SQD on N₂ (Worked Example)

qc/algorithms

A concrete, end-to-end run of Sample-Based Quantum Diagonalization (SQD) on molecular nitrogen (N₂), stretched to 2.0 Å (roughly double its ~1.1 Å equilibrium bond length, chosen because a stretched bond has stronger electron correlation and a bigger gap between the cheap Hartree-Fock guess and the true ground state). The pipeline: PySCF builds the molecule in a cc-pvdz basis (28 spatial orbitals total), freezes the 2 inert core orbitals, and leaves an active space of 26 orbitals with 5 electrons per spin — a Hilbert space of $\left(\genfrac{}{}{0ex}{}{26}{5}\right)=65,780$ configurations per spin sector, over 4 billion states total, far beyond brute-force diagonalization. Key insight: the lab makes “quantum utility” a measurable, falsifiable claim rather than a slogan — it defines a purely classical brute-force baseline, then shows the quantum-sampled subspace beats it once augmented, which is the actual empirical bar for “the quantum computer contributed something classical enumeration alone could not.”

State preparation and hardware mapping

The LUCJ ansatz circuit is built from CCSD amplitudes (ffsim.UCJOpSpinBalanced.from_t_amplitudes) and prepared via Jordan-Wigner encoding on $2\times 26=52$ qubits (26 for spin-α, 26 for spin-β). The same circuit is mapped to two different IBM device topologies to compare hardware connectivity, directly answering the “which topology fits this circuit better” question:

• Heron (ibm_kingston): the α and β qubit chains only touch at every 4th site — only 5 hardware-native α–β crossing pairs are available ([(p, p) for p in range(0, 26, 4)], truncated to 5).
• Nighthawk (ibm_miami): the α and β rails sit adjacent to each other, so up to 24 α–β pairs can be made hardware-native — far denser cross-spin connectivity than Heron, at the cost of needing a custom initial_layout that keeps each same-spin chain connected and keeps α–β pairs close.

Sampling and configuration recovery

2,000 shots are drawn from the mapped circuit; each shot is a 52-bit string splitting into an α-half and β-half, one proposed electron configuration. On real hardware, it’s normal for zero of the 2,000 raw shots to have the correct particle number (5 electrons per spin) — bit-flip noise reliably pushes samples out of the physical sector. Configuration recovery repairs this using the concrete weight function $W_{0\to 1}(\text{occ})=e^{\text{occ}}-1$ (favor filling orbitals that are usually occupied) and its mirror $W_{1\to 0}(\text{occ})=W_{0\to 1}(1-\text{occ})$, seeded from the Hartree-Fock occupancy prior and refined each iteration using the diagonalizer’s own orbital-occupancy output.

The self-consistent loop and the reference-subspace result

The full loop (recover → subsample into batches → build a Slater-determinant subspace via qiskit_addon_sqd.fermion.solve_sci_batch → diagonalize → feed occupancies back) runs for several iterations, tracking the best (lowest) energy seen. On its own, this typically lands below Hartree-Fock but above CCSD — proof the quantum samples carry real signal, but not yet chemistry-grade. The lab then builds a purely classical baseline: enumerate the 300 lowest-excitation-rank Slater determinants around the Hartree-Fock reference (brute-force, no chemical insight, via itertools.combinations) and diagonalize in that fixed set alone. This classical reference subspace typically beats pure-sampling SQD — proof that hardware noise is dropping chemically important near-HF configurations that a systematic classical enumeration would keep. The final step pins the 100 best classical reference determinants as a guaranteed floor and lets quantum sampling fill the remaining subspace slots each iteration. If this reference-augmented SQD result beats the classical-only reference subspace, that is quantum utility demonstrated empirically: the quantum device contributed configurations that brute-force classical enumeration could not find on its own, and that contribution measurably lowered the energy (and typically tightens the run-to-run variance too, since the pinned floor stabilizes every batch).

Related

• Sample-Based Quantum Diagonalization (SQD) — the general method this is a worked instance of
• The LUCJ Ansatz — the state-preparation circuit used here, including the CCSD-seeded orbital rotations
• Quantum Chemistry — QPE on H2 (Worked Example) — the QPE-flavored worked example on a much smaller molecule; contrast the fault-tolerant-style precision-vs-depth tradeoff there with SQD’s noisy near-term sampling-plus-classical-cleanup approach here
• Quantum Utility vs Quantum Advantage — the general framing of what “quantum utility” means; this note is a concrete instance of it
• The Ground-State Problem

Self-Check

• Why is it unsurprising (not a bug) that zero of the 2,000 raw hardware shots have the correct particle number before configuration recovery?
• What exactly does it mean for reference-augmented SQD to beat the classical reference subspace, and why is that the specific bar for “quantum utility” rather than just “SQD beat Hartree-Fock”?
• Why would the Nighthawk topology allow far more native α–β interaction pairs than Heron, and what does that cost in the initial-layout design?