What These Numbers Mean
Every column on the leaderboard answers a narrow question, and most of them are easy to over-read. This page says what each one measures, what it is measured against, and where it stops.
The short version
An entry is one quantum chemistry calculation, run end to end, with every input recorded so it can be rebuilt. The gap says how close it got to the right answer for its own problem. The CCSD(T) column says how a good classical method does on the same problem — usually better. The gate and T-gate columns price the circuit on today’s hardware and on a future error-corrected machine respectively.
The accuracy columns
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.
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 certified entry, so it does not discriminate between them. Each entry carries this definition in its own beats_classical_definition field.
The cost columns
These price two different eras of hardware, and a circuit can look cheap on one and expensive on the other.
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.
The three views
The tabs re-rank the same entries and show the columns relevant to each question:
- Accuracy — ranked by gap. Shows the classical baseline, so you can see the distance to CCSD(T).
- Cost — ranked by hardware cost. Shows gate counts, depth and T-gate estimates.
- Balanced — a combined score, and the only view that shows accuracy and cost on the same row. This is the view to use if you want to judge whether an approach is worth its resources.
What these numbers do not tell you
Stated plainly, because each is easy to over-read:
- These are exact statevector simulations, not hardware runs. There is no gate noise, no readout error and no device topology. A circuit that reaches 2 mHa here would not reach 2 mHa on a real device today.
- The shot budget is not in the cost columns. Gate counts price one circuit execution; a real VQE run needs millions to hundreds of millions of executions, and how those are spent changes the answer substantially. That cost is measured separately in our work on shot allocation.
- A small gap is not a chemical prediction. It means the algorithm solved its own problem well. Basis-set and active-space error sit on top and are usually much larger.
- Classical methods still win at these sizes. Nothing here is a claim of quantum advantage, and the CCSD(T) column is there so that is visible at a glance.
Every entry records its full provenance — Hamiltonian, ansatz, optimiser, seed, package versions and code commit — so any number here can be rebuilt from scratch. See the methodology for how entries are produced and certified, or the leaderboard to read the numbers themselves.