What These Numbers Mean

The absolute difference between the VQE energy and the exact ground state energy of the same qubit Hamiltonian, found by exact diagonalisation (CASCI) in the same active space.

It is not measured against experiment, and not against a complete-basis-set limit. That is deliberate. A real molecule’s energy error comes mostly from the basis set and the choice of active space, which every method in the table shares equally. Measuring against the exact answer for the same problem isolates the part the quantum algorithm is actually responsible for. It is a strict test of the algorithm, not a claim about chemistry.

1.6 mHa is chemical accuracy — about 1 kcal/mol, roughly the point at which a computed reaction energy becomes useful to a chemist.

10 mHa is the QEncode certification threshold. It is a deliberately looser bar, set so that an entry can be certified as a well-executed, reproducible calculation without having to also be chemically useful. An entry can be certified and still not be accurate enough to make a prediction with. The colour of the bar tells you which side of each line you are on, and entries under 1.6 mHa carry a Chem. accuracy badge.

How much room an entry has before it stops certifying, shown as a share of the 10 mHa threshold. An entry certified at 9.98 mHa and one certified at 0.001 mHa are both “certified”; they are not the same. Under 20% is flagged as thin.

Margin says how far the energy can move, not how far it will. Every certified entry has been re-run on a second machine with the identical pinned software, seeds and thread count. Most move by less than a microhartree; the largest movement is 1.6 × 10⁻² Ha. Three entries certified at 3.8, 4.4 and 4.5 mHa regenerate at 11.8, 10.9 and 20.4 mHa on the other processor and no longer certify. Each still reproduces exactly on the machine that made it, which is what its provenance attests, so they are flaggedfragile rather than withdrawn. The full table and the cause: docs/CROSS_MACHINE.md.

Every certified entry in the suite stopped the moment its gap cleared the threshold. For UCCSD and hardware-efficient runs that is a multistart loop halting at the first restart that certifies — most used one restart of ten. For ADAPT-VQE it is the operator loop halting at the first operator count that certifies. The cell shows how much budget was used when it stopped. The seven research entries show full: they used everything and never certified.

This changes what the accuracy column means. The gap of an early-stopped run is where the rule fired, not the smallest the configuration can reach. Measured: H₄ under ADAPT-VQE certifies with a single operator at 9.94 mHa and stops; run to convergence it reaches 0.05 mHa with twenty-two. Every UCCSD entry re-run with a gradient-based optimiser improved, C₄H₄ from 7.9 mHa to 10⁻⁷ mHa, and N₂ from a research entry at 10.8 mHa to a certified one at 0.05 mHa. An ADAPT entry at 9.94 mHa and a UCCSD entry at 10⁻⁴ mHa are therefore not two points on one scale, and this column is here so that a reader comparing them knows it. The measurements: docs/SUITE_BALANCE.md.

The classical optimiser the run used, because it is what decides whether a thin margin is dangerous. A gradient-free optimiser (COBYLA) chooses its next step by comparing two nearly equal energies, so a difference in the thirteenth decimal — threaded arithmetic, a different BLAS, a different NumPy — can flip a comparison and send the run into a different local minimum. A gradient-based one (L-BFGS-B) removes that comparison from the optimiser step, and this was believed to make it immune. It does not. Two L-BFGS-B entries lost certification across machines, because their multistart loop stops at the first attempt that certifies, and “is this attempt good enough” is itself a comparison of nearly equal numbers. What matters is whether anydecision the run makes compares near-equal quantities, not whether the optimiser consumes gradients.

The ansatz matters as much as the optimiser. ADAPT-VQE selects its operators by analytic gradient, so its structure is gradient-determined even with a COBYLA inner optimiser. Measured on H₄, where both variants exist for the same molecule and environment: ADAPT/COBYLA moved 3.4 × 10⁻⁸ Ha across environments; HEA/COBYLA moved 8.8 × 10⁻⁴ Ha — 25,595 times more. The chip turns amber for the amplifying combination: gradient-free optimiser on an unstructured ansatz.

Where an entry has actually been re-run on another environment, a Re-run badge shows the measured outcome — robust, marginal (it passed, but moved further than its own margin and survived on the sign), or fragile. Where it has not, but the margin is thin and the configuration amplifies, it shows At risk: a prediction, not a measurement.

Every certified energy is an exact statevector result: the algorithm’s quality with no device error at all. The Noise column measures what the same circuit — same parameters, nothing re-optimised — returns as a density matrix with a depolarizing channel after every gate, at rates near good current superconducting hardware (5 × 10⁻⁴ per one-qubit gate, 5 × 10⁻³ per two-qubit gate). It is a penalty in millihartree: how far the energy rose.

It is a bias, not a variance. Shot noise averages away with more samples. Gate noise drives the state toward the maximally mixed state, whose energy is the mean of the spectrum, so it can only push the energy up and no number of shots removes it. On this suite the smallest penalty for a multi-qubit circuit is 22 mHa against a 10 mHa bar, and deep UCCSD circuits lose more than a hartree. Hover a value for the gap under noise and for the residual after zero-noise extrapolation, the standard mitigation, which recovers the shallow circuits and not the deep ones.

Measured, not certified. No entry’s status depends on this column. It is absent for entries above ten qubits, where exact density-matrix simulation is out of reach, and for two entries whose stored circuit cannot be rebuilt from the record. Method, conventions and every record: docs/NOISY_TIER.md.

The correlation energy that CCSD(T) — the classical gold standard for molecules this size — recovers on the identical problem.

CCSD(T) is not beaten here. It is cheaper and, on the full system, more accurate than anything in this table. The column exists so you can see the scale of the correlation energy the quantum calculation is trying to capture.

The “Beats Classical” flag is weaker than it sounds, and it is worth reading carefully. It is set when the VQE error is smaller than the magnitude of the correlation energy CCSD(T) recovers — |EVQE − ECASCI| < |Ecorr(CCSD(T))|. That says the VQE error is small on the scale of correlation energy. It is not a claim that VQE outperforms CCSD(T), and it is currently true for every entry in the suite — certified or not — so it does not discriminate between them, and it is not the certification criterion. Each entry carries this definition in its own beats_classical_definition field.

Two-qubit gate count and circuit depth after transpilation. On today’s noisy hardware the two-qubit gate is the dominant error source, so this is what decides whether a circuit is runnable now.

Blank cells are intentional. UCCSD circuits are built from exponential Pauli operators that stay symbolic until they are compiled for a specific hardware target. A raw pre-transpilation count would not be comparable to a real one, so it is left empty rather than filled with a misleading number.

An estimate of the non-Clifford cost of solving the same Hamiltonian on an error-corrected machine. In that setting Clifford gates are comparatively cheap and T gates dominate, so this count — not the two-qubit count — sets the runtime.

It is derived, not measured. It comes from classical post-processing of the stored Pauli decomposition, assuming qubitized quantum phase estimation: from the Hamiltonian one-norm λ = Σ|ha|, reaching precision ε needs ⌈πλ/2ε⌉ applications of a walk operator; each costs about 2L+μ Toffoli gates for L Pauli terms; each Toffoli is four T gates.

Wall-clock runtime is deliberately not published. Converting T gates to hours requires assuming a code distance and a physical error rate, and those assumptions would dominate the answer — a single number would say more about the assumption than about the molecule. Logical qubit counts are the honest stopping point.

An entry carries the badge when no other entry for the same molecule is better on both accuracy and hardware cost at once.

It marks a genuine trade-off. An entry without the badge is beaten on both axes by something else in the table, so there is no set of priorities under which it is the right choice for that molecule. Roughly half the suite is strictly dominated in this sense.