Computation has historically been framed in Newtonian terms: time and space as separate, absolute measures of algorithmic cost. Yet modern algorithms—from randomized heuristics to quantum circuits—demand a relativistic view where time, space, energy, entropy, and coherence form a unified manifold. This work introduces computational relativity: a geometric theory of algorithms built on spacetime geodesics, entropy trade-offs, and quantum coherence dynamics. We show how classical complexity results like the Hopcroft-Paul-Valiant theorem and recent improvements by Williams emerge naturally as specific geodesic types in this framework. The theory extends through thermodynamic principles to encompass stochastic algorithms and energy consumption, then to quantum coherence for quantum computing applications. This progression motivates living algorithms—self-monitoring systems that dynamically optimize their computational trajectories in real-time. We demonstrate applications across machine learning, quantum computing, robotics, and optimization, concluding with a comprehensive algorithm compendium that classifies computational methods by their geometric properties and resource trade-offs.
Computation has historically been framed in Newtonian terms: time and space as separate, absolute measures of algorithmic cost. Yet modern algorithms—from randomized heuristics to quantum circuits—demand a relativistic view where time, space, energy, entropy, and coherence form a unified manifold. This work introduces computational relativity: a geometric theory of algorithms built on spacetime geodesics, entropy trade-offs, and quantum coherence dynamics. We show how classical complexity results like the Hopcroft-Paul-Valiant theorem and recent improvements by Williams emerge naturally as specific geodesic types in this framework. The theory extends through thermodynamic principles to encompass stochastic algorithms and energy consumption, then to quantum coherence for quantum computing applications. This progression motivates living algorithms—self-monitoring systems that dynamically optimize their computational trajectories in real-time. We demonstrate applications across machine learning, quantum computing, robotics, and optimization, concluding with a comprehensive algorithm compendium that classifies computational methods by their geometric properties and resource trade-offs.
The geometric theory of computational spacetime models algorithmic efficiency as geodesics in a five-dimensional manifold (S,T,H,E,C) spanning space, time, memory hierarchy, energy, and quantum coherence. Classical ``wormholes'' - localized reparameterizations that bypass regions of high action - capture many known shortcuts (e.g., spectral sparsifiers, spanners, FFT/NTT, JL embeddings). This paper proposes seven new wormhole classes that leverage quantum--classical hybrids, topological structure, distributed consensus geometry, neuromorphic dynamics, differential privacy, and federated optimization. For each class we provide a formal setup, entry tolls, shortcut statements, structural preconditions, and resource trade-offs in the (S,T,H,E,C) framework. All complexity improvements are stated under explicit assumptions (e.g., QRAM-style access, partial synchrony, low-distortion embeddings, well-conditioned quantum Fisher information, sparsity, and idealized sketching/compression bounds). The analysis yields convergence conditions, approximation/stability bounds, and a comparative matrix to guide algorithm design in heterogeneous compute environments.
We establish a unified framework for understanding the fundamental physical limits of computation by developing a geometric theory of resource trade-offs. Our approach treats computation as a physical process constrained by thermodynamics, quantum mechanics, and relativity, leading to a multidimensional capacity region in resource space. We prove that any algorithm achieving error ε on problem class P must satisfy joint constraints across five dimensions: space S, time T, bandwidth H, energy E, and coherence C. The capacity region is characterized by fundamental physical constants: Landauer’s principle (kBT ln 2 per bit erasure), the Margolus-Levitin quantum speed limit (2E/(π~) operations per second), the Bekenstein bound (maximum information density), and statistical sampling limits. We provide explicit constructions showing these bounds are achievable within constant factors for canonical problems including matrix multiplication, covariance estimation, and Gaussian process inference. This framework unifies classical complexity theory with quantum information and thermodynamics, providing the first general theory of multi-resource computational limits with applications to algorithm design, hardware optimization, and fundamental physics.
The third law of thermodynamics forbids cooling a physical system to absolute zero in a finite number of operational steps. Although this unattainability principle has been quantified for specific state-to-state transitions, a universal, dynamics-independent bound for implementing a state-agnostic reset map remains elusive. In this work, we unveil the fundamental limits of physical map implementation by deriving a trade-off relation based on geometric complexity. By analyzing continuous paths of maps on a geometric manifold, we prove that the geometric complexity of any classical stochastic map or quantum channel is bounded from below by its execution error. As a consequence, we show that achieving zero error in a state-reset operation requires a divergent geometric complexity -- a unified measure that naturally incorporates disparate physical resources, including infinite time, energetic cost, or control bandwidth. This unattainability principle holds universally across both classical and quantum regimes, establishing a strict geometric limit on the physical realization of reset operations in thermodynamic control and quantum computation.
Abstract For over a century, theoretical physics has treated General Relativity and Quantum Mechanics as fundamentally incompatible: one posits a continuous, deterministic spacetime manifold; the other, a discrete, probabilistic Hilbert space. The Algorithmic Reality Model (ARM) resolves this schism by identifying a single computational mechanism—the Zeno Threshold (Zα)—as the universal bound governing both quantum collapse and spacetime geometry. ARM treats physical reality as the macroscopic projection of a finite-bandwidth computational substrate operating on a timeless, pre-geometric state (the Singleton). By enforcing Landauer's principle of irreversible erasure, ARM derives the entirety of the gravitational sector from seven pre-geometric, quantum informational axioms. Key Derivations & Unifications Emergent Spacetime: Spatial geometry emerges from the Quantum Fisher Information Metric (QFIM) of the observer's state space, while time emerges from the Tomita-Takesaki modular flow. The Lorentzian signature (-,+,+,+) is algebraically forced by the compact/non-compact structure of their isometry generators. The Holographic Scalar (The 3→1 Unification): Applying the Nyquist-Shannon sampling theorem to the holographic boundary yields the exact Bekenstein-Hawking factor of 1/4 as a universal 2D anti-aliasing limit. This simultaneously fixes the area-law coefficient and structurally derives Newton's constant as G = c3ξmin2 / ℏ. Systematic Boundary Deviations: The framework quantitatively resolves macroscopic anomalies as computational resolution limits. Dark energy emerges as the minimum Landauer maintenance cost of the cosmological event horizon, and the MOND acceleration scale (a0) emerges as a thermodynamic noise floor, algebraically locked by the structural identity a02 = (8πG/3)ρΛ. Thermodynamic State Truncation: Wavefunction collapse, the black hole information paradox, and EPR non-locality are rigorously unified as manifestations of the computational substrate's finite Landauer memory budget. Computational Computability Because ARM is built on discrete informational axioms rather than continuous manifolds, the entire framework is natively computable. The structural derivations in this paper—including emergent geodesic lensing, the Bekenstein-Hawking threshold, and the MOND noise floor—have been explicitly verified via sparse Hermitian lattice simulations. The complete open-source computational validation suite is available at: github.com/srdrymn/algorithmic-reality-model-validations This manuscript represents the master synthesis and culmination of the Algorithmic Theory of Reality (ATR) foundational framework.
We introduce a unified physical framework for algorithmic tradeoffs based on Space (S), Time (T), Energy (E), and Entropy (H) --- abbreviated as STEH. This framework integrates classical and quantum perspectives, and yields rigorous lower bounds on computational processes by linking memory, runtime, thermodynamic cost, and uncertainty. We derive a set of inequalities governing the achievable resource combinations and illustrate how they sharpen classical results such as Landauer's principle and quantum speed limits. The STEH tradeoff formalism provides a systematic approach to study computation as a constrained physical process, highlighting resource complementarity and impossibility results that shape the limits of algorithms.
We propose a fundamentally new paradigm for quantum computation based on the intrinsic geometry of quantum state space. Rather than specifying computation as a sequence of discrete gates applied over external time, Geometric Quantum Computing (GQC) defines computation as geodesic evolution through Hilbert space, where the computational 'time' emerges from the Fubini-Study metric itself. This framework inverts the standard relationship between quantum dynamics and time: instead of viewing decoherence as degradation of quantum information over time, we recognize that decoherence is time—the environment's information processing. We derive the theoretical foundations of GQC, demonstrate that it naturally circumvents barren plateaus by operating directly in state space rather than parameter space, and propose experimental protocols for implementation. The framework suggests that optimal quantum computation is achieved not by fighting decoherence but by engineering the effective geometry of state space to make desired computations correspond to natural geodesic flow.