We propose gTRQC, a curvature—coupled extension of Triangulated Relativistic Quantum Computation, that embeds entanglement—equilibrium principles into finite-dimensional GKSL dynamics on causal triangulations. Each spacelike slice carries matter Hilbert spaces and a vertex curvature surrogate K(v) obtained from 2D angle deficits of an auxiliary triangulation of the slice vertices. For TRQC balls B we define <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S_{\textrm{gen}}(B)=\alpha \,\textrm{Area}^{\textrm{disc}}(\partial B;K_\Sigma )+S(\rho _B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mtext>gen</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>α</mi> <mspace width="0.166667em" /> <msup> <mtext>Area</mtext> <mtext>disc</mtext> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi>B</mi> <mo>;</mo> <msub> <mi>K</mi> <mi mathvariant="normal">Σ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mi>B</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and postulate its stationarity under first-order variations of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\rho _B$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ρ</mi> <mi>B</mi> </msub> </math></EquationSource> </InlineEquation> and K at fixed discrete volume. Using the finite-dimensional entanglement first law, this yields vertex-local linear-response relations linking curvature perturbations to modular-energy responses; for quadratic counterterms they become discrete Poisson/Helmholtz equations. Backreaction is realized by augmenting each slice with finite-dimensional geometry registers whose pointer basis labels curvature levels, and by constructing joint matter—geometry Gorini–Kossakowski–Lindblad–Sudarshan generators that are completely positive trace preserving, causally factorized, and no-signaling. In our explicit models the geometry remains nearly diagonal in this basis, so curvature-label dynamics reduces to a classical continuous-time Markov relaxation while retaining quantum-compatible channel semantics. We provide a variational characterization: for local Laplacian+mass counterterms and given modular-energy source J (with the volume constraint fixing the zero mode), the curvature response is the unique minimizer of a strongly convex slice functional <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {F}_\Sigma (\cdot ;J)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">F</mi> <mi mathvariant="normal">Σ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>;</mo> <mi>J</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and we give a constructive Gorini–Kossakowski–Lindblad–Sudarshan Gibbs-sampler dynamics whose stationary distribution concentrates near this minimizer. At the level of entire histories, we define canonical energy as the second variation of relative entropy, coinciding with quantum Fisher information for coupled perturbations, and derive stability criteria and refinement/triangulation-compatibility results under boundedness and commuting-locality assumptions. Within a stated scope—scalar curvature surrogates and finite-dimensional Markovian generators—gTRQC is positioned not as quantum gravity but as a simulation-ready backreaction architecture relevant, e.g. to non-inertial quantum sensing and gravimetry.

Curvature-coupled triangulated relativistic quantum computation: entanglement equilibrium, geometry registers, and discrete curvature-response relations

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2026年4月4日

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Javier Villalba-Díez,
Claudia Nuber,
Joaquín Ordieres-Meré

摘要

We propose gTRQC, a curvature—coupled extension of Triangulated Relativistic Quantum Computation, that embeds entanglement—equilibrium principles into finite-dimensional GKSL dynamics on causal triangulations. Each spacelike slice carries matter Hilbert spaces and a vertex curvature surrogate K(v) obtained from 2D angle deficits of an auxiliary triangulation of the slice vertices. For TRQC balls B we define $S_{\textrm{gen}}(B)=\alpha \,\textrm{Area}^{\textrm{disc}}(\partial B;K_\Sigma )+S(\rho _B)$ $S_{gen} (B) = α {Area}^{disc} (\partial B; K_{Σ}) + S (ρ_{B})$ and postulate its stationarity under first-order variations of $\rho _B$ $ρ_{B}$ and K at fixed discrete volume. Using the finite-dimensional entanglement first law, this yields vertex-local linear-response relations linking curvature perturbations to modular-energy responses; for quadratic counterterms they become discrete Poisson/Helmholtz equations. Backreaction is realized by augmenting each slice with finite-dimensional geometry registers whose pointer basis labels curvature levels, and by constructing joint matter—geometry Gorini–Kossakowski–Lindblad–Sudarshan generators that are completely positive trace preserving, causally factorized, and no-signaling. In our explicit models the geometry remains nearly diagonal in this basis, so curvature-label dynamics reduces to a classical continuous-time Markov relaxation while retaining quantum-compatible channel semantics. We provide a variational characterization: for local Laplacian+mass counterterms and given modular-energy source J (with the volume constraint fixing the zero mode), the curvature response is the unique minimizer of a strongly convex slice functional $\mathcal {F}_\Sigma (\cdot ;J)$ $F_{Σ} (\cdot; J)$ , and we give a constructive Gorini–Kossakowski–Lindblad–Sudarshan Gibbs-sampler dynamics whose stationary distribution concentrates near this minimizer. At the level of entire histories, we define canonical energy as the second variation of relative entropy, coinciding with quantum Fisher information for coupled perturbations, and derive stability criteria and refinement/triangulation-compatibility results under boundedness and commuting-locality assumptions. Within a stated scope—scalar curvature surrogates and finite-dimensional Markovian generators—gTRQC is positioned not as quantum gravity but as a simulation-ready backreaction architecture relevant, e.g. to non-inertial quantum sensing and gravimetry.