We explicitly carry through to the Hubbard endpoint the product-state (Hartree-Fock-type) case that Waite et al. state is available within their framework. The closest route is their Theorem 7 with a subset of cardinality one. We give explicit constants, track each guiding spinor through the reduction, and provide a finite description of the resulting site-local Slater determinant. The target is a repulsive Fermi-Hubbard Hamiltonian on a full open rectangular square grid, with spatially uniform positive hopping and repulsion, local vector fields, and the entire fixed-number sector Nf = M. With inverse-polynomial squared overlap and normalized decision precision, the resulting promise problem is PromiseBQP-complete; hardness holds on the subclass with a unique sector ground state and an inverse-polynomial spectral gap. The guiding spinors are retained from their creation-time penalty ground states and rounded to normalized dyadic-pair descriptions. The proof includes explicit global perturbation bounds and preparation and energy-estimation costs. MSC2020: Primary 68Q12; Secondary 81P68, 81V70. The archive contains the complete LaTeX source, vector figures, numerical test data, build instructions, a manifest, and SHA-256 digests. AI-use disclosure: The natural-language proofs, diagnostic code, partial formal proofs, and manuscript were generated using GPT-6 Astra and Claude Fable 5.1 under the author's direction on the research question and acceptance criteria. AI review comprised four independent audits and one heterogeneous adversarial audit across the two model families; completed human mathematical proof reviews number zero. Formal checking is peripheral: five source-side and five finite-endpoint declarations have Lean kernel proofs, Schrieffer-Wolff composition is zero of eight original targets, and one overgeneral transfer signature has a singleton counterexample. The full uniform completeness theorem is not formally proved. The author supplied the presentation revisions and remains responsible for final acceptance of the work. Appendix C distinguishes the earlier high-precision numerical example from the new certified Arb interval calculation.
We explicitly carry through to the Hubbard endpoint the product-state (Hartree-Fock-type) case that Waite et al. state is available within their framework. The closest route is their Theorem 7 with a subset of cardinality one. We give explicit constants, track each guiding spinor through the reduction, and provide a finite description of the resulting site-local Slater determinant. The target is a repulsive Fermi-Hubbard Hamiltonian on a full open rectangular square grid, with spatially uniform positive hopping and repulsion, local vector fields, and the entire fixed-number sector Nf = M. With inverse-polynomial squared overlap and normalized decision precision, the resulting promise problem is PromiseBQP-complete; hardness holds on the subclass with a unique sector ground state and an inverse-polynomial spectral gap. The guiding spinors are retained from their creation-time penalty ground states and rounded to normalized dyadic-pair descriptions. The proof includes explicit global perturbation bounds and preparation and energy-estimation costs. MSC2020: Primary 68Q12; Secondary 81P68, 81V70. The archive contains the complete LaTeX source, vector figures, numerical test data, build instructions, a manifest, and SHA-256 digests. AI-use disclosure: The natural-language proofs, diagnostic code, partial formal proofs, and manuscript were generated using GPT-6 Astra and Claude Fable 5.1 under the author's direction on the research question and acceptance criteria. AI review comprised four independent audits and one heterogeneous adversarial audit across the two model families; completed human mathematical proof reviews number zero. Formal checking is peripheral: five source-side and five finite-endpoint declarations have Lean kernel proofs, Schrieffer-Wolff composition is zero of eight original targets, and one overgeneral transfer signature has a singleton counterexample. The full uniform completeness theorem is not formally proved. The author supplied the presentation revisions and remains responsible for final acceptance of the work. Appendix C distinguishes the earlier high-precision numerical example from the new certified Arb interval calculation.
We give a self-contained explicit counterexample for the existential unregularized bounded-region fermion question on printed page 371 of Oberwolfach Report 7/2025, with pointwise operator-norm topology distinguished from strong-operator topology on page 365. In dimension three, one fixed smooth nonnegative radial compactly supported pair potential and one fixed interaction ball give a fixed smooth normalized annihilation mode whose Heisenberg norm differences tend to the maximal value 2. Explicit packed finite-particle Slaters and times pi/(2N+1-kappa) witness the same vector-norm limit, a stronger vector asymptotic, and changing normal-state values tending to 4. Both diagonal-density and normal-ordered interactions are covered with ordered-pair coefficient 1 and no ultraviolet smearing. A fixed smooth-cutoff commutator, Slater second moment and two-sector Duhamel comparison give error O(N^(-1/6)+N^(-1/3)). The kinetic energy acts on all of R³. Continuity on every fixed vector and in every fixed normal state remains valid. Siebert's arXiv:2609.34402v1, submitted 28 September 2026 before the original 7 October candidate, supplies the earlier high-density cluster and localized phase mechanism. Its stated theorem concerns full-space difference potentials, but its proof directly adapts to this bounded symmetric cutoff kernel and yields the qualitative existential conclusion. Its Proposition 2.6 explicitly gives maximal norm 2 for the full-space constant-number reference. The present note gives an explicit free-Slater transfer to the fixed spatial cutoff. A separately checked audit-time energy bootstrap also recovers the cutoff maximal limit through the earlier localized phase mechanism. These cutoff implications are deductions made in the audit, not literal cutoff theorems attributed to the prior paper. No first solution, first quantitative refinement, historically certified novelty, or continued open status through October 2026 is claimed. The bounded primary-source priority audit records access and indexing limitations. This five-page unrefereed research note is accompanied by its identical authored PDF/LaTeX source, unchanged 26-file historical candidate and independently authored analytic, operator-algebra, computational, source/model, priority and priority-mapping audits. The fresh computational family has 2,668 exact finite predicates and 558 floating diagnostics; finite tests supplement the universal written proof, and numerical diagnostics are not rigorous interval enclosures or formal proof certificates. Historical optimized-assertion and inventory weaknesses are repaired by externally pinned closed regular-file validation and isolated nonoptimized checker execution, while all original bytes and the original 1/5 candidate count are preserved. Historical result outputs are not claimed byte-identical to current replays, and runtime source-byte checks are explicitly omitted. AI tools were used extensively in development, verification, adversarial review and writing. Independent AI-assisted review is not human or journal peer review; neither is claimed. No copied third-party source bodies are distributed. No all-potential, all-region, dimension-one/two, fixed-density/energy, thermodynamic, fixed-state-discontinuity or CAR-invariance result is asserted.
We prove the ground-state energy of the periodic, half-filled 4×4 repulsive Hubbard model with t = 1 to at least 23 decimal places at U = 1, 2, 3, 4, 5, 6, 7, 8, 10, 12 and 16: every enclosure has width below 10⁻²⁴. The double occupancy, the spin correlations and the staggered structure factor have certified interval widths below 10⁻¹⁰, and the kinetic energy below 10⁻⁹. Hypercube, spin and pseudospin symmetries reduce the problem to a 19,186-dimensional block that contains the ground state by Lieb's theorem and a character argument. A binary64 Cholesky factorization, verified using a rigorous rounding-error bound evaluated in exact rational arithmetic, certifies spectral separation within this block; exact Rayleigh quotients and residuals then give Temple enclosures of the energy and bounds on the angle to the ground state, which certify the observables. Separation certificates at both ends and a perturbative trial family enclose E₀(U) for every U in [3.9, 4.1], with uniform width at most 1.6 × 10⁻⁹. Two independent implementations replay the certificates, the interlacing and Temple steps are proved in Lean 4, and all computations run on one laptop. At U = 4 and 8, exact-arithmetic brackets that use neither Lieb's theorem nor floating point are also given, and the enclosures bound their deficits. At U = 4 the certified energy rounds to −13.62185 and excludes the published value −13.62192 [Physical Review B 94, 085103]. Contents of the bundle paper/ — the manuscript (Markdown and PDF), its figures, and supplementary/ with five videos generated from the certificate files. rayleigh/symmetry/ — the enclosure pipeline (block dimensions, exact basis, certificates, observables, the interval certificate) and recheck/, the independent Rust reference checker; certificates/ holds every certificate, trial vector, run log, checker log and the clean-replay report. sos/, rayleigh/demo/, cluster/ — the exact-arithmetic bracket certificates and the expected outputs of their checkers (golden_ files); the bracket checkers belong to the authors' certification software and are not included. formal/ — the Lean 4 statements (rank-one interlacing, Temple's inequality, the sum-of-squares remainder bound). tests/ — the generators of every table, figure and video of the manuscript. replay.sh, REPLAY.md — the automated replay and the step-by-step commands. AUDIT_AXIOMS.txt — #print axioms of the formal statements. SHA256SUMS — checksums of every file. Verification PYTHON=<python with numpy and scipy> ./replay.sh rebuilds the symmetry block from the Hamiltonian, reproduces every certificate from the stored integer vectors, runs the Rust reference checker, recomputes the observables and checks the manuscript's tables, and writes REPLAY_REPORT.md. It needs about 12 GB of memory and 16 GB of free disk, and takes about half an hour on the hardware above; its work directory is removed at the end. REPLAY.md lists the commands step by step; rayleigh/symmetry/certificates/REPLAY_REPORT_2026-10-14.md is a clean replay of this material. Not included: the two 662,547,600-byte full-sector vectors of the upper bracket certificates (their SHA-256 digests are in rayleigh/demo/). Integrity: shasum -a 256 -c SHA256SUMS inside the bundle; the SHA-256 of the tarball is in the .sha256 file beside it. Environment Python: 3.13.7 with numpy 2.5.4, scipy 1.18.1, matplotlib 3.11.2 (producer-requirements.txt) Rust: 1.90.0 (reference checker; cargo build --release in rayleigh/symmetry/recheck) Lean: leanprover/lean4:v4.33.1 with Mathlib v4.33.1 (formal statements) Hardware: Apple M3 Pro, 18 GiB (all production and replay) License: CC BY-NC 4.0. Commercial use requires written permission from Origin Neural AI (labs@smartledger.solutions).
Learning compact, interpretable descriptions of quantum many-body states is an important task in quantum science. We study the task of learning the Slater determinant with maximum fidelity to an arbitrary fermionic many-body state, with motivation from both Hartree-Fock methods and agnostic tomography. Given an n-fermion wavefunction built from m fermionic modes, we provide classical and quantum algorithms returning a Slater determinant with fidelity within ε of maximal in time mpoly(n,1/ε). We prove matching hardness lower bounds, assuming standard complexity conjectures, along some parameter axes. Given access to quantum copies, we prove this can be accomplished with poly(m,n,1/ε) copies of ρ. We also show that above a fidelity of 2/3 any stationary point is the unique global maximum while below 2/3 the optimization landscape can have spurious stationary points, and hence 2/3 marks a transition point in the optimization landscape for this problem. We apply the algorithm to the Fermi-Hubbard model, extracting the closest Slater determinant from neural quantum state solutions. Together, our results provide algorithmic tools with provable guarantees in understanding fermionic many-body systems with classical or quantum simulation.
Finding the ground state energy of electrons subject to an external electric field is a fundamental problem in computational chemistry. We prove that this electronic-structure problem, when restricted to a fixed single-particle basis and fixed number of electrons, is QMA-complete. Schuch and Verstraete have shown hardness for the electronic-structure problem with an additional site-specific external magnetic field, but without the restriction to a fixed basis. In their reduction, a local Hamiltonian on qubits is encoded in the site-specific magnetic field. In our reduction, the local Hamiltonian is encoded in the choice of spatial orbitals used to discretize the electronic-structure Hamiltonian. As a step in their proof, Schuch and Verstraete show a reduction from the antiferromagnetic Heisenberg Hamiltonian to the Fermi-Hubbard Hamiltonian. We combine this reduction with the fact that the antiferromagnetic Heisenberg Hamiltonian is QMA-hard to observe that the Fermi-Hubbard Hamiltonian on generic graphs is QMA-hard, even when all the hopping coefficients have the same sign. We then reduce from Fermi-Hubbard by showing that an instance of Fermi-Hubbard can be closely approximated by an instance of the Electronic-Structure Hamiltonian in a fixed basis. Finally, we show that estimating the energy of the lowest-energy Slater-determinant state (i.e., the Hartree-Fock state) is NP-complete for the Electronic-Structure Hamiltonian in a fixed basis.
We consider two bosons in the one-dimensional Bose–Hubbard model with hopping amplitude J and constant on-site repulsion U, both subject to the same weak, static, stationary Gaussian lattice potential. For a repulsively bound pair of total quasimomentum K we derive in closed form the second-order (golden-rule) kernel for transitions into the interacting two-particle scattering continuum at transferred momentum q. The kernel is strictly positive throughout the kinematically open region and vanishes as a square root at its edges. Its continuum factor coincides with that of the known dissociation rate under hopping modulation, while the static density coupling contributes a momentum-dependent form factor and a folded on-shell support. For flat high-pass disorder spectra the kernel gives an exact classification of when dissociation and elastic backscattering vanish at order W² in the disorder strength. For π/2 < |K| < π and U < 4J sin(|K|/2), a critical interaction U* = 4J√(−cos K) separates pairs whose second-order dissociation can be removed by a finite spectral gap from those for which it cannot, and there is a window in which backscattering is closed while dissociation stays open. As the gap edge q0 approaches the largest open transfer qc, the coefficient vanishes as Δ^(3/2), Δ^(1/2) or Δ, with Δ = qc − q0, or tends to a nonzero plateau, depending on the geometry of the open region and on whether the pass-band height or the on-site variance is held fixed. Exact finite-ring simulations of the bound-band population reproduce the predicted enhancement and suppression for the two normalizations. This preprint includes the complete analytic proofs and numerical-check appendices. It is not an accepted journal publication. The PDF is unchanged from the author’s final manuscript dated 15 October 2026. Supporting data and code: 10.5281/zenodo.23390681. Public repository with the manuscript, source, figures and reproduction instructions: https://github.com/SHIN09-git/hubbard-pair-correlated-disorder. Generative AI was used extensively in the research and manuscript preparation. The manuscript and the repository’s AI-use record disclose the reported model roles, human verification scope and limitations. JIN HANZHE is the sole human author and takes responsibility for the work.
We study the computational complexity of the Guided Local Hamiltonian problem: given a local Hamiltonian H together with a classical description of a guiding state that has non-negligible overlap with the ground state of H, estimate the ground-state energy within inverse-polynomial precision. This setting captures real-world scenarios in quantum chemistry and many-body physics, where trial states derived from classical heuristics can be used to guide quantum algorithms. We identify families of physically-motivated guiding states for which the computational hardness of ground-state energy estimation persists in the guided setting. Extending prior results for semi-classical subset states, we prove BQP-hardness for classes including fixed-weight states, matrix product states, Gaussian states, and Fendley states. Our hardness results are obtained via refined Feynman-Kitaev circuit-to-Hamiltonian constructions that explicitly expose the structural role of the guiding state in the reduction. Complementing these results, we give a constructive proof of BQP containment when the guiding state admits a polynomial-size classical description, establishing BQP-completeness for the canonical formulation of the problem. Our results show that quantum advantage persists for the newly introduced state classes, and classical methods also remain viable when said guiding states admit appropriate descriptions. Together, our results identify a Goldilocks zone of guiding states that are efficiently preparable, succinctly described, and sample-query accessible, within which quantum advantage for ground-state estimation can be meaningfully assessed. We additionally formalise the Guided Fermi-Hubbard Hamiltonian problem and prove BQP-completeness on 2D square and triangular lattices, both with and without magnetic fields, when provided with an appropriate fermionic guiding state.