The computational difficulty of Coulomb energy in the full continuum

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Family 275 · Mathematical physics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Given fixed nuclei and a number of electrons, determine the lowest possible electronic energy. This is a basic problem of quantum chemistry. Its mathematical definition is simple enough to write on one line, yet the electrons may occupy an enormous space of correlated wavefunctions.

Complexity theory asks a worst-case question: can instances of a general difficult quantum problem be encoded into this apparently rigid Coulomb system? The rigidity matters. A physical nuclear potential is a sum of attractive point-charge potentials; one cannot simply choose arbitrary matrix entries or an arbitrary external field.

Family 275 reports two reductions establishing QMA-hardness for the full continuum problem. M1 allows positive integer nuclear charges written in binary. M2 goes further by using charge one at every nucleus. Neither supplies an orbital basis that restricts the electron states. The crucial task is to control all continuum states, including those that a finite-dimensional approximation would omit.

2. Background and essential definitions

The electronic operator

Let nuclei have distinct positions $R_1,\ldots,R_M\in\mathbb R^3$ and positive charges $Z_1,\ldots,Z_M$. With nuclei clamped in place, the electronic Hamiltonian in atomic units is

$$H=-\frac12\sum_{i=1}^{N}\Delta_{x_i} -\sum_{i=1}^{N}\sum_{\alpha=1}^{M}\frac{Z_\alpha}{|x_i-R_\alpha|} +\sum_{1\leq i<j\leq N}\frac1{|x_i-x_j|}.$$

The terms are kinetic energy, electron–nucleus attraction and electron–electron repulsion. Each electron also has a two-dimensional spin space. The admissible Hilbert space is $\bigwedge^N L^2(\mathbb R^3;\mathbb C^2)$: exchanging two electrons’ positions and spins reverses the wavefunction’s sign.

The operator is defined through its lower-bounded closed energy form on antisymmetric wavefunctions with square-integrable first derivatives. This fixes its realization despite the Coulomb singularities. No total-spin sector is selected. The target is

$$E_0=\inf\operatorname{Spec}H.$$

An infimum need not be attained by a normalizable ground eigenvector. The decision problem is meaningful even without a binding assumption. Nuclear–nuclear repulsion is omitted because the stated convention is electronic energy with clamped nuclei.

A promise problem

Rather than demand an exact real number, the input gives rational thresholds $a<b$. A YES instance has $E_0\leq a$; a NO instance has $E_0\geq b$. Nothing is required when the energy lies between them. The separation is an accuracy promise, not the spacing between physical energy levels.

The total input length $L$ counts bits used for rational positions, charges where applicable, thresholds and electron number. Rational numbers have signed binary numerators and positive binary denominators. Electron number is unary: describing $N$ uses order $N$ symbols. This prevents an exponentially large electron count from being hidden in a short binary input.

QMA is the class of promise problems whose YES answers admit polynomial-size quantum witnesses checked by a polynomial-time quantum verifier, with bounded error. QMA-hardness means every such problem can be transformed into the Coulomb decision problem by a deterministic classical polynomial-time map preserving YES and NO answers. It is a reduction statement, not an unconditional proof that all algorithms take exponential time.

3. Development history and previous partial results

Electronic-structure complexity developed through several related but distinct formulations. Liu, Christandl and Verstraete established QMA-completeness of the finite-mode representability problem for reduced density matrices. General fermionic interaction coefficients were important there; requiring actual Coulomb coefficients imposes additional structure.

Schuch and Verstraete proved hardness for Hubbard models with local magnetic fields and developed continuum realizations using designed scalar and spin-dependent potentials. Their route through effective spin interactions and control of complementary one-particle states are methodological predecessors. The nuclear-only potential here is a more restrictive input class.

O’Gorman, Irani, Whitfield and Fefferman proved QMA-completeness for electronic structure in a supplied finite orbital basis. Their published work includes positive unit nuclear charges, but the supplied basis remains part of the problem. It explicitly raises the complete-space question.

Why is that distinction substantial? If $P$ projects onto chosen orbitals, the variational principle gives

$$\inf\operatorname{Spec}H\leq \min\operatorname{Spec}(PHP|_{\operatorname{Ran}P}).$$

A low projected energy supplies a valid trial state for the continuum. A high projected energy does not exclude a lower state outside the chosen space. Thus a finite-basis hardness reduction can transfer its YES instances while failing to transfer its NO instances.

The present manuscripts start from a field-free square-lattice Heisenberg hardness result of Cubitt, Montanaro and Piddock. They use singlet mediators associated with Piddock and Montanaro and the superexchange mechanism introduced by Anderson. The claimed new continuum constructions realize the effective interactions using positive point nuclei while bounding the omitted states from below.

M2’s unit-charge theorem logically implies hardness of the larger binary-charge class. Its proof nevertheless starts directly from the same spin source rather than invoking M1’s hardness theorem. M1 retains a distinct construction involving exponentially large charges and calibrated hydrogenic tunneling.

4. What the new manuscripts claim

Binary-encoded positive charges

M1 uses distinct rational positions $R_\alpha\in\mathbb Q^3$, positive integer charges $Z_\alpha$ encoded in binary, unary $N$, and rational thresholds with $b-a\geq L^{-1}$. It claims this promise problem is QMA-hard under deterministic classical polynomial-time many-one reductions.

The reduction can produce the stronger threshold separation $b-a\geq1$, while all output descriptions have polynomial bit length. Charges and inverse nuclear separations may have exponentially large numerical values. Binary notation makes those values compatible with polynomial-length descriptions.

There is no supplied basis, magnetic field, electronic state or independently adjustable external potential. The minimization is over the full spinful antisymmetric continuum space and every spin sector. The theorem does not assert membership in QMA.

Only unit nuclear charges

M2 claims the same type of hardness with

$$Z_\alpha=1\qquad\text{for every nucleus }\alpha.$$

Its reduction outputs polynomially many nuclei and electrons, distinct rational positions with polynomial bit lengths, and rational thresholds separated by at least one. The only variable output data are the nuclear positions, electron number and thresholds.

No neutrality, strict binding, normalizable ground state, molecular spectral gap or lower nuclear-separation promise is imposed. In particular, “unit charges” does not mean ordinary stable hydrogen molecules under familiar chemical conditions. These are general promised instances of the specified operator.

Both statements are about approximation of the spectral infimum to the promised decision precision. The number one in the constructed threshold separation is not a lower bound on any excitation energy.

5. Why the advance matters

Importance: 5/5, provisional. The family claims a worst-case hardness result for the unrestricted Coulomb continuum with nuclear-only external data, including the strong unit-charge restriction. It closes a conceptual gap between encoding difficulty in a selected basis and encoding it in the actual variational problem over all wavefunctions.

The unit-charge result shows that large nuclear charge is not essential to the claimed complexity phenomenon. The geometric arrangement and collective electronic interactions suffice in M2’s construction. M1, meanwhile, contributes a different quantitative route for controlling extremely small tunneling amplitudes and high-accuracy one-well energies.

The practical interpretation has limits. Worst-case hardness does not say that every molecule is difficult or that successful approximate chemistry methods cannot work on useful families. It says that a uniformly efficient solver for every promised instance would also solve every problem in QMA through these reductions. The rating concerns the significance of that claim if correct, not an independent proof audit.

6. The main idea

Build a finite spin problem inside electron motion

Both reductions begin with a hard Heisenberg spin Hamiltonian with signed couplings on a sparse square-lattice layout. Strongly coupled ancillary spin pairs have a singlet ground state, meaning total spin zero. Coupling other spins to suitable members of these pairs produces effective interactions with the desired signs using positive exchange terms.

Next, a repulsive Hubbard model supplies those exchanges. It has localized spatial sites, electron hopping between sites and an energy penalty for double occupancy. At half filling, one electron per site is favored. Virtual excursions through doubly occupied configurations produce an effective spin interaction proportional to the square of the hopping amplitude divided by an occupancy-energy scale.

Single occupancy is obtained through this low-energy analysis. It is not imposed as a restriction on continuum trial states. Keeping that distinction is necessary for a genuine full-space reduction.

M1: large charges and tiny tunneling

M1 chooses a large integer scale $Z_*$ of exponential magnitude but polynomial binary length. In coordinates $y=Z_*x$, primary nuclear wells resemble hydrogen ground states. Weaker secondary nuclei tune isolated-well energies to high accuracy.

Small geometric displacements calibrate the required tunneling ratios. Absolute closeness to a hydrogenic orbital is insufficient when the overlap itself is exponentially small: the reduction needs relative accuracy in that overlap. M1 develops relative tail estimates and a certified polynomial-time eigenvalue computation with exponentially small error.

The effective energy uses a shift and division by $Z_*$. Errors must remain controlled after the large physical rescaling, and all numerical precision must be describable and computable with polynomially many bits. These requirements are part of the reduction, not free numerical assumptions.

M2: a positive density replaced by unit nuclei

M2 instead uses a polynomially large dilation $y=\lambda x$. After dividing energy by $\lambda^2$, a physical unit nucleus has coupling $1/\lambda$. The construction first designs a nonnegative continuous nuclear charge density that creates the needed wells and transverse confinement.

A broad positive slab supplies enough background density to keep the modified density nonnegative. Adjustments to well depths compensate for long-range direct electron repulsion. This direct repulsion is retained in the effective model rather than discarded at polynomial scales.

A transport of density and an equal-mass cubature rule replace the continuous density by point masses of exactly $1/\lambda$. Cubature is numerical integration by weighted samples; equal masses here ensure that every physical nucleus has charge one after rescaling. Rational rounding is included in the error estimates, and the point positions remain distinct.

Exclude every omitted continuum state

Both approaches need a lower bound on the one-particle complement of the selected modes. Localization provides an energy penalty there. Bounds on electron interactions control coupling between selected and omitted states; completing a square makes the resulting energy loss manageable.

Together with trial states from the selected modes, this produces matching lower and upper comparisons between the continuum infimum and the effective finite model. The lower comparison is what transfers NO instances. Explicit scalar energy shifts and rational threshold adjustments then preserve the original promise under the complete classical reduction.

7. Scope, limitations, and open questions

Neither manuscript claims QMA membership, so “QMA-complete” would overstate the result. Infinite-dimensional witness encoding and verification raise questions beyond a hardness reduction.

The input models have no neutrality, separation or binding promise. Hardness under additional chemically motivated constraints is a distinct question. M1’s exponentially large charges cannot be replaced by individually listed unit nuclei without potentially exponential output size; M2 requires its separate polynomial-scale construction.

The theorem also does not give a ground-state preparation algorithm, a dynamical simulation hardness statement or a universal claim about physically accessible chemical processes. Those tasks have different input and output requirements. The precise conclusion is a worst-case decision reduction for the clamped-nuclei electronic spectral infimum.

8. Sources and further reading

The Schuch–Verstraete and O’Gorman–Irani–Whitfield–Fefferman papers linked in section 3 explain earlier designed-potential and supplied-basis formulations. The source manuscripts below specify the continuum domains, numerical encodings, finite spin inputs and error budgets. Their claims are reported as stated.

Source manuscripts

• M1. OpenAI, Continuum Coulomb hardness with binary nuclear charges (September 24, 2026).
• M2. OpenAI, QMA-hardness of continuum Coulomb energy with unit nuclear charges (September 24, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.