This preprint studies learning an unknown bounded-degree n-qubit Hamiltonian in a known Pauli basis from independent copies of a single-temperature Gibbs state. For fixed locality, interaction degree and fixed positive rational inverse temperature and coefficient bound, it gives a polynomial-resource algorithm using product single-qubit Pauli measurements, without geometry, a gap, correlation decay or a minimum nonzero coefficient. The output has rational coefficients, support inside the true support and total coefficient error at most the requested tolerance with the stated success probability. The temperature-dependent exponents are extremely large; the result is asymptotic. The refinement certificate is formalized in Lean conditionally on explicit analytic inputs. Initialization, iteration, solver, sampling and full resource bounds remain paper proofs. A companion source note preserves the separate hidden-Majorana discussion and its open calibration problem; it is not a proof of that broader learner. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.
This preprint studies learning an unknown bounded-degree n-qubit Hamiltonian in a known Pauli basis from independent copies of a single-temperature Gibbs state. For fixed locality, interaction degree and fixed positive rational inverse temperature and coefficient bound, it gives a polynomial-resource algorithm using product single-qubit Pauli measurements, without geometry, a gap, correlation decay or a minimum nonzero coefficient. The output has rational coefficients, support inside the true support and total coefficient error at most the requested tolerance with the stated success probability. The temperature-dependent exponents are extremely large; the result is asymptotic. The refinement certificate is formalized in Lean conditionally on explicit analytic inputs. Initialization, iteration, solver, sampling and full resource bounds remain paper proofs. A companion source note preserves the separate hidden-Majorana discussion and its open calibration problem; it is not a proof of that broader learner. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.
We present an efficient protocol for learning an unknown k-local Lindblad generator on n qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed k and a supplied weighted interaction-strength bound αˉ, the protocol estimates all Hamiltonian and dissipative Pauli--GKSL coefficients with total coefficient error at each site at most ε with probability at least 1−δ using Ok(αˉ2n2k−2ε−2log(n/δ)) samples and polylogarithmically many distinct evolution times. This guarantee requires no degree, sparsity, or tail promise. A convex optimization converts these estimates into a valid k-local Lindblad generator with diamond-norm error at most ε using Ok(αˉ2n2kε−2log(n/δ)) samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound r and a sufficiently small sitewise subthreshold tail Ok(ε), coefficient recovery requires only Ok(αˉ2r2ε−2log(n/δ)) samples, without a supplied threshold or coefficient locations and without a coefficient-gap assumption. In particular, exactly sparse bounded-degree models have logarithmic sample dependence on n. We also provide guard-band support recovery and complementary supplied-candidate guarantees, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds.
This preprint proves a sharp purity bound for arbitrary complex pure states in tensor products of antisymmetric pair spaces, across the physical cut consisting of the first system from each pair versus the second system from each pair. For every finite number m of pairs and local dimensions at least two, purity is at most 2^(-m) and entanglement entropy is at least m bits, with equality characterized by products of normalized decomposable two-forms. It derives entanglement of formation m for m copies of the normalized antisymmetric state and, using the published regularization theorem, entanglement cost one ebit for every local dimension at least two. The archive provides original formal sources and a precise account of their scope; it does not claim a complete Lean translation of every ordinary LOCC execution. Research directed and curated by Matthew Liston. The mathematics, proofs, formalization and manuscript preparation substantially involve AI systems; paper-specific attribution and checking limits are disclosed in the paper and archive. No independent human proof verification, external endorsement or peer review is claimed. Licensing: original manuscript and documentation CC BY 4.0; original code MIT. The inherited QICLean SchattenNorm helper retains Apache 2.0 and its original attribution. The archive provides the exact file-level license mapping. Manuscript revision: 22 October 2026. Research director's note — Matthew Liston: This project is motivated by philosophical exploration and curiosity about AI-assisted research. My aim in sharing these results is to make them available for scrutiny, correction and further development, so that others can build on whatever proves useful to mathematics and science. Correspondence: atemporalwalk@gmail.com.
Abstract. We study the problem of learning a local quantum Hamiltonian [Formula: see text] given copies of its Gibbs state [Formula: see text] at a known inverse temperature [Formula: see text]. Anshu et al. [2020 IEEE 61st Annual Symposium on Foundations of Computer Science, pp. 685–691] gave an algorithm to learn a Hamiltonian on [Formula: see text] qubits to precision [Formula: see text] with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem [ Alhambra, PRX Quantum, 4 (2023), 040201 ; Anshu and Arunachalam, Nature Rev. Phys., 6 (2023), pp. 59–69 ], with prior work only resolving this in the limited cases of high temperature [ Haah, Kothari, and Tang, Markov field on finite graphs and lattices, 1971 ] or commuting terms [ Anshu et al., Efficient learning of commuting Hamiltonians on lattices, 2021 ]. We fully resolve this problem, giving a polynomial time algorithm for learning [Formula: see text] to precision [Formula: see text] from polynomially many copies of the Gibbs state at any constant [Formula: see text]. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multivariate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian.
We prove uniform QMA membership for the spectral-infimum decision problem for the full spinful antisymmetric Coulomb Hamiltonian with unit nuclei, distinct rational nuclear positions, unary electron number and inverse-polynomial promise gap. Nuclear repulsion is omitted. Positions and thresholds use binary rational encoding; the result assumes neither binding nor a spectral gap, geometric separation, bounded physical diameter, a selected finite basis, a fixed spin sector or an efficiently prepared low-energy state. The proof combines quantitative localization including escaped electrons, Hardy-controlled Coulomb caps, a finite kinetic cutoff, exact shifted sampling with L2 error control, and a polynomial-size verifier on arbitrary witness states. Together with the hardness theorem of the pinned source manuscript, this gives QMA completeness for the same problem. Classical sector localization, finite-basis electronic complexity, smooth-box discretization and earlier real-space simulation estimates are credited. The accompanying Lean support proves selected finite sampling, representation-average and scalar error interfaces; it does not formalize the full continuum or quantum-complexity theorem. No extension to exponentially large binary nuclear charges or worldwide priority claim is made. AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review or worldwide-priority certification. Identified OpenAI source inputs at revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb are stipulated true. Lean verifies scoped supporting results only, not the full theorem. Original prose: CC BY 4.0; original code and Lean: MIT; third-party rights preserved. See the supporting archive for precise assumptions, coverage, provenance and corrections.
We study the problem of learning a local quantum Hamiltonian H given copies of its Gibbs state ρ = e−β H/(e−β H) at a known inverse temperature β>0. Anshu, Arunachalam, Kuwahara, and Soleimanifar gave an algorithm to learn a Hamiltonian on n qubits to precision with only polynomially many copies of the Gibbs state, but which takes exponential time. Obtaining a computationally efficient algorithm has been a major open problem, with prior work only resolving this in the limited cases of high temperature or commuting terms. We fully resolve this problem, giving a polynomial time algorithm for learning H to precision from polynomially many copies of the Gibbs state at any constant β > 0. Our main technical contribution is a new flat polynomial approximation to the exponential function, and a translation between multi-variate scalar polynomials and nested commutators. This enables us to formulate Hamiltonian learning as a polynomial system. We then show that solving a low-degree sum-of-squares relaxation of this polynomial system suffices to accurately learn the Hamiltonian.
First preprint and reproducibility companion for bounded exact recovery of a nonzero trace-determinant tensor over an unknown binary extension field under corruption supported on a small common covector space. Includes the manuscript PDF and source, complete prose proofs, executable recovery and independent certificate checks, pinned partial Lean proofs, validation receipts, and three AI-model review reports with feedback dispositions. For s >= 2R+5, at most 8*2^(4R)*h trials return a verified compatible representation with probability at least 1-2^(-h). Accepted outputs are checked exactly; unique identification of a planted field is not claimed. The cubic-only bound does not establish the broader quantum application or a BPP/BQP result. Lean verification is partial. The complete recovery theorem, its probability and resource bounds, and the Python implementation are not fully formally verified. The reviews are AI-assisted internal reviews, not independent human peer review. Licenses: manuscript text and PDF are CC-BY-4.0; software and Lean sources are Apache-2.0. See LICENSE, LICENSES/CC-BY-4.0.txt and PROVENANCE.md. The package-level Other (Open) designation preserves these per-component licenses.