Certifying N₂: QEncode Benchmarks the Triple Bond

Every quantum chemistry benchmark suite eventually has to reckon with N₂. The nitrogen molecule appears in DARPA's QB-GSEE target list, in every major VQE paper as a stretch goal, and in textbooks as the archetype of a strongly correlated system. It is hard for a specific, well-understood reason: the triple bond — one σ and two π bonds — creates a six-orbital active space where the Hartree-Fock reference is a poor starting point and standard single-reference methods struggle.

We added N₂ to the QEncode Suite v4.0 catalog at the start of this year. It took until Suite v4.1 to certify it. This post explains what made it hard, what we changed, and what the certified result means.

Why N₂ is different

The molecules we certified in v4.0 — H₂, HF, LiH, BeH₂, H₂O, NH₃ — all share a property: Hartree-Fock molecular orbitals cleanly partition the active space. You freeze the core, pick the frontier orbitals, and the VQE circuit has a sensible starting point. N₂ breaks this pattern.

The [6e, 6o] active space for N₂ covers the full triple-bond manifold: σ, σ*, πₓ, πₓ*, πᵧ, πᵧ*. In the cc-pVDZ basis, Hartree-Fock does not cleanly separate these six orbitals from the rest of the virtual space. The π and σ* orbitals mix with higher virtuals, and the resulting active space is poorly conditioned for VQE. The circuit starts far from the CASCI minimum and COBYLA, a gradient-free optimizer, cannot navigate there reliably from that starting point.

This is not a deficiency of the benchmark — it is the point. N₂ exposes the difference between a well-set-up VQE calculation and one that is just running the default pipeline.

The fix: CASSCF orbital optimization

The solution is to pre-optimize the molecular orbitals before building the qubit Hamiltonian. CASSCF — Complete Active Space Self-Consistent Field — runs an iterative procedure that rotates the orbital basis to minimize the energy of the active space directly. For N₂, this means the six orbitals passed into the VQE circuit are already the best possible representation of the triple-bond manifold rather than the generic Hartree-Fock canonical orbitals.

In practice the CASSCF and CASCI energies for N₂ are identical to twelve decimal places (-109.0899581524 Ha) — confirming that the orbital basis is fully converged before a single VQE evaluation runs. The qubit Hamiltonian is then built from these optimized orbitals, giving UCCSD a sensible starting point.

We added CASSCF support to the generator in Suite v4.1 as a first-class flag:

python scripts/generate_entry_v4.py \
  --molecule N2 --mapping jordan_wigner \
  --ansatz-type uccsd --orbital-opt casscf \
  --multistart 1 --max-iter 10000

12 qubits, Z2 tapering, 404 parameters

N₂ in the JW encoding with a [6e, 6o] active space requires 12 qubits — one per spin-orbital. Before running VQE we apply Z2 symmetry tapering, which identifies four conserved symmetry sectors and eliminates them, reducing the circuit to 8 qubits. The tapered Hamiltonian has 378 Pauli terms. Exact diagonalization confirms the ground state energy at -109.0899581524 Ha, matching CASCI to 1.1 × 10⁻¹³ Ha — the taper is exact.

On the ansatz side, UCCSD for a [6e, 6o] system generates all symmetry-allowed single and double excitation operators. After tapering, this produces 404 variational parameters. For comparison, H₂ UCCSD has 4 parameters and LiH has around 20. The jump to 404 changes the optimization problem fundamentally.

Why HEA failed — and what it tells us

Before committing to UCCSD, we ran the hardware-efficient ansatz with 4 layers (40 parameters) and 30 random restarts at 500 iterations each. Every single restart hit the iteration limit without converging. The best result after 15,000 function evaluations was a gap of 0.12 Ha — twelve times the certification threshold.

This is a meaningful data point, not a failure to discard. HEA is a generic circuit with no chemistry built in. For small, weakly correlated molecules it is competitive with UCCSD. For N₂, the rugged energy landscape with 40 free parameters and no physical structure means COBYLA gets trapped. The benchmark is doing its job: distinguishing ansatz families under real conditions.

The HEA result is recorded in the Research tab of the leaderboard as a validated (not certified) entry — honest, reproducible, and informative.

The certified result

With CASSCF orbitals, UCCSD, and 10,000 COBYLA iterations starting from the HF reference (all-zero parameters, corresponding to the Hartree-Fock state), the optimizer reached a VQE energy of -109.0879426090 Ha against a CASCI reference of -109.0899581524 Ha.

The gap of 2.015 mHa is certified — well under the 10 mHa threshold. It is also close to chemical accuracy (1.6 mHa), though not quite there. For a [6e, 6o] strongly correlated system on 8 qubits with a gradient-free optimizer, this is a solid result. The entry carries the beats_classical flag: the VQE correlation energy exceeds the CCSD(T) correlation energy, meaning the quantum ansatz captures more electron correlation than the gold-standard classical approximation on this active space.

What CASSCF changes — and when it matters

CASSCF orbital optimization is now a first-class option in the QEncode generator but is not the default. For most molecules in the suite — H₂, HF, BeH₂, H₂O, NH₃ — HF orbitals work well and CASSCF adds computation time without improving the result. N₂ is different because the triple bond creates near-degeneracy in the orbital energies that HF cannot resolve.

The practical rule: if a molecule has multiple bonds of the same type (triple bonds, conjugated π systems, transition metal d-orbitals), CASSCF orbital optimization is recommended. For single-bond dominated systems, HF is sufficient. This is noted in the molecule catalog and surfaced as the CASSCF badge on the leaderboard for entries that used it.

Relevance to DARPA QB-GSEE

DARPA's Quantum Benchmarking program (QB-GSEE) targets ground-state energy estimation for molecules where classical methods are insufficient. N₂ appears explicitly in their target list as a near-term benchmark candidate. Our certified result at cc-pVDZ with a [6e, 6o] active space is directly comparable to the QB-GSEE target specification.

The QEncode certification framework — provenance hash, CASCI reference, trust level, signed entry — provides exactly the kind of verifiable, reproducible evidence that program-level benchmarking requires.

What comes next: benzene

The natural next step is benzene — the first aromatic molecule in the suite. Benzene has a [6e, 6o] π active space with the same qubit count as N₂ but more complex correlation structure from the conjugated ring. It also has direct relevance to pharmaceutical chemistry, where aromatic rings appear in nearly every drug molecule.

We expect CASSCF orbital optimization to be equally critical for benzene, and we will need the GPU backend (lightning.gpu) for practical runtime at this scale. The Suite v4.2 milestone is a certified benzene entry on the public leaderboard.

The N₂ result and the new generator capabilities — CASSCF, GPU backend, checkpoint restart, early-stop certification — are available now in the open-source repository. All entries are reproducible with a single command using the pinned requirements-v4.txt environment.