This paper explores the application of adaptive geometric topology to quantum hyperbolic computation (QHC). We propose a novel framework for QHC that leverages quantum properties to enhance efficiency and security. The core of this approach lies in the dynamic validation of geometric topology structures during computation, enabling a more robust and adaptive system. We detail the algorithm's methodology, the rationale behind the chosen design choices, and the potential advantages over existing QHC methods. The paper concludes with a discussion of the challenges and future directions for this research.
Gauge-Higgs theory in flat spacetime provides a prime theoretical model of multiple phases of matter with distinct entanglement structures. More recently, there has been interest in hyperbolic space and the influence its extensive boundaries have on these structures. Entanglement transitions between non-equilibrium phases, such as topological phases, can be realized by the competition of non-commutating measurements. Here we study the entanglement phases of gauge-Higgs models on hyperbolic lattices described by measurement-only dynamics. We chart 2D phase diagrams by employing topological entanglement entropy and introducing a generalized measurement-induced entanglement tailored for hyperbolic Higgs phases. Utilizing duality, we find a phase absent in Euclidean space, wherein the bulk lattice hosts separated clusters of symmetry-protected topological order. Transitions between this "archipelago" phase and neighboring topological and trivial phases feature a critical exponent
δ belonging to a mean-field universality class. Our essential tool is a 3D resource state inspired by measurement-based quantum computation that simulates the spacetime evolution of the 2D quantum state with a third direction of entanglement. This allows us to analyze spacetime "sponges" of entanglement, which map to classical percolation theory and allow us to infer values of critical points on our phase diagrams' boundaries.