control of splitting and perturbation errors [33]; curvature continuity of the maps follows from bounded-perturbation theory [34,35]. Intrinsic curvature on a triangulated surface is captured by vertex angle deficits: for a vertex v, the sum of interior angles around vfalls short of 2πby an amount δ(v), and the discrete Gaussian curvature density is K(v) =δ(v)/Av for a dual area Av>0[36]. This Regge-style discretization is intrinsic to the induced piecewise-Euclidean metric 123 Triangulated relativistic quantum computation Page 5 of 21 2 and satisfies a discrete Gauss–Bonnet identity on closed meshes; in our implementation, the metric is induced by a latent embedding, but a global embedding is not required in principle vK(v)Av=vδ(v) =2πχ()[37]. TRQC leverages these facts in two ways. First, it uses an orthogonally gauged latent embedding to compute K(v): latent coordinates zv∈Rddetermine a geometric triangulation and angles, but all physically relevant quantities are invariant under z→ zR for R∈O(d)=Rd×dRR=Id[38]. Second, it promotes holonomy to a programmable resource: link unitaries Uij implement parallel transport, and plaquette Wilson loops W=e∈∂ Ueencode curvature-like phases that integrate seamlessly into coherent couplings [39]. Several mature lines of work inform TRQC and motivate its synthesis: •Quantum dynamics on curved substrates [40]. Continuum models of particles constrained to curved surfaces predict curvature-induced potentials and bound states, while photonic and polaritonic platforms realize synthetic curvature and gauge fields. These approaches emphasize coherent Hamiltonian effects but rarely address CPTP, causally factorized open-system dynamics at the channel level. •Graph and hyperbolic lattices [41]. Hyperbolic tilings and negatively curved graphs host unusual dispersion and topological features; tensor-network models connect such tilings to holographic encodings. Existing treatments operate coherently or at the level of abstract networks, not as curvature-modulated channel calculi with remeshing invariance. •Finite-element and discrete exterior calculus [42]. Finite-element and discrete exterior calculus techniques approximate Schrödinger and Poisson equations on nonflat geometries and inform conductance predictions. These are powerful numerics but do not furnish a CPTP/no-signaling semantics for distributed quantum operations or asynchronous schedules. •Graph-theoretic “curvature” [43]. Ollivier–Ricci and Forman–Ricci curvatures quantify expansion and transport properties in graphs. They provide surrogates when smooth geometry is absent (e.g., fractals), yet they are not typically tied to Lindbladian rate modulation with complete-positivity guarantees. •Relativistic quantum networking and QKD [44]. Space-based and high-mobility architectures face propertime desynchronization and light-cone constraints. Protocol design in these regimes benefits from a causal, order-independent scheduler and from geometry-aware noise budgets. •Quantum error correction on non-Euclidean tilings [45]. Hyperbolic and other curved codes exploit expansion properties for improved distances and decoding; integrating curvature-aware noise models into these codes is a natural next step. What is missing from the preceding ecosystem is a single, end-to-end framework that: 1. Computes an intrinsic, gauge-invariant curvature field on spacelike slices via angle deficits; 2. Injects curvature into both coherent couplings and dissipative rates while preserving CPTP structure and nosignaling for spacelike separation; 3. Composes channels along a causal triangulation with rigorous order-independence within slices and monoidality across slices; 4. Remains triangulation-invariant under commuting-locality and converges to a triangulation-independent limit under refinement (Lie--Trotter); 5. Provides a priori bounds that display explicit curvature dependence (quantum speed limits; Duhamel-type perturbation inequalities). TRQC is designed to close this gap. It packages the geometric, algebraic, and analytic ingredients into a coherent calculus that is compatible with realistic, moving, and curved architectures while retaining the formal guarantees demanded by mathematical foundations. 3 Methodology Let Tbe a finite, oriented, D–dimensional simplicial complex with vertex set Vand edge set E. When defining slice-wise Gaussian curvature, we explicitly use the set F=2(T)of 2-simplices on a 2D spacelike slice T; 123 2 Page 6 of 21 J. Villalba-Díez, J. Ordieres-Meré this does not restrict the ambient dimension Dand is standard in Regge-style discretizations. We write B(H)for the algebra of all linear operators on H;throughout this section and the paper we work with finite-dimensional local Hilbert spaces (cf. Assumption 1), so with d=dim H, one has B(H)∼ =Md(C). We also write D(H)for the set of density operators on H(positive, trace-one). A time labeling is a map t:V→R, such that each oriented edge e=(u→v) ∈Esatisfies t(u)<t(v). The relation uviff there exists a directed path from uto vis a partial order on Vand extends to simplices by inclusion. Definition 3.1 (TRQC spacetime and slices)ATRQC spacetime is a pair (T,t)as above. An antichain is a subset ⊂Vwhose vertices are pairwise incomparable. A spacelike (Cauchy) antichain is a maximal antichain. For a slice , attach a finite-dimensional Hilbert space Hv≃Cdvto every v∈and define the slice space H=v∈Hv. For U⊆, we write HU=v∈UHv. Definition 3.2 (Local channels and paths) For each e=(u→v) ∈E,lete:B(Hu)→B(Hv)be CPTP. For a directed path γ=e1···ekwith ei=(vi−1→vi),setγ=ek◦···◦e1. Atriangulated cobordism W :is a subcomplex whose boundary is ∂W=with the induced orientation; we write E(W)for its internal edges. A within-step (or spacelike) set of edges is a family {eα=(uα→ vα)}⊂E(W), such that: (i) uα= uβand vα= vβfor α= β. (ii) All endpoints are pairwise incomparable under the time labeling tand lie within the same step window, separated by a guard band δm>0, so that no directed path connects any endpoints with t-difference <δ m. (iii) The corresponding channel factors act on disjoint tensor legs, i.e., they are tensored with identities outside Huαand Hvα. These conditions identify spacelike separation with the causal graph induced by tand guarantee commutation within the step. 3.1 Latent curvature estimator and orthogonal gauge Curvature is computed intrinsically on each spacelike slice from angle deficits at vertices of an auxiliary geometric graph built on latent coordinates. This estimator is independent of any specific physical embedding of the degrees of freedom and is invariant under global orthogonal gauges. Fix d∈N.Alatent embedding is a map ϕ:D(H)→Rdassociating with a chosen set of local features (e.g., reduced density matrices, classical labels, or external metadata) a collection of points zv∈Rd,v∈. Write R∈O(d)={R∈Rd×d:RR=Id}for an orthogonal transform acting by zv→ zvR; this is the orthogonal gauge. Let Gbe a geometric (piecewise-Euclidean) triangulation of {zv:v∈}using the ambient Euclidean metric in Rd(e.g., Delaunay). Each triangle is flat, and its interior angles are computed in its own plane via the law of cosines; the vertex angle deficit sums these per-face angles and is purely intrinsic to the piecewise-Euclidean metric. For a triangle ={i,j,k}with vertices zi,zj,zk∈Rd, define interior angles θv, by the Euclidean law of cosines and let Abe its Euclidean area. Let Av>0 be an orthogonally invariant dual area associated with v(e.g., mixed Voronoi/barycentric). Because all distances and angles used by Gare Euclidean in the ambient space, global orthogonal transforms z→ zR preserve edge lengths and per-face angles; consequently, δ(v) and K(v) are O(d)-invariant. Definition 3.3 (Angle deficit and discrete Gaussian curvature)Forv∈,set δ(v) =2π− ∈star(v) θv,,K(v) =δ(v) Av .(1) 123 Triangulated relativistic quantum computation Page 7 of 21 2 Proposition 3.1 (O(d)-invariance of the estimator) For any R ∈O(d), both δ(v) and K (v) are invariant under z→ z R. We state this explicitly as a proposition, since it is invoked repeatedly (e.g., in remeshing checks and Gauss–Bonnet diagnostics). Remark 1 (Scale convention) While δ(v)is scale-invariant, Avscales like length2,soK(v) =δ(v)/Avdepends on the overall scale of z. We adopt the per-slice normalization := |E|−1e∈Ee=1 (rescale zto unit mean edge length), which fixes the units of K. Alternatives are to work with the dimensionless K(v) := δ(v)/(Av/A), or to treat the scale as a physical choice tied to the platform and absorb it into γe,j(κ). Proof Angles are functions of inner products of edge vectors, which are preserved by O(d);Aand any orthogonally defined Avare likewise invariant. Hence, δ(v) and K(v) are unchanged.  For Theorem 3.2, we assume is a closed 2D simplicial complex (triangulated surface), so that each edge belongs to exactly two triangles. Theorem 3.2 (Discrete Gauss–Bonnet on closed slices) If is a closed, oriented 2D simplicial complex (each edge belongs to exactly two faces) and the geometric triangulation Grealizes the same combinatorial triangulation by Euclidean triangles, then  v∈ δ(v) =2πχ() and  v∈ K(v)Av=2π χ(). (2) Proof Each has angle sum π, and hence, vvθv, =π|F|. Therefore  v δ(v) =2π|V|−π|F|=2π(|V|−|E|+|F|)=2π χ(), (3) using 3|F|=2|E|for triangulations without boundary and Euler’s identity. The second equality follows by definition of K(v). Lemma 3.3 (Discrete Gauss–Bonnet with boundary) If has boundary, then  v∈ δ(v) + e⊂∂ θext(e)=2π χ(), where θext(e)are the exterior turning angles along the boundary. Equivalently, vK(v)Av+e⊂∂ θext(e)= 2πχ()for any orthogonally invariant choice of dual areas. Remark 2 On slices with boundary, exterior turning angles appear; see the discrete boundary version stated in the lemma following Theorem 3.2. Remark 3 (Curvature as an external control field) The latent embedding ϕ:D(H)→Rdis introduced to describe how one may construct curvature fields from features of interest. All structural theorems in this paper assume that the curvature values {K(v)}and the derived {κe}are provided as exogenous (state-independent) controls on each step. If one lets ϕdepend on the evolving quantum state and feeds back κe(ρ) into the generator, the evolution becomes nonlinear in ρ; complete-positivity and contractivity then require a separate analysis not undertaken here. 3.2 Curvature-modulated generators and well-posedness For e=(u→v) ∈E(W)whose application is localized near a slice , define a local curvature average κe=1 |N(u)| w∈N(u) K(w), (4) 123 2 Page 8 of 21 J. Villalba-Díez, J. Ordieres-Meré where N(u)⊂is a finite neighborhood. Default: the closed star of uin G, with uniform averaging. Weighted variants (e.g., inverse-area or distance weights with outlier trimming) are also admissible and remain O(d)-invariant. In all cases, Proposition 3.5 applies with the same structure; the Lipschitz constant scales with the chosen weights through the bound on jγ e,jevaluated at the corresponding κe. Definition 3.4 (Curvature-modulated GKSL generator) Fix bounded self-adjoint operators He,0,He,1on Huand noise operators {Le,j}j.Letγe,j:R→[0,∞)be locally bounded and continuous (and differentiable on compact sets when invoking Proposition 3.5) rate functions. For κ∈R, define L(κ) e(ρ) =−i[He,0+κHe,1,ρ]+ j γe,j(κ)Le,jρL† e,j−1 2{L† e,jLe,j,ρ}.(5) Remark 4 (Concrete norm bounds) In finite dimension and for the 1→1 norm, one has adH1→1≤2Hand DL1→1≤2L2. Substituting these in Eq. (6) yields an explicit curvature Lipschitz constant depending only on He,1,Le,j, and bounds on γ e,jover I. Within the step τe,wetakeHe,0,He,1and the rates γe,j(κ) as constant (time-independent). If time dependence within a step is required, replace the exponential by a time-ordered exponential; all statements continue to hold with the usual Grönwall-type bounds. For a proper-time step τe>0, we implement transport via a Stinespring dilation (κ) e(ρu)=TrE eVe(eτeL(κ) e,in (ρu⊗|0 0|Ee)) V† e, where the subscript “in” indicates that L(κ) e,in acts on B(Hu⊗HEe)(the input leg and its ancilla). The isometry Ve:Hu⊗HEe→Hv⊗HE etypes the map from the u-leg to the v-leg. (If dim Hv≥dim Hu, one may equivalently choose a partial isometry Ue:Hu→Hvand take Ee=E e=C.) Remark 5 (Norm conventions and contractivity) All superoperator norms · 1→1are taken on the trace class (B1,· 1); CPTP maps are contractions in · 1and hence have 1→1≤1. The diamond norm · is used in finite dimension and obeys =1 for CPTP . For notational convenience, we set adH(ρ) := [H,ρ]for bounded H, and DL(ρ) := LρL†−1 2{L†L,ρ} for any noise operator L(the standard Lindblad dissipator). Proposition 3.4 (CPTP well-posedness and curvature continuity) For every κ∈R, the typed channel (κ) ein Definition 3.4 is CPTP. If γe,jare locally bounded and continuous in κ, then κ→ (κ) eis norm-continuous on bounded intervals (in fact, locally Lipschitz under the hypotheses of Proposition 3.5). Proof The GKSL form with nonnegative rates generates a uniformly continuous quantum dynamical semigroup; thus, t→ etL(κ) eis CPTP for t≥0. Continuity in κfollows from boundedness of He,1, continuity of γe,j, and the Trotter–Kato perturbation theory for bounded generators.  Proposition 3.5 (Curvature Lipschitz control) Let · 1→1denote the induced trace-norm operator norm. For bounded He,1and rates with bounded derivatives on a compact interval I ⊂R sup κ,κ∈I(κ) e−(κ) e1→1 |κ−κ|≤τe⎛ ⎝adHe,11→1+ j sup ξ∈I γ e,j(ξ) DLe,j1→1⎞ ⎠eτeCI,(6) for a constant CIdepending only on bounds of L(κ) eover κ∈I, where adH(ρ) =[H,ρ]and DL(ρ) =LρL†− 1 2{L†L,ρ}. 123 Triangulated relativistic quantum computation Page 9 of 21 2 The bound in Eq. (6) is state-independent and hence uniform over inputs ρ∈B(Hu)with ρ1≤1, which facilitates a priori step-size selection in curvature-swept simulations. Proof Differentiate eτeL(κ) ein κvia the Duhamel formula and bound by submultiplicativity in · 1→1;integrate from κto κ. 3.3 Global channels, causality, and no-signaling Fix a total order ≺on E(W)extending the partial order induced by t.Theglobal TRQC channel is the ordered product W= −−−→  e∈E(W) (κe) e.(7) With a fixed global typing (Definition 3.5), each edge map is extended to the full slice algebra by identity on untouched legs ιe:B(H)→B(H\{u}⊗Hv), ιe=Id\{u}⊗(κe) e.(8) Because each edge map (κe) eis typed from B(Hu)to B(Hv)via the fixed transport isometry (or Stinespring dilation) in Definition 3.4, the ordered product is type-consistent along the triangulated cobordism. Definition 3.5 (Global typing and dimension compatibility) For each spacelike slice , fix an ordered list of tensor legs H=v∈Hv. An edge e=(u→v) is typed as a CPTP map e:B(Hu)→B(Hv), realized via a finite-dimensional Stinespring/Kraus dilation. Composition along a directed path requires that codomains/domains match on successive legs, and that extensions by identities always act on the fixed complement H\{u}of the current slice typing. We audit trace preservation and complete-positivity via Choi positivity at each step (guaranteed here by the GKSL form). Lemma 3.6 (Tensorial commutation on disjoint factors) If :B(HA)→B(HA)and :B(HB)→B(HB) are CPTP, then ( ⊗IdB)◦(IdA⊗) =(IdA⊗) ◦( ⊗IdB)(9) on B(HA⊗HB). Proof Both sides act as X→ ( ⊗)(X)by associativity of ⊗and functoriality at the channel level.  Theorem 3.7 (Order-independence, monoidality, and causal independence) (i) Wis CPTP and independent of the chosen topological ordering ≺. (ii) For cobordisms W1:01and W2:12,W2◦W1=W2◦W1. (iii) If W =WAWBwith spacelike separation and tensor-disjoint supports at each step (no edge in WAshares a vertex with an edge in WBwithin the same spacelike layer), then W=WA⊗WB. Proof (i) Adjacent swaps in two linear extensions of the partial order exchange incomparable edges whose actions are on disjoint factors in that step and thus commute by Lemma 3.6. (ii) Concatenation of cobordisms corresponds to composition of channels. (iii) Spacelike separation implies actions restricted to tensor factors; iterated use of Lemma 3.6 yields the product form.  123 2 Page 16 of 21 J. Villalba-Díez, J. Ordieres-Meré trap geometry is non-Euclidean or time varying. The causal product formula aligns with digital-analog hybrid control schedules. •Curved/fractal nanostructures and metamaterials. In excitonic or magnonic transport, angle deficits modulate both hopping and dephasing; in fractal geometries, a graph-curvature surrogate can be substituted when smooth embeddings are unavailable, preserving the channel semantics. Limitations and scope. The guarantees of TRQC rest on modeling choices whose scope should be explicit: •Latent embedding and estimator dependence. Although the curvature estimator is orthogonally gauge-invariant, it depends on the latent embedding and neighborhood selection. PCA-based tangent planes and Delaunay triangulations are numerically convenient but can be sensitive to noise and near-degeneracies. In practice, robustification (e.g., angle clamping, aspect-ratio regularization, and outlier trimming) is advisable, and boundary terms must be handled explicitly. •Markovian closure. The GKSL form assumes effective Markovianity. Long-delay optical paths, memory in reservoirs, or feedback can induce non-Markovian effects. A practical remedy is to enlarge the system with delay lines or ancilla edges, so that the enlarged dynamics is Markovian; otherwise, one must resort to time-nonlocal master equations, losing some of TRQC’s simple compositionality. When γe,j(κ) are inserted phenomenologically, one should check compatibility with KMS/detailed-balance if a thermal microscopic derivation is intended. •Calibration of curvature-rate maps. The functions γe,j(κ) encode device physics and environment. Without experimental calibration, they are phenomenological. Identifiability (recovering γe,jfrom observed channels) is a nontrivial inverse problem; priors from materials science or wave optics help, but a systematic Bayesian treatment is still open. •Noncommuting within-slice generators. When generators fail to commute on a spacelike step, first-order Trotterization introduces commutator errors. While refinement controls these, hardware constraints may limit time resolution. Strang or higher-order splittings reduce bias but can be costly; adaptive layerings that minimize noncommutativity are a promising compromise. •Topological changes and defects. Node failures or link blockages induce local surgery on the triangulation. TRQC survives local Pachner moves, but true topological changes (e.g., component splitting/merging) require explicit boundary and interface handling, including proper reinitialization of curvature budgets and holonomy phases. Because the framework comes with invariants, it supports rigorous test suites: •Geometric checks. On closed slices whose geometric triangulation realizes the same combinatorial triangulation, verify vK(v)Av=2πχ(); with boundary include the exterior-angle term. Track compliance with these hypotheses under remeshing. •CPTP and no-signaling audits. Numerically confirm that channels preserve trace and positivity for random inputs; test spacelike factorization by swapping within-slice orderings and comparing reduced states. •Perturbation scalings. Vary curvature by a small δκ and verify that channel deviations scale linearly in tδκ consistent with Duhamel-type bounds. •Convergence under refinement. Plot channel distance between successive refinements; exponential or algebraic decay consistent with commutator norms indicates that the Trotter limit is reached. Several research directions emerge naturally: •Continuum limits. Under what conditions does TRQC converge to a well-defined GKSL evolution on a curved manifold as mesh size and step-size vanish in tandem? Can one quantify the dependence on curvature gradients and holonomy strength? •Inverse geometry. Given process-tomography data across a slice, to what extent can one reconstruct κor a graph-curvature surrogate, and what are fundamental identifiability limits in the presence of noise? •Curvature as a computational resource. Do curvature and holonomy expand or contract complexity classes for distributed tasks (e.g., clock synchronization, consensus, and secure randomness expansion)? Can speed limits with curvature dependence translate into lower bounds on communication or gate counts? 123 Triangulated relativistic quantum computation Page 17 of 21 2 •Fault tolerance on curved substrates. How do curvature-modulated noise rates impact thresholds and decoder performance for homological and LDPC codes on non-Euclidean tilings? Is there an optimal curvature profile for a given code family? •Non-Markovian extensions. Can one preserve causal factorization and a form of triangulation invariance for time-nonlocal master equations, perhaps via auxiliary memory edges or kernel factorizations? •Holonomy control and gauge design. What systematic methods select Uij and plaquette targets to synthesize desired band topology or transport features while respecting device constraints and minimizing dephasing? TRQC elevates curvature, holonomy, and causal structure to programmable resources while holding fast to quantum information’s compositional laws. Its mathematical guarantees—CPTP evolution, no-signaling across spacelike regions, O(d)-invariance (with explicit scale convention) and commuting-refinement invariance under stated hypotheses, and explicit curvature-dependent bounds—translate into actionable practices for modeling and control on moving and curved platforms. The path forward is clear: calibrate curvature-rate maps on real hardware, integrate adaptive meshing and higher-order splittings into compilers, and explore curvature-aware error correction and metrology. If successful, these efforts will turn the geometry of the world from a nuisance into an ally for robust quantum technologies. 5 Conclusion and future work TRQC elevates curvature, holonomy, and causal order to first-class citizens in quantum computation and communication. The framework rests on five pillars: (i) a causal, triangulated scaffold for channel composition; (ii) an intrinsic, orthogonally gauged curvature estimator from vertex angle deficits, backed by discrete Gauss–Bonnet; (iii) curvature-modulated GKSL generators that guarantee CPTP evolution for all curvature values; (iv) causal factorization and order-independence within spacelike steps, implying no-signaling across spacelike regions; and (v) triangulation invariance under commuting-locality together with refinement invariance in a Lie–Trotter limit. Beyond structural results, TRQC supplies curvature-dependent quantum speed limits and contractive perturbation bounds within a curvature-modulated causal channel semantics, providing design-level guidance for schedules and tolerances. A priority is to close the loop between geometry and control. Three complementary thrusts are natural: •Adaptive meshing and scheduling. Online quality metrics (e.g., minimum angle, aspect ratio, and commutator norms within a slice) trigger local retriangulation and re-layering. Order-independence and commuting-locality ensure that such updates do not alter the global channel, while Trotter bounds quantify residual errors when noncommuting terms persist. •Curvature-aware optimal control. Incorporate curvature fields {K(v)}and holonomies {Uij}into objective functionals for gate synthesis, routing, and entanglement distribution. Constraints include CPTP preservation, proper-time budgets, and bounds from curvature-dependent speed limits. •Calibration of curvature-rate maps. Treat γe,j(κ) as control-informed parameters learned from experiments. Bayesian or variational identification loops can balance hardware priors with TRQC’s Lipschitz and positivity constraints to recover rate–curvature laws in situ. Where no smooth manifold exists (fractals, disordered networks), substitute graph-curvature proxies for K: •Ollivier/Forman drivers. Map edgeor vertex-based Ricci-like curvatures to TRQC via Ke←Avg{K(endpoints/ faces)}, preserving channel semantics. Investigate convergence to angle-deficit curvature in random geometric graphs and the impact on transport exponents. •Hybrid surrogates. Combine local embedding-based deficits (when available) with graph curvature elsewhere, yielding robust fields on mixed-quality data without sacrificing gauge invariance where an embedding is trustworthy. 123 2 Page 18 of 21 J. Villalba-Díez, J. Ordieres-Meré •Benchmark phenomena. Predict and verify curvature-controlled crossovers in mean-square displacement, participation ratios, and mobility edges on Sierpi´nski-type and hyperbolic designer lattices, using TRQC’s curvatureaware noise and holonomy controls. Geometry and relativistic timing should be first-class optimization variables in compilers: •Cost models. Extend standard gate-time and error budgets by adding curvature penalties, holonomy targets, and proper-time schedules. Use commutator-based Trotter surrogates as splitting-cost proxies for within-slice noncommutativity. •Scheduling under light-cone constraints. Optimize layerings that minimize spacelike commutators and curvature-induced dephasing while meeting latency and synchronization goals in moving-node networks. •CPTP-preserving numerics. Integrate scaling-and-squaring for GKSL exponentials and exact Kraus realizations where available, ensuring positivity by design during compilation and simulation. Two theoretical frontiers invite careful development: •Memoryful dynamics. Represent delays and reservoirs as explicit ancilla edges and nodes to lift non-Markovian processes into a Markovian TRQC supergraph, retaining causal factorization. Analyze limits in which coarsegraining reproduces effective time-nonlocal kernels while preserving no-signaling. •Continuum limits. Establish rigorous convergence from triangulated TRQC evolutions to GKSL-type dynamics on curved manifolds as mesh size and step-size vanish together. Quantify dependence on curvature magnitude and gradients, and identify counterexamples that require higher-order geometric corrections. Curvature-aware noise models invite domain-specific advances: •QEC on curved codes. Study thresholds and decoder performance for homological and LDPC codes on hyperbolic or mixed-curvature tilings with curvature-modulated error rates. Seek curvature profiles that maximize code capacity under fixed resources. •Distributed metrology. Use ∂H/∂κ as the generator of parameter encoding to derive Fisher-information bounds for gravimetry and inertial sensing with moving networks, accounting for proper-time schedules. •Band topology via holonomy. Design link unitaries and plaquette phases to synthesize target Chern numbers or protected edge modes while minimizing curvature-driven dephasing predicted by TRQC. To facilitate adoption, we advocate an open benchmark suite comprising: canonical meshes with known χ() and closed-form K; randomized mobility traces with reproducible latent embeddings; and reference channels for flat and entanglement-breaking limits. Each artifact should include diagnostics for CPTP preservation, Gauss–Bonnet consistency, refinement convergence, and spacelike factorization. Ethical guidance should accompany releases, noting dual-use implications of geometry-aware scheduling and control in critical infrastructure. By welding intrinsic discrete geometry, relativistic causality, and open-system quantum dynamics into a single calculus, TRQC provides a principled route to modeling and controlling quantum information in the world as it is: curved, moving, and asynchronous. The next phase is inherently collaborative, combining mathematical analysis, compiler engineering, and hardware calibration. In turning curvature and proper time from constraints into resources, we anticipate not only more faithful simulations but also qualitatively new protocols that exploit geometry for robustness, efficiency, and discovery. Acknowledgements The authors want to recognize that this research has been partially supported by the Ministerio de Ciencia e Innovación of Spain (Grant Ref. PID2022-137748OB-C31 funded by MCIN/AEI/10.13039/501100011033) and “ERDF A way of making Europe”. JVD wants to acknowledge funding from the Hochschule Heilbronn and the Dieter Schwarz Stiftung through the program “Innovative Sonderprojekte im Bereich Forschung und Bildung 2025” to the HyQCA Project, which supports strategic, high-excellence research initiatives at the interface of digital technologies, AI, and quantum computing science. During the preparation of this work, the author(s) used Writefull to edit and polish the grammar. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication. JVD would like to dedicate this work to Prof. Juan Ignacio Villalba-Sánchez, esteemed professor of exact mathematics and differential calculus at the Universidad de Alcalá, Spain, beloved uncle, and tireless chess player, whose example of intellectual discipline and curiosity has profoundly shaped this research. 123 Triangulated relativistic quantum computation Page 19 of 21 2 Author contributions JVD designed the theoretical framework, developed the mathematical results, and authored the manuscript. JVD gratefully acknowledges the insightful contributions of Prof. Joaquín Ordieres Meré for his expert guidance. Funding Open Access funding enabled and organized by Projekt DEAL. Data Availability A numerical proof-of-concept (PoC) accompanies this article as Supplementary Material. All details are explained there, including the GitHub public repository [46] as well as the interpretation for the provided folders. It implements the TRQC framework on representative graph families (sphere,geometric2d,Erd[Pleaseinsertintopreamble]s–Rényi,andscale-free)forN∈ {6,8,10,12}and rounds {2,3}, reporting final-step and peak-over-steps arrival probabilities, participation ratios, and runtimes. The PoC confirms the theoretical invariants—complete-positivity, order-independence, remeshing robustness, and curvature-driven transport behavior—and reproduces the expected O(4N)scaling. Declarations Conflict of interest The authors declare that there are no conflict of interest, financial or otherwise, that could be perceived to influence the results or interpretation of this work. No competing financial interests, personal relationships, or affiliations have affected the conduct of the research, the preparation of the manuscript, or its potential publication. All funding sources and institutional supports are fully acknowledged within the manuscript. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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