The Ladder of Gaps Information D; The Stability Ladder and Fault-Tolerant Quantum Computation An Axiomatic Derivation from Information Conservation to Threshold Gap
This paper systematically applies the stability ladder axiomatic system to faulttolerant quantum computation. Starting from two foundational axioms—the conservation of logical quantum information and the locality of quantum operations—itre-derives the necessary existence of a fault-tolerance threshold. The core argumentshows that under the constructive completeness condition that a quantum computing system must support arbitrarily deep reliable computation, hierarchical encoding is the minimal sufficient structure that coordinates the axioms. This structureinherently possesses a superlinear compression property of the error propagationmapnearthe origin, which via a fixed-point analysis necessarily yields a strictly positive threshold gap. This conclusion repositions the fault-tolerance threshold froma numerical discovery of specific code constructions to an analytical consequence ofthe axiomatic system, providing a unified framework for quantum fault-tolerancetheory rooted in information ontology. Using the concatenation of surface codesand the threshold theorem as an example, this paper establishes a logical divisionof labor between axiomatic derivation and constructive analysis: the former answers ”why the threshold must exist,” while the latter answers ”how the thresholdis concretely realized.”
This paper systematically applies the stability ladder axiomatic system to faulttolerant quantum computation. Starting from two foundational axioms—the conservation of logical quantum information and the locality of quantum operations—itre-derives the necessary existence of a fault-tolerance threshold. The core argumentshows that under the constructive completeness condition that a quantum computing system must support arbitrarily deep reliable computation, hierarchical encoding is the minimal sufficient structure that coordinates the axioms. This structureinherently possesses a superlinear compression property of the error propagationmapnearthe origin, which via a fixed-point analysis necessarily yields a strictly positive threshold gap. This conclusion repositions the fault-tolerance threshold froma numerical discovery of specific code constructions to an analytical consequence ofthe axiomatic system, providing a unified framework for quantum fault-tolerancetheory rooted in information ontology. Using the concatenation of surface codesand the threshold theorem as an example, this paper establishes a logical divisionof labor between axiomatic derivation and constructive analysis: the former answers ”why the threshold must exist,” while the latter answers ”how the thresholdis concretely realized.”
This paper applies the four-tier stability ladder axiomatic system to classicaland quantum information coding theory, demonstrating that under the joint constraints of the information conservation design goal and the physical constraint oflocal noise, a coding system must necessarily exhibit the structural feature of anonzero error correction radius. The core argument uses a proof by contradictionto rigorously show: if the minimum distance of the code is less than 3, then localerrors can propagate without attenuation in the code space, causing mixing of logical states and directly violating the information conservation axiom. At the formalfoundation level, a precise mathematical definition of coding systems is established,and a bidirectional equivalence theorem between the information conservation axiom and the Knill–Laflamme condition is proved, with a complete classification ofboundary cases. Using the quantum surface code as an example, the ontologicalfoundation of the quantum error correction threshold is reconstructed from thestability axioms, and the framework is extended to general stabilizer codes andclassical linear codes. The entire derivation does not rely on any specific code construction, decoding algorithm, or explicit calculation of channel capacity, providingan axiomatic explanatory structure for the universality of the error correction radius in coding theory.
This paper places the condensed-matter quantum many-body system withinthe framework of a four-tier stability ladder axiomatic system, and systematicallydemonstrates the existence of a finite gap in the bulk excitation spectrum of topological order and the pairing gap of charged superconductor quasiparticles. Thecore logical chain shows that under the condition that the degenerate ground-statespace carries nonlocal information encoding, if the system satisfies causal closurewith local interactions, then in the thermodynamic limit the excitation spectrummust possess a strictly positive energy threshold; otherwise, the axiom of information conservation would be violated by local perturbations. Unlike known stability theorems (gap implies stability), this paper proves the converse proposition information encoding structure implies the existence of a gap. By rigorouslydistinguishing between the broad and narrow definitions of topological long-rangeentanglement, we place gapless metallic phases, quantum critical points, non-Fermiliquids, etc., outside the legitimate boundary of the axiomatic framework, explicitlystating that information conservation in these systems is achieved by alternativemechanisms such as topological invariants of the Fermi surface. The exact solutionof the Kitaev honeycomb model and BCS mean-field theory are used to verify thelogical self-consistency of the derivation, and the elimination of Goldstone modes bythe Anderson-Higgs mechanism is precisely pinpointed as the watershed betweencharged superconductivity and neutral superfluidity. The appendices systematically organise the Majorana fermion representation, the area law of topological entanglement entropy, Gor’kov Green’s functions, and quasi-adiabatic continuationtechniques, providing complete mathematical support for the axiomatic derivation.
This paper constructs a four-tier axiomatic system of the stability ladder toreveal the underlying logic of phase transitions in statistical mechanics. Startingfrom information conservation (L0) and short-range interactions (L1), we rigorously prove, using DLR measure theory and large deviation principles in the thermodynamic limit, that the coexistence of macroscopically distinguishable states insystems of spatial dimension not less than two necessarily requires a strictly positiveenergy barrier in the free energy landscape (L2). In one-dimensional systems, information protection is instead realized via topological boundary conditions. Takingthe two-dimensional Ising model as a paradigm, we reconstruct the ontological basisof the existence of a phase transition in Onsager’s exact solution from the axioms,and extend the framework to general lattice systems and quantum many-bodymodels. The derivation does not rely on mean-field approximations or specific calculations of the renormalization group, thereby providing a necessary explanationfor the universality of phase transitions based on the logic of information conservation. In critical dynamics and quantum phase transitions, the barrier conceptis generalized to a dynamical barrier or quantum gap, demonstrating the broaderapplicability of the stability ladder.
The far-deep isomorphism proposition asserts that systems with infinite surfaceseparation must converge in deep topology, but previously lacked quantitative predictions across different systems. This paper rigorously applies this proposition tothe cross-system comparison of topological quantum error-correcting codes and derives a new conditional conjecture: if two topological codes are drastically differentin surface structure but converge to the same equivalence class at the deep anyoncategory level, and if the functoriality axiom together with the Lipschitz continuitycondition holds, then their optimal fault-tolerance thresholds necessarily approachthe same value in the thermodynamic limit, and the deviation of thresholds atfinite sizes is bounded above by the deep topological distance. This paper takesthe square-lattice toric code and the hexagonal-lattice color code as a test pairfor the deep-convergence group, and the Z2 toric code with the Z3 toric code asa control group for the deep-divergence case. A complete numerical test schemeand a statistical operationalization procedure are provided, together with explicitcriteria for strong and weak falsification as well as a systematic diagnosis protocol after falsification. This framework advances the far-deep isomorphism from anepistemological proposition in category theory to an operational general programof physical universality.
We apply the Information-Theoretic Unification (ITU) framework (Terada 2026, concept DOI 10.5281/zenodo.20109209; current version v2.0.0 at 10.5281/zenodo.20133709) to fault-tolerant quantum computing. Phase 5 of ITU established that bulk locality in AdS/CFT is mathematically equivalent to a quantum error correction code (QECC); here we exploit the reverse direction — using the ITU axiom δS = δ⟨K⟩ to derive optimal QECC architectures for practical quantum computation. This is a Tier 1 application paper, complementary to the Tier 0 core ITU theory. It is the first in a planned series of focused application papers (ITU + AI, ITU + cancer biology, ITU + economics, etc.), each as an independent Zenodo deposit linked via isSupplementTo relations to the core ITU concept DOI. Phase 43: [[5,1,3]] perfect quantum code — the discrete realization of bulk locality in AdS₃/CFT₂ (Pastawski-Yoshida-Harlow-Preskill HaPPY code building block) — verified as a complete fault-tolerant primitive: 4 stabilizers, code distance d = 3, all 15 single-qubit Pauli errors yield distinct syndromes, decoder lookup table of size 16 maps every syndrome to its corrective operation, and the encoded logical error rate is below the bare-qubit rate at physical error rates up to 25%. Phase 44: Toric/surface code threshold simulated for L = 3, 5, 7 (18, 50, 98 qubits) under independent bit-flip noise. Empirical threshold p_c ≈ 0.045 with a greedy nearest-neighbor decoder (theoretical perfect-MWPM threshold ~0.103); exponential error suppression with lattice size confirmed. The toric code is the discrete realization of AdS₃/CFT₂ bulk locality at finite size. Phase 45: Magic state distillation (Bravyi-Kitaev 5-to-1 protocol) verified using the same [[5,1,3]] code from Phase 43. Fit slope = 1.98 (theoretical 2.0; quadratic suppression confirmed), coefficient c ≈ 10, threshold ε_th ≈ 0.129. Iterated distillation converges to machine precision (10⁻¹⁶) in 4-6 rounds below threshold. This completes the path from Clifford to universal fault-tolerant computation. Phase 46: NISQ-era resource estimates for 11 hardware platforms (IBM Heron R1, IBM Condor, IBM Flamingo, IBM Kookaburra, Google Willow, Google projected 2026, IonQ Forte, IonQ Tempo, Quantinuum H2, Quantinuum H3 projected, Rigetti Ankaa-3). At physical error rate p = 10⁻³ (current best), standard surface code requires distance d = 11 (302 physical qubits per logical qubit). A hypothetical ITU-optimised code with 2× threshold improvement reduces this to d = 9 (202 physical qubits per logical qubit), a 40-60% overhead reduction. IBM Kookaburra (2026, 4158 qubits) is projected to support 13 logical qubits with standard codes, 20 with ITU-optimised codes. Central thesis: the same single information-theoretic axiom that yields emergent spacetime (ITU Phases 1-32), life (Phases 33-39), and consciousness (Phases 41-42) also yields practical quantum-computing primitives ready for near-term hardware deployment. ITU is not merely a theoretical TOE but a constructive framework that can be applied to engineering problems. Includes 4 theory documents (Markdown), 4 Python numerical experiments, 4 result figures, 4 JSON summaries, paper metadata (CITATION.cff, .zenodo.json, README). Total runtime ~30 seconds on a modern laptop.
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉ for distance-d codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed T gate on the [[16,4,4]] BB code, T-type rotations on the [[64,8,8]] BB code, and CCZ-, CS-, and T-type operations on the [[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields CrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.