This note connects two claims in the Quantumograph program. The Computational Reducibility Theorem shows that a finite unitary quantum graph is, in principle, exactly forecastable from one spectral slice: the state at one time together with the spectrum of the evolution operator. The Fundamental Limit of Quantum Computation (FLQC) adds a physical constraint: any realizable quantum device has a finite phase resolution, so exact spectral data and exact phase gates cannot be implemented with arbitrary precision. Realism, in this setting, is the position between these two statements. The universe is computable in principle, but nature imposes a nonzero floor on the precision with which that computability can be instantiated. The note derives the phase floor, compares it with the phase resolution required by QFT- and Shor-type circuits, and gives a concrete experimental signature on superconducting quantum processors.
This note connects two claims in the Quantumograph program. The Computational Reducibility Theorem shows that a finite unitary quantum graph is, in principle, exactly forecastable from one spectral slice: the state at one time together with the spectrum of the evolution operator. The Fundamental Limit of Quantum Computation (FLQC) adds a physical constraint: any realizable quantum device has a finite phase resolution, so exact spectral data and exact phase gates cannot be implemented with arbitrary precision. Realism, in this setting, is the position between these two statements. The universe is computable in principle, but nature imposes a nonzero floor on the precision with which that computability can be instantiated. The note derives the phase floor, compares it with the phase resolution required by QFT- and Shor-type circuits, and gives a concrete experimental signature on superconducting quantum processors.
A Principled Limitation of Wolfram's Computational Irreducibility for Finite Unitary Quantum Systems. The sufficiency of a single slice to predict the entire evolution. After diagonalization, prediction cost is independent of the time horizon. Time is not a fundamental step-by-step process—it emerges from the spectrum of the evolutionary operator. Therefore, the "future" is not ontologically inaccessible: it is already fully encoded in the spectral structure of the present. Computational reducibility is not a trick involving substitution into a formula. It reflects the fact that in Quantumograph, time and spectrum are the same thing, described in different languages.
Computational Reducibility Theorem.Against S. Wolfram's Principle of Computational Irreducibility:The sufficiency of a single slice to predict the entire evolution.Time is not a fundamental step-by-step process—it emerges from the spectrum of the evolutionary operator. Therefore, the "future" is not ontologically inaccessible: it is already fully encoded in the spectral structure of the present. Computational reducibility is not a trick involving substitution into a formula. It reflects the fact that in Quantumograph, time and spectrum are the same thing, described in different languages.A Blow to Cryptography?Implications and Practical Outlook. The main consequence is not a universal cryptanalytic breakthrough, but a precise separation between principled and operational predictability. Exact microstate prediction remains exponentially costly, whereas coarse-grained observables admit polynomial forecasting. This makes the framework relevant for model reduction, spectral inference, and device-level analysis, while the cryptographic threat landscape remains governed by dedicated quantum algorithms and by the ongoing transition to post-quantum cryptography.
Abstract Wolfram's Principle of Computational Irreducibility asserts that for most non-trivial discrete systems there exists no analytical shortcut: to determine the state at time T, one must simulate all T steps. For finite-dimensional unitary graph systems, the evolution admits an exact spectral representation for a specific class of systems: finite quantum graphs with unitary evolution. Knowledge of the spectral slice - the quantum state |Ψ(t₀)⟩ and the spectrum of the evolution operator Û - fully and exactly determines the state at any moment T without stepping through intermediate times. The theorem is a principled, not a practical result: algorithmic complexity O( ) remains exponential. Practical significance is achieved in the coarse-graining limit at scale ℓ >> a, where the problem reduces to polynomial complexity. Three conditions - finiteness, unitarity, linearity - constitute the exact domain of applicability.
This revision withdraws the claim, present in earlier circulated versions of this preprint, that FLQC renders Shor's algorithm physically unrealisable for RSA-2048 and sets an obstruction threshold at n* ≈ 100 bits (formerly §3.1). That argument rested on an invalid premise: that a single physical rotation gate must directly realise an exponentially small ideal phase angle. This is contradicted by established results in quantum-circuit compilation (Solovay–Kitaev/Ross–Selinger synthesis into a fixed discrete gate set; Coppersmith's approximate quantum Fourier transform) and by an explicit, quantitative treatment of exactly this question already in the peer-reviewed literature (Fowler & Hollenberg, Phys. Rev. A 70, 032329, 2004), which shows that factoring integers of thousands of bits requires no controlled rotation finer than - roughly 28 orders of magnitude coarser than the Δθmin estimate used throughout this work. The corrected argument is given in §3.1 and Part V.. Numerical simulation phase-resolution test protocol with qiskit: https://github.com/SergejMaterov/FLQC-Protocol
This work studies the operational limitations on the physical realization of exact mathematical limits in quantum theory. Focusing on the zero-temperature limit, we analyze the extent to which ground-state preparation can be achieved using quantum operations constrained by finite physical resources. We introduce a general class of physically realizable quantum channels characterized by bounded interaction strength, finite implementation time, and finite control resolution. Within this resource-theoretic setting, we derive bounds on spectral localization using quantum speed limits and variance propagation arguments. These bounds imply that arbitrarily sharp spectral concentration cannot be achieved by finite-resource operations in finite time. As a consequence, the preparation of states arbitrarily close to the exact ground state is subject to resource-dependent limitations. In particular, the achievable trace distance to the ground state is bounded from below by a quantity determined by the available energy, control precision, and operational time. The result is independent of specific coarse-graining models or measurement schemes. Instead, the limitation follows from general dynamical and informational constraints associated with finite resources in quantum operations. This perspective provides a resource-theoretic interpretation of the third law of thermodynamics. Within this framework, the unattainability of absolute zero can be understood not only as a dynamical constraint but also as a consequence of operational bounds on spectral localization. These considerations also imply practical limits for ground-state preparation, quantum annealing protocols, and low-temperature quantum metrology, where finite resources impose irreducible accuracy floors. The analysis is interpreted within the broader program of Finite-Closure Geometry (FCG), which distinguishes between mathematical closure operations (such as exact limits) and the physically realizable projections produced by finite-resource dynamics. In this view, exact limits function as ideal mathematical constructs that guide theoretical descriptions but cannot be implemented exactly by finite physical processes. Related work This work complements and extends the results of Operational Non-Representability of the Zero-Temperature Limit under Finite Energy Resolution (Menéndez, 2025). The earlier paper demonstrates non-representability through explicit finite-resolution coarse-graining channels. The present work instead analyzes the problem from a more general resource-theoretic perspective, where the obstruction arises from dynamical bounds on quantum operations implemented with finite resources. In this sense, finite-resolution models appear as particular realizations within a broader class of resource-constrained dynamics. Together, the two works provide both an explicit operational example and a general dynamical perspective on the limitations associated with exact quantum limits. Conceptual index of the Finite-Closure Geometry (FCG) research program Finite-Closure Geometry (FCG): A Structural Program for Global Non-ClosureDOI: 10.5281/zenodo.18001440 Keywords quantum channels, finite resources, zero-temperature limit, third law of thermodynamics, quantum speed limits, spectral localization, ground-state preparation, quantum thermodynamics, operational limits, trace distance, resource theory, finite-closure geometry, quantum foundations, mathematical limits
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.