A constructive solution to the $n$-representability problem

AI summary

Overview

Research area: Quantum chemistry and quantum information — specifically the fermionic quantum marginal problem (the $n$-representability problem) and semidefinite-programming hierarchies for certifying two-particle reduced density matrices (2-RDMs).

Technical level: Advanced. The paper is written in the language of fermionic operator algebra, Gram-matrix representations of sum-of-squares cones, and convex duality; readers need comfort with second quantization and semidefinite programming.

Scope (one sentence): The paper gives an explicit, computable construction of the complete family of two-body witnesses at every level of a weighted sum-of-squares hierarchy, proves the hierarchy terminates at a level equal to the particle number, and demonstrates the construction on a Richardson pairing Hamiltonian in the three-fermion sector.

What This Paper Is About

The $n$-representability problem asks which two-particle reduced density matrices can actually come from a legitimate ensemble of $n$ fermions — the set of valid 2-RDMs is characterized by enumerating "witnesses," operators that are nonnegative on the $n$-particle subspace but may be negative elsewhere. The paper turns an existing, largely abstract completeness proof into a concrete symbolic recipe: build a positive-semidefinite Gram matrix over fermionic monomials and impose linear nullspace constraints that exactly cancel every term of ladder degree greater than four, so the resulting operator is purely at-most-two-body.

Key Contributions

1. Constructive nullspace recipe for all two-body SOS witnesses. The authors define the degree-$d$ sum-of-squares cone $\Sigma_d$ over fermionic ladder monomials and write it in Gram form $\Phi_d(G) = \vec{o}_d^{\dagger} G \vec{o}d$ with $G \succeq 0$, then derive the linear truncation equations $A{\mathrm{trunc}}(d),\mathrm{vec}(G)=0$ whose nullspace removes all high-degree terms. Theorem II.1 proves this nullspace construction captures every at-most-two-body operator admitting a degree-$d$ SOS representation.

2. Weighted extension with particle-number multipliers. Section III augments the cone with multiplier terms $f_z r + r f_z^{\dagger}$ where $r = N - \eta\mathbb{1}$, giving the weighted cone ${}{\mathrm{wn}}\Sigma_d$, and derives the combined augmented nullspace condition $\begin{bmatrix} A{\mathrm{SOS}} & A_{\mathrm{sym}} \end{bmatrix} (\mathrm{vec}(G), z)^T = 0$ that allows joint cancellations between the SOS part and the multiplier part.

3. Completeness of the weighted SOS at level $d=n$. The construction establishes completeness in two stages (stated in the introduction and proved in Section III, with Theorem III.1 stated as "Completeness of the weighted SOS nullspace"): at $d=n$ the weighted SOS cone contains every Hermitian, particle-conserving, at-most-two-body operator nonnegative on the $n$-particle subspace, and the hierarchy terminates at the level equal to the number of particles.

4. Reduction of multiplier degree and recovery of known v2RDM conditions. The paper proves that particle-number multipliers of degree at most two suffice to preserve the supremal energy bound at every level $d \geq 2$; higher-degree multiplier contributions can be absorbed into the SOS part and degree-two multipliers with an arbitrarily small decrease in the lower-bound energy certificate. It also shows canonical v2RDM constraints — the DQG conditions and the $\mathrm{DQGT_1T_2T_2'}$ conditions — appear as restricted, non-general subsets of the wSOS construction.

5. Open-source implementation. The degree-$d$ weighted SOS two-particle witness generator is provided as a component of the p†q fermionic computer algebra library, and the authors note the code extends to arbitrary weighted SOS with minor modifications and that the nullspace construction is not fermion-specific.

Main Findings

• Exhaustive capture of two-body witnesses: The nullspace of the truncation matrix contains exactly the Gram matrices whose terms of degree greater than four cancel, so no admissible at-most-two-body SOS witness is missed (Theorem II.1, proved via mutual inclusion of the cone $C_d$ and $\Sigma_{d,\leq 2}$).
• Unweighted SOS is insufficient for fixed particle number: Corollary II.2 notes every unweighted witness is globally positive semidefinite on Fock space and hence valid on $\mathcal{H}p$, but the authors explicitly remark that $\Sigma{d,\leq 2}$ is incomplete for fixed-particle simulations because some valid $p$-particle witnesses have negative eigenvalues on sectors with $N \neq p$.
• Exact energy for a three-fermion Richardson Hamiltonian: Solving the associated semidefinite program numerically recovers the exact ground-state energy of a Richardson-type pairing Hamiltonian in the three-fermion sector within solver accuracy. (No numerical energy value or basis size is reported in the supplied text.)
• Three-positivity alone adds little: The authors report that approximate 3-positivity does not improve the lower bound certificate substantially beyond 2-positivity for this model — an expected result that highlights the gap in traditional v2RDM constraints — whereas the complete degree-three weighted SOS formulation recovers the exact energy.
• Multiplier degree can be capped at two: Degree-at-most-two particle-number multipliers preserve the supremal energy bound at every level $d \geq 2$ of the complete fermionic SOS hierarchy.
• Duality bookkeeping made explicit: Converting the SOS program to its dual yields a primal program with unrestricted Lagrange multipliers $u_6$ and $u_8$ for the constraints $q_6 = 0$ and $q_8 = 0$; only the assembled moment matrix $M(x,u) = M_0(x) + A^{}(u)$ is constrained positive semidefinite. Because $\langle A^{}(u), G\rangle = 0$ for admissible $G$, the witness cannot distinguish $M_0(x)$ from $M_0(x) + A^{*}(u)$, giving an equivalence class of lifted RDM solutions.
• Agreement with prior work: The proofs reinforce the solution put forward independently by Mazziotti (2012, 2023, 2026), and identify the nullspace condition of Mazziotti (2024) as the requirement for instantiating the dual formulation, and the additional variance term of Mazziotti (2026) as precisely the particle-number multiplier term in weighted SOS.
• Relation to Nakata's result: The paper notes Nakata's recent use of weighted sum-of-squares to show that a 2-RDM satisfying the $(2,q)$-conditions also satisfies the $q$-positivity conditions.
• Contraction relations from multipliers: The numerical study affords a derivation of how contraction relations between RDMs emerge from the multiplier polynomials in wSOS.

Methodology in Plain English

The authors work entirely at the level of operator algebra rather than explicit wavefunctions. They pick a set of elementary fermion creation and annihilation operator products up to some degree $d$, and consider all possible sums of squares of those operators. Any such sum of squares is automatically nonnegative on the whole Fock space, and it can be written compactly as a "Gram matrix" sandwiched between vectors of monomials — the Gram matrix being positive semidefinite is what encodes the sum-of-squares structure.

The problem is that a generic sum of squares contains six- and eight-body terms, which a 2-RDM cannot evaluate. So they normal-order every product, collect the coefficients of each high-degree operator, and demand those coefficients vanish. That demand is a set of linear equations in the Gram matrix entries, so the admissible set is simply "positive semidefinite matrices inside a linear subspace" — a semidefinite program. To let witnesses be negative on the wrong particle-number sectors (which is essential, since valid witnesses for $n$ particles need only be nonnegative on that sector), they add particle-number multiplier terms of the form $f(N - \eta\mathbb{1}) + (N-\eta\mathbb{1})f^{\dagger}$ and allow the multiplier and the sum of squares to cancel each other's high-degree terms jointly. They then do the algebra symbolically with the p†q library, using index-permutation and contraction-ordering bookkeeping to map normal-ordered products back onto the Gram matrix entries, so that the nullspace equations are generated generically rather than for one fixed orbital count.

For the numerical check they build this hierarchy for a Richardson pairing Hamiltonian and solve the resulting semidefinite program.

Why This Matters

The 2-RDM is far more compact than the wavefunction yet contains everything needed for molecular energies and most equilibrium chemical properties; the U.S. National Research Council in 1995 named $n$-representability one of the ten most prominent research challenges in quantum chemistry. The problem is known to be QMA-complete, so practical progress comes from hierarchies of necessary conditions — and this paper supplies a systematic, software-backed way to generate the complete set of two-body conditions at each level rather than guessing individual constraints.

• Quantum chemistry: tighter lower bounds on ground-state energies than traditional v2RDM conditions such as DQG and $\mathrm{DQGT_1T_2T_2'}$ provide.
• Condensed matter simulation: lower-bound certificates for fermionic lattice models, with the Richardson pairing model as the demonstrated example.
• Algorithm and software development: a symbolic generator of witnesses that can be used to design efficient computational methods instead of hand-deriving each new constraint.
• Beyond fermions: the nullspace construction is not fermion-specific and applies to other outer approximation methods.

Industry relevance: the work originates from Google Quantum AI together with Florida State University, and the implementation ships inside an open-source fermionic algebra library, positioning it as infrastructure for benchmarking quantum algorithms and classical certification tools rather than a standalone result.

Future Directions

• Scaling the semidefinite programs: the paper builds a complete hierarchy but does not report solving it beyond the three-fermion Richardson sector; how the construction behaves for larger particle numbers and realistic orbital sets remains open.
• Exploiting symmetry structure: the authors note that the block structure of $G$ can be extended to additional symmetries such as $S_z$ (discussed in their Appendix D, reproducing the block structure of Ref. [31]), suggesting reduced-cost formulations as a direction.
• Bridging to approximate positivity conditions: since approximate 3-positivity barely improved on 2-positivity in the demonstration, understanding exactly which wSOS levels give meaningful gains over standard v2RDM constraints is an open practical question.
• Efficient lower-bound certificates: the stated motivation is to cast the general solution in a symbolic form amenable to computation "to aid in the designing of efficient computational methods for lower bound certificates" — turning the completeness proof into a usable computational tool is the natural next step.

Target Audience

Researchers in electronic structure theory and quantum chemistry working on reduced density matrix methods; mathematicians and computer scientists working on the quantum marginal problem, non-commutative polynomial optimization, and the NPA hierarchy; quantum information theorists interested in QMA-completeness and correlation certification; and scientific software developers who want a working symbolic implementation of fermionic sum-of-squares witnesses. The paper is self-contained by design — the authors state they found the existing solution difficult to follow and wrote this account to add expository value — but the operator-algebra formalism makes it most useful to readers already comfortable with second quantization and semidefinite programming.

Authors’ abstract

We present an instantiation of the constructive solution to the fermionic quantum marginal problem by building the complete family of two-body witnesses at each level of a weighted sum-of-squares hierarchy. Our construction combines positive-semidefinite Gram matrices with explicit linear nullspace constraints that exactly cancel all higher-body contributions following a similar idea to the existing solution but made explicit in open source software. We demonstrate the weighted sum-of-squares construction numerically by solving the associated semidefinite program and recovering the exact ground-state energy of a Richardson Hamiltonian in the three-fermion sector. We further prove that higher-degree particle-number multipliers are unnecessary when the full sum-of-squares generating algebra is used.