The Physical Impossibility of Quantum Computational Speedup

Abstract

The Physical Impossibility of Quantum Computational Speedup: Two Independent Arguments from Quantum-Geometry Dynamics Quantum computing promises computational advantages that are, in the most celebrated cases, exponential. Shor's algorithm factors integers exponentially faster than the best known classical algorithm; Grover's algorithm searches an unstructured database with a quadratic speedup. These results are mathematically rigorous within their assumed framework. This paper asks not whether the mathematics is correct but whether the physical assumptions underlying it are. I argue that Quantum-Geometry Dynamics (QGD) — an axiomatic framework for physics derived from the discreteness of space and the kinetic nature of matter — establishes two independent foundational preclusions of quantum computational speedup. These are not engineering obstacles or decoherence problems that better technology can overcome. They are consequences of the physical structure of discrete space, and no improvement in qubit isolation, error correction, or fabrication technique can supply a physical resource that does not exist. The first preclusion concerns physical states. In QGD, matter is composed of preons⁺ that propagate through quantum-geometrical space by preonic leaps, each of which is a definite transition from one spatial unit to the next in the direction of the momentum vector. A preon(+) always occupies a definite position. There is no ontological superposition. What quantum mechanics represents as a superposed state is, in QGD, an epistemic description of our ignorance of the system's actual preonic microstate. Since ontological superposition is the physical resource on which quantum parallelism depends, its absence means that no physical system can instantiate the state required for exponential speedup. The second preclusion concerns physical processes. Causal succession in discrete space is categorically and structurally irreversible: every preonic leap has a definite direction, and no physical process can reverse the causal order of preonic states. This is not thermodynamic irreversibility arising from large numbers but a property of every individual preonic transition. Since the quantum circuit model is built on reversible unitary gates, and reversible computation has no physical instantiation in a discrete-space framework, the operational basis of quantum computing is equally precluded. I show how both arguments bite at specific steps in Shor's algorithm and Grover's algorithm. In Shor's algorithm, the first preclusion destroys the state preparation step and the second destroys the quantum Fourier transform. In Grover's algorithm, the first preclusion destroys the uniform superposition and the second destroys the diffusion operator and oracle. Either preclusion alone is sufficient; together they attack quantum speedup at both ends — what goes in and how it is processed. The black hole information paradox, correctly understood within QGD, provides independent confirmation: the paradox dissolves precisely because physical states are always definite and physical evolution is categorically irreversible — the same two features that preclude quantum speedup. The Uniqueness Theorem for QGD further strengthens the claim: if the two-axiom discrete-space structure is the unique viable foundational description of physical reality, these preclusions are not features of one possible framework among many but consequences of the physical structure of the universe. This paper supersedes the earlier paper Quantum Computing Under Quantum-Geometry Dynamics (Zenodo, DOI: 10.5281/zenodo.19522137) and should be read in its place. This paper is part of the Minimal Physically Derivable Theory (MPDT) programme.