1 · The essentials
These manuscripts study how difficult it is to estimate the lowest possible energy of electrons interacting with fixed atomic nuclei in three-dimensional space. The electrons are allowed to range throughout space, rather than being restricted to a supplied list of possible shapes. The results place this task among problems at least as hard as a class of demanding quantum computation problems. The main manuscript allows nuclei with positive whole-number charges that can be very large but are written compactly. A companion manuscript obtains the same kind of difficulty even when every nucleus has charge one. In both settings, the construction produces two energy cutoffs separated by at least one. The main result has a Lean formalization, but the supplied information does not establish that this verification covers every claim in both manuscripts.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine trying to find the lowest point in a landscape whose shape is determined by the locations and charges of fixed markers. These results concern how difficult that search can be, even with identical markers; the actual landscape is one of possible many-electron states, not ordinary terrain.
Why it matters
The results locate computational difficulty in the continuum Coulomb energy problem itself, with minimization over the full spinful fermionic space. In the unit-charge setting, this difficulty does not require a supplied orbital basis, a magnetic field, or an additional external potential. That manuscript also imposes no binding assumption.
2 · Going further
The precise target is the electronic Coulomb spectral infimum in three dimensions: the infimum of energy over all allowed antisymmetric continuum states and spin sectors. The main manuscript considers positive integer nuclear charges encoded in binary, distinct rational nuclear positions, and an electron count encoded in unary. Its result is QMA-hardness of the energy approximation problem, established through a deterministic classical polynomial-time reduction with energy-threshold separation at least one.
The construction described in the main abstract can use exponentially large nuclear charges while keeping every output's bit length polynomial. The abstract identifies the reduction and its output guarantees, but does not give the intermediate constructions or estimates behind the argument. Thus the supplied text supports a description of the reduction's role, not a more detailed account of its mechanism.
The companion manuscript reaches QMA-hardness with clamped unit-charge nuclei. Its reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. It retains the full spinful fermionic continuum space, supplies no orbital basis, magnetic field, or additional external potential, and assumes no binding. The main result has a Lean formalization; the supplied material does not specify whether the companion result is also formalized.
Useful notions
- Energy minimization and infima
- Continuum electronic states and spin
- Fermionic antisymmetry
- Computational reductions and QMA-hardness
- Binary, unary, and rational input encodings
What was known before / what is new
Before
The supplied text gives no historical account of earlier results or previously open questions. It presents two related formulations of the continuum Coulomb energy problem: one allowing binary-encoded positive integer nuclear charges, and another restricting every nuclear charge to one.
New
The main result establishes QMA-hardness with distinct rational nuclear positions, unary electron count, and minimization over all antisymmetric continuum states and spin sectors; it has a Lean formalization. The companion manuscript shows QMA-hardness under the stronger restriction to unit-charge nuclei, with polynomially many nuclei and electrons. Both abstracts specify deterministic classical polynomial-time reductions and energy-threshold separation at least one.
Limits and assumptions
The claims concern the electronic Coulomb energy infimum in three dimensions under the stated input encodings and threshold guarantees. The main setting permits exponentially large charges, although outputs have polynomial bit length; the companion setting uses fixed unit-charge nuclei and polynomially many particles. The companion does not impose binding, so its statement should not be read as requiring a bound electronic system. The main result has a Lean formalization, but the supplied information does not delineate its coverage of individual claims or establish formal verification of the companion manuscript. No remaining open problem is identified in the supplied text.
Mini glossary
- Energy infimum —
- The greatest lower bound on the energies of all allowed states, without requiring a state that attains it.
- QMA-hardness —
- A computational hardness classification indicating that the target problem is at least as hard, through reductions, as problems in the quantum complexity class QMA.
- Spinful fermionic continuum space —
- The space of electron states with continuous spatial coordinates and spin, subject to antisymmetry under exchange of electrons.
- Polynomial-time reduction —
- A transformation from one computational problem to another whose running time is bounded by a polynomial in the input length.
- Clamped nuclei —
- Nuclei whose positions are held fixed in the electronic energy problem.
- Threshold separation —
- The difference between the two energy cutoffs used in the reduction, specified here to be at least one.
Keywords : Continuum Coulomb energy · QMA-hardness · Unit-charge nuclei
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Continuum Coulomb hardness with binary nuclear charges
24/09/2026We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.
QMA-hardness of continuum Coulomb energy with unit nuclear charges
24/09/2026We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.