This paper establishes a geometric sufficient framework for the quantum polynomialtime class BQP on compact spin manifolds. The lattice skeleton carries legalcomputation paths, the fermionic Fock space on the lattice spin fibres carries thequantum register, and the Jordan–Wigner correspondence maps fermionic modes toqubits. The core observation is that the Jordan–Wigner non-locality of single-modefermion operators degenerates into constant-depth circuit blocks under edge-localcouplings on the skeleton, thereby preserving compatibility with the locality of thestandard quantum circuit model. On this basis, we establish quantitative relations among three geometric control quantities: a positive scalar curvature lowerbound fixes the Dirac spectral gap via the Friedrich constant; the spin connectioncurvature, as an infinitesimal quantum gate, determines the universality of thegate set through its Lie algebra generation property; the Dirac spectral gap andbounded curvature together control the resource consumption of adiabatic initialstate preparation. Furthermore, under the spin condition, polynomial spectralgap, polynomial transmission depth, holonomic universality, readout resolution,and a compilation axiom, we prove that the corresponding decision problem belongs to BQP. The converse direction proves that any polynomial-size quantumcircuit family can be constructively embedded as a family of spin computationmanifolds satisfying the above conditions. Quantum phase estimation is explicitlygeometrically realised as a positive example, while Grover’s unstructured searchis excluded by the structural constraint of the polynomial depth condition. Thepaper does not assume P= BQP, does not assume NP ⊂ BQP, and does not claimany new complexity class separation.
This paper establishes a geometric sufficient framework for the quantum polynomialtime class BQP on compact spin manifolds. The lattice skeleton carries legalcomputation paths, the fermionic Fock space on the lattice spin fibres carries thequantum register, and the Jordan–Wigner correspondence maps fermionic modes toqubits. The core observation is that the Jordan–Wigner non-locality of single-modefermion operators degenerates into constant-depth circuit blocks under edge-localcouplings on the skeleton, thereby preserving compatibility with the locality of thestandard quantum circuit model. On this basis, we establish quantitative relations among three geometric control quantities: a positive scalar curvature lowerbound fixes the Dirac spectral gap via the Friedrich constant; the spin connectioncurvature, as an infinitesimal quantum gate, determines the universality of thegate set through its Lie algebra generation property; the Dirac spectral gap andbounded curvature together control the resource consumption of adiabatic initialstate preparation. Furthermore, under the spin condition, polynomial spectralgap, polynomial transmission depth, holonomic universality, readout resolution,and a compilation axiom, we prove that the corresponding decision problem belongs to BQP. The converse direction proves that any polynomial-size quantumcircuit family can be constructively embedded as a family of spin computationmanifolds satisfying the above conditions. Quantum phase estimation is explicitlygeometrically realised as a positive example, while Grover’s unstructured searchis excluded by the structural constraint of the polynomial depth condition. Thepaper does not assume P= BQP, does not assume NP ⊂ BQP, and does not claimany new complexity class separation.
This paper studies the cross-problem in the geometrization of computationalcomplexity: whether geometric signatures of classical hardness and geometric sufficient conditions for quantum polynomial time are necessarily mutually exclusive. Targeting unsatisfiable formula families with exponential resolution lowerbounds, we examine whether their computational manifolds can simultaneouslysatisfy quantum-side conditions such as spin structure, polynomial Dirac spectralgap, and polynomial transmission depth. First, the naive argument that positiveRicci obstruction transfers via scalar curvature to spectral gap absence is showninvalid. Theorem A shows that such manifolds, as connected sums of S1×S2, carrypositive scalar curvature metrics by the Gromov–Lawson surgery theorem and obtain a constant positive spectral gap via the Friedrich inequality; the spectral gapis not the locus of incompatibility. Second, we give two genuine cross-constraints attwo levels. Theorem B is an unconditional scale incompatibility: faithful skeletonencoding requires the number of modes to equal the shortest resolution refutation length, whose exponential growth is incompatible with the polynomial modebudget of the quantum-side framework. Theorem C is a conditional decision incompatibility: if there exists an encoding family satisfying the quantum sufficientconditions that decides unsatisfiability of the formula family in polynomial depth,then NP ⊆ BQP; its contrapositive states that under the standard conjectureNP ̸⊆ BQP, such encodings do not exist. Moreover, Theorem D proves that thescale-invariant Ricci curvature defect K · L2 = Ω(n2) persists for all metrics onthis class of manifolds, so that the hard topology cannot be smoothed by any metric choice. This paper neither assumes nor proves NP ̸⊆ BQP; all conditionalconclusions explicitly mark their dependencies.
The Berry phase is a fundamental geometric quantity for characterizing the geometry and topology of quantum many-body systems. Previous work established a super-polynomial quantum advantage in Berry phase estimation at inverse-polynomial precision, given an ansatz state approximating the initial ground state. However, whether this hardness persists for Berry phases quantized by symmetry, which can be distinguished at constant precision, remained open. In this work, we characterize the computational complexity of quantized Berry phase estimation. First, we prove that deciding whether the Berry phase is exactly 0 or π is BQP-complete when the spectral gap is inverse-polynomially small. This result implies that distinguishing the quantized Berry phase still has a super-polynomial quantum advantage, assuming BPP=BQP. To prove this result, we introduce an encoding that maps the output of any quantum computation to a Berry phase that is exactly 0 or π. Second, we extend this hardness to physically motivated Hamiltonians on a 2D square lattice, including Heisenberg and XY interactions. Third, we provide a classical polynomial-time algorithm for geometrically local Hamiltonians on fixed-dimensional lattices with constant spectral gaps. Therefore we identify that the spectral gap is a key resource for the classical tractability, rather than precision. These results establish a super-polynomial quantum advantage for computing a quantized topological invariant that is unchanged under symmetry- and gap-preserving deformations, and provide new steps towards practical quantum advantage in quantum physics.
Five-Part Validation-Grade Suite on Universal Spectral Determinism and Non-Commutativity Residues: An Unconditional Resolution and Precision-Certified Certification of the \text{BQP} \not\subseteq \text{PH} Separation via Topological, Variational, and Information-Theoretic Invariants --- This 5-package resolution suite delivers an unconditional, non-relativizing, and non-algebrizing proof establishing the strict mathematical separation of Bounded-Error Quantum Polynomial-Time (\bm{\text{BQP}}) from the classical Polynomial Hierarchy (\bm{\text{PH}}) and Polynomial Time (\bm{\text{P}}). Rather than approaching the problem through symbolic logic or localized circuit lower bounds—which are structurally constrained by historic complexity barriers—this framework shifts the question into the continuous, precision-certified domains of Differential Geometry, Algebraic Topology, Category Theory, Information Thermodynamics, and Matrix Perturbation Theory. By analyzing the intrinsic geometric and numerical properties of complexity reductions under finite-precision constraints, the suite isolates a hard computational singularity. This singularity serves as an absolute structural boundary, proving that classical alternating configurations lack the algebraic and dimensional capacity to track non-local multi-particle quantum updates. Individual and Interlinking Functional Dynamics The 5-package suite operates as a mathematically closed, self-contained replication pipeline, where each package executes a specific analytical or numerical enforcement that feeds directly into the next. Package A: Analytic Foundations & Definitions • Individual Mechanics: This package translates the discrete states of algorithmic complexity into the continuous geometry of Riemannian manifolds. It formalizes the structural constraints of the Unitary Manifold (\bm{\mathcal{M}_{\text{BQP}}}) against the Classical Hypercube Manifold (\bm{\mathcal{M}_{\text{PH}}}). By utilizing Ricci flow to evaluate the state space reduction, it identifies a non-invertible Phase-Collapse Singularity (a "Neck-Pinch" obstruction), mathematically proving that the projection fails to preserve the underlying manifold structure. • Interlinking Pipeline: It establishes the primary mathematical universe, smooth structures, and geometric boundaries used by all subsequent packages to measure computational state deformations. Package B: Geometric Mapping & The Final Proof • Individual Mechanics: Building on the continuous manifold foundations, this package details the exact differential-geometric embedding of the computational manifolds utilizing the Fubini-Study metric (\bm{g_{\text{FS}}}). It formally characterizes the absolute dimensional mismatch between exponential quantum phase spaces and polynomial classical coordinate systems. By calculating the metric pullback field via the mapping Jacobian matrix (\bm{J_{\Psi}}), it demonstrates that the scalar curvature (\bm{R}) blows up and diverges as \bm{\mathcal{O}(2^n)}, establishing a geometric impossibility for polynomial reduction. • Interlinking Pipeline: It bridges the smooth geometric structures of Package A with the global topological invariants needed to escape localized oracle dependencies. Package C: Categorical Asymmetry • Individual Mechanics: This package elevates the separation proof into the abstract realm of Category Theory, defining the Quantum Monoidal Category (\bm{\mathbf{Quant}_n}) and the Classical Alternating Category (\bm{\mathbf{Alt}_n}). It frames the candidate complexity reduction as a natural transformation between computational functors. By applying the Yoneda Lemma, the text proves that such a natural transformation cannot exist because the quantum and classical domains possess fundamentally incompatible limit and colimit preservation profiles, ensuring they cannot be mapped without violating the Mac Lane Pentagon Coherence Invariant. • Interlinking Pipeline: By mapping the exact functorial boundaries, Package C transforms geometric and topological asymmetry into strict algebraic constraints, setting up the exact limits required to analyze information-theoretic capacities. Package D: The Entropic Seal • Individual Mechanics: This package models computation as a continuous thermodynamic channel, mapping quantum density operators within the Von Neumann entropy space against classical joint probability distributions within the Shannon alternating space. By invoking the Data Processing Inequality alongside Landauer's Principle, it argues that compressing exponential quantum entanglement capacity into polynomial classical bounds triggers an Entropic Seal Singularity. This singularity is characterized by an unphysical, exponential thermodynamic dissipation requirement (\bm{\Delta Q \sim \Omega(2^n)}), which physically precludes the containment of \bm{\text{BQP}} within \bm{\text{PH}}. • Interlinking Pipeline: This package creates the thermodynamic boundary conditions—demonstrating that a polynomial classical reduction violates physical laws. It defines the continuous informational flow equations that serve as the direct mathematical input for the final numerical validation engine. Package E: Error Analysis & Replication Framework • Individual Mechanics: Package E serves as the definitive numerical and empirical verification asset. It evaluates the complete reduction within a finite-precision digital environment using Moore-Rump Interval Enclosures to rigorously track non-linear error accumulation. Through Singular Value Decomposition (SVD) of the Morphic Coherence Matrix, it documents that the minimum singular value collapses exponentially (\bm{\sigma_{\min} \sim \mathcal{O}(2^{-n})}), driving an explosion of the condition number (\bm{\kappa \sim \Omega(2^n)}). When propagated, the target classical enclosures expand past unity (\bm{\omega \ge 1}), causing a total loss of tracking capability. • Interlinking Pipeline: Package E closes the entire suite. By tracking the relaxation kinetics of the non-commutativity residue, it isolates the power-law halting of the stable execution time-step (\bm{\Delta t \to 0}) under a critical break exponent \bm{\alpha = 1.0000 \pm 0.0001}. This bridges the theoretical abstractions of Packages A–D with an exact, error-bounded, replicable digital certificate (The Numerical Interlock Break). How the Suite Resolves, Validates, Seals, and Enables Replication 1. Resolves: The suite completely bypasses the Relativization, Algebrization, and Natural Proofs barriers. It does not construct localized classifiers over boolean functions; instead, it establishes a global functional space exclusion inside a Bounded-Variation Numerical Space \bm{\mathcal{V}_{\text{mach}}}. It proves that a polynomial classical reduction requires an impossible infinite-energy or zero-precision mapping. 2. Validates: Validation is handled dynamically by monitoring invariant spectral properties, metric pullbacks, and the convergence kinetics of the naturality residue. The system does not rely on heuristic approximations; it enforces absolute mathematical containment via strict interval-enclosure walls. 3. Seals: The resolution is sealed by the simultaneous convergence of independent physical and mathematical invariants: the phase-collapse neck-pinch, the scalar curvature divergence, the Mac Lane coherence failure, the Landauer dissipation wall, and the finite-time numerical halting. The complexity separation is locked because breaking the proof would require violating the fundamental laws of matrix calculus, differential geometry, and thermodynamics. 4. Enables Replication: Package E contains fully pre-wired, compilable LaTeX source code environments, exact citation keys anchored directly to foundational peer-reviewed literature (e.g., Golub & Van Loan, Page, Rump, Aaronson), and explicit numerical stability criteria. Independent replication teams can immediately deploy automated verification scripts to track the SVD spectrum, the scalar curvature metrics, and the \bm{\alpha = 1} critical break exponent, ensuring a completely transparent peer-to-peer review process. --- Note: The accompanying Agnostic Replication Kit (ARK) and Standard Academic Core (SAC) 17-package operational suite will be uploaded in the forthcoming version release to enable down-stream cross-institutional simulation and formal peer review.
Geometric quantum phases enable the construction geometric quantum gates, whose action only depend on the evolution path in state space. Geometric gates may offer increased robustness to control parameter noise. In this report, the conditions under which non-adiabatic geometric and holonomic quantum gates preserve their geometric character under sequential composition are analysed. Grover’s search algorithm requiring non-commuting operations is used as a representative example. The basic concepts of qubits, quantum gates and quantum circuits are explained. The concept of geometric phases is introduced and derivations of Abelian and non-Abelian non-adiabatic geometric phases are provided (Aharonov-Anandan and Anandan phase respectively). Construction of geometric and holonomic universal quantum gate sets with the aforementioned geometric phases are briefly described.Explicit calculation of two non-commuting Abelian and non-adiabatic geometric time evolutions show that the successively combined operation is not geometric. A proof of successive combinations of non-Abelian and non-adiabatic geometric time evolutions preserving their non-adiabatic Abelian and geometric properties is given. The report concludes that it is in general not possible to perform quantum circuits completely geometrically with a geometric universal quantum gate set. However, performing quantum computations with gates from a holonomic universal gate set is geometric and holonomic throughout the computation.
Quantum entanglement is a basic resource of quantum theory, but its usual definition assumes a fixed decomposition into subsystems. We develop an algebro--geometric framework to address quantum entanglement. For the Severi--Brauer scheme SB(A)→X of an Azumaya algebra and a factorization type d, we prove that a global locus of product states exists exactly when the associated torsor under PGLn reduces to the stabilizer Gd of the Segre variety. Thus entanglement in families is measured by a geometric obstruction to global subsystem structure.
Beyond this reduction criterion, we identify the cohomology that carries computable entangling obstructions. After determinant and incidence data L⊂R⊂K are chosen, the relevant classes lie in the mapping cone relative cohomology E∙(BL→BR). Their absolute images on X, modulo classes induced from BK, give determinant entangling obstructions beyond the Brauer class. We construct the moduli of subsystem structures as the quotient P/Gd and realize it as a locally closed locus in the relative Hilbert scheme, with a compactification by degenerations of product state loci. In the bipartite case, a chosen subsystem structure globalizes the Schmidt rank stratification to a flat filtration with base change compatible resolutions and fiberwise constant numerical invariants. Examples show that reducibility can depend on the underlying torsor, not only on the Brauer class, with an interpretation by entangling monodromy.
We continue the study of the circuit class GC^0, which augments AC^0 with unbounded-fan-in gates that compute arbitrary functions inside a sufficiently small Hamming ball but must be constant outside it. While GC^0 can compute functions requiring exponential-size circuits, Kumar (CCC 2023) showed that switching-lemma lower bounds for AC^0 extend to GC^0 with no loss in parameters. We prove a parallel result for the polynomial method: any lower bound for AC^0[p] obtained via the polynomial method extends to GC^0[p] without loss in parameters. As a consequence, we show that the majority function MAJ requires depth-d GC^0[p] circuits of size 2Ω(n1/2(d−1)), matching the best-known lower bounds for AC^0[p]. This yields the most expressive class of non-monotone circuits for which exponential-size lower bounds are known for an explicit function. We also prove a similar result for the algorithmic method, showing that E^NP requires exponential-size GCC^0 circuits, extending a result of Williams (JACM 2014). Finally, leveraging our improved classical lower bounds, we establish the strongest known unconditional separations between quantum and classical circuit classes. We separate QNC^0 from GC^0 and GC^0[p] in various settings and show that BQLOGTIME is not contained in GC^0. As a consequence, we construct an oracle relative to which BQP lies outside uniform GC^0, extending the Raz-Tal oracle separation between BQP and PH (STOC 2019).