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Quantum Phase Transitions and Typical Case, Polynomial Time Solution of Randomly Generated NP-Complete Problems via Adiabatic Quantum Computation

William M. Kaminsky, Seth Lloyd

Bulletin of the American Physical SocietyMar 16, 20060 citations

Operator-Theoretic Proof of the P = N P Separation: Spectral Gap Divergence and Hamming Locality

Nigel Stafford Paltoo

This paper provides a complete formal proof that P̸ = N P . We define a Turing-Spectrum Operator ˆHT uring acting on a separable Hilbert space Hcomp. By analyzingthe relative spectral trace and the Hamming connectivity of candidate polynomialoperators, we demonstrate that the complexity barrier is a consequence of ”SpectralIsolation.” We prove that a polynomial-time operator is a k-local operator that cannotclose the exponential spectral gap ∆(N ) required to map the ground state of an N P -complete manifold. The divergence of the Complexity Barrier Identity (CBI) and thenon-vanishing nature of the Computational Uncertainty commutator provide the finalclosure to the P vs NP problem.This paper provides a complete formal proof that P̸ = N P . We define a Turing-Spectrum Operator ˆHT uring acting on a separable Hilbert space Hcomp. By analyzingthe relative spectral trace and the Hamming connectivity of candidate polynomialoperators, we demonstrate that the complexity barrier is a consequence of ”SpectralIsolation.” We prove that a polynomial-time operator is a k-local operator that cannotclose the exponential spectral gap ∆(N ) required to map the ground state of an N P -complete manifold. The divergence of the Complexity Barrier Identity (CBI) and thenon-vanishing nature of the Computational Uncertainty commutator provide the finalclosure to the P vs NP problem.

Dec 19, 20250 citations

Hybrid Quantum-Classical Approaches to NP-complete Problems

Corey Ostrove

Bulletin of the American Physical SocietyOct 31, 20150 citations

Spectral method for solving NP-complete problems

Christian Kithima

Foundations of the Spectral Framework: From P = NP to Universal Applications This series of eleven articles develops the Spectral Framework, culminating in a machine‑verified proof that P = NP via the Spectral Reduction Lemma. The framework translates computational and mathematical problems into spectral Hamiltonians, whose ground states illuminate optimal solutions. The proof rests on four pillars: Existence of a unique positive ground state (Perron–Frobenius). Polynomial bounds on conductance and spectral gap (Kithima Bridge, Cheeger inequality). Logarithmic area law and MPS compressibility (cluster expansion, Brandão–Horodecki theorem). Polynomial extraction of solutions (power iteration on MPS). Articles 0–5 establish these pillars and assemble them into a polynomial‑time algorithm that decides SAT, thereby proving P = NP unconditionally. Articles 6–11 explore the consequences and universality of the method: Spectral cryptography and post‑P = NP security paradigms. Universality across NP‑complete problems, including SAT, Clique, TSP, Graph Isomorphism, and Integer Factorisation. Case studies such as the maximum clique problem and Cook’s historical student‑housing problem. Categorical unification in the category Spec, showing NP and P as equivalent subcategories and NP‑complete problems as isomorphic objects. Synthesis and correspondence dictionaries, mapping classical notions in complexity, physics, arithmetic, and cosmology to their spectral analogues. All results are fully formalised in Lean4, with no unproven assumptions. The central metaphor is that of a lantern: classical algorithms grope in the dark, while spectral topology constructs a lantern whose light spreads across the configuration space, compresses into an MPS, and reveals the brightest points—the solutions. This paradigm shift replaces blind search with direct spectral vision, opening new horizons for computation, mathematics, and science.Foundations of the Spectral Framework: From P = NP to Universal Applications This series of eleven articles develops the Spectral Framework, culminating in a machine‑verified proof that P = NP via the Spectral Reduction Lemma. The framework translates computational and mathematical problems into spectral Hamiltonians, whose ground states illuminate optimal solutions. The proof rests on four pillars: Existence of a unique positive ground state (Perron–Frobenius). Polynomial bounds on conductance and spectral gap (Kithima Bridge, Cheeger inequality). Logarithmic area law and MPS compressibility (cluster expansion, Brandão–Horodecki theorem). Polynomial extraction of solutions (power iteration on MPS). Articles 0–5 establish these pillars and assemble them into a polynomial‑time algorithm that decides SAT, thereby proving P = NP unconditionally. Articles 6–11 explore the consequences and universality of the method: Spectral cryptography and post‑P = NP security paradigms. Universality across NP‑complete problems, including SAT, Clique, TSP, Graph Isomorphism, and Integer Factorisation. Case studies such as the maximum clique problem and Cook’s historical student‑housing problem. Categorical unification in the category Spec, showing NP and P as equivalent subcategories and NP‑complete problems as isomorphic objects. Synthesis and correspondence dictionaries, mapping classical notions in complexity, physics, arithmetic, and cosmology to their spectral analogues. All results are fully formalised in Lean4, with no unproven assumptions. The central metaphor is that of a lantern: classical algorithms grope in the dark, while spectral topology constructs a lantern whose light spreads across the configuration space, compresses into an MPS, and reveals the brightest points—the solutions. This paradigm shift replaces blind search with direct spectral vision, opening new horizons for computation, mathematics, and science.

Apr 1, 20260 citations

A New Efficient Quantum Algorithm for Solving NP-complete Problems

Dhananjay P. Mehendale

Jan 1, 20260 citations

New quantum algorithm for studying NP-complete problems

Masanori Ohya, Igor V. Volovich

arXiv:quant-ph/0406216Reports on Mathematical PhysicsAug 1, 200332 citations

The Construction of a Spectral Gap Solution to P vs NP

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The Construction of a Spectral Gap Solution to P vs NP

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