Public rankings include only certified entries with official trust filtering (2026-09-09).
Entry counts can differ by molecule when some configurations are not yet certified. All runs use the cc-pVDZ basis set.
What these numbers mean → what the gap is measured against, why CCSD(T) is shown, and how T-gate estimates are derived.
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47 certified entries match current filters
Ranked by lowest |EVQE − ECASCI| error gap. Chemical accuracy threshold: 1.6 × 10⁻³ Ha.
| Rank | Molecule | Mapping | Ansatz | Error Gap | Margin | Stop | Noise | CCSD(T) corr. | Status | |
|---|---|---|---|---|---|---|---|---|---|---|
H2 cc-pvdz | Parity | UCCSD COBYLA | 1.05 × 10⁻⁹ | 100% | 1/10 | +0.4 | 3.46 × 10⁻² | |||
H2 cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 1.05 × 10⁻⁹ | 100% | 1/10 | +0.4 | 3.46 × 10⁻² | |||
H2 cc-pvdz | Bravyi-Kitaev | UCCSD COBYLA | 1.05 × 10⁻⁹ | 100% | 1/10 | +0.4 | 3.46 × 10⁻² | |||
| #4 | HF cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 1.83 × 10⁻⁹ | 100% | 1/10 | +0.5 | 2.11 × 10⁻¹ | ||
| #5 | HF cc-pvdz | Jordan-Wigner | HEA COBYLA | 4.76 × 10⁻⁹ | 100% | 1/10 | +1.4 | 2.11 × 10⁻¹ | ||
| #6 | HF cc-pvdz | Bravyi-Kitaev | UCCSD COBYLA | 5.85 × 10⁻⁹ | 100% | 1/10 | +0.5 | 2.11 × 10⁻¹ | ||
| #6 | HF cc-pvdz | Parity | UCCSD COBYLA | 5.85 × 10⁻⁹ | 100% | 1/10 | +0.5 | 2.11 × 10⁻¹ | ||
| #8 | H2 cc-pvdz | Parity | HEA COBYLA | 8.59 × 10⁻⁹ | 100% | 1/10 | +1.2 | 3.46 × 10⁻² | ||
| #8 | H2 cc-pvdz | Bravyi-Kitaev | HEA COBYLA | 8.59 × 10⁻⁹ | 100% | 1/10 | +1.2 | 3.46 × 10⁻² | ||
| #10 | HF cc-pvdz | Bravyi-Kitaev | HEA COBYLA | 1.27 × 10⁻⁸ | 100% | 1/10 | +1.4 | 2.11 × 10⁻¹ | ||
| #10 | HF cc-pvdz | Parity | HEA COBYLA | 1.27 × 10⁻⁸ | 100% | 1/10 | +1.4 | 2.11 × 10⁻¹ | ||
| #12 | H2 cc-pvdz | Jordan-Wigner | HEA COBYLA | 1.36 × 10⁻⁸ | 100% | 1/10 | +1.2 | 3.46 × 10⁻² | ||
| #13 | BeH2 cc-pvdz | Jordan-Wigner | HEA COBYLA | 5.43 × 10⁻⁸ | 100% | 1/10 | +34.4 | 6.90 × 10⁻² | ||
| #14 | BeH2 cc-pvdz | Parity | HEA COBYLA | 5.43 × 10⁻⁸ | 100% | 1/10 | +34.4 | 6.90 × 10⁻² | ||
| #15 | H2O cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 1.44 × 10⁻⁷ | 100% | 1/10 | +893 | 2.16 × 10⁻¹ | ||
| #16 | water_dimer cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 1.93 × 10⁻⁶ | 100% | 1/10 | +1332 | 4.35 × 10⁻¹ | ||
| #17 | BeH2 cc-pvdz | Parity | UCCSD COBYLA | 2.40 × 10⁻⁶ | 100% | 1/10 | +466 | 6.90 × 10⁻² | ||
| #18 | LiH cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 2.86 × 10⁻⁶ | 100% | 1/10 | +2720 | 3.11 × 10⁻² | ||
| #19 | BeH2 cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 6.65 × 10⁻⁶ | 100% | 1/10 | +263 | 6.90 × 10⁻² | ||
| #20 | NH3 cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 3.24 × 10⁻⁵ | 100% | 1/10 | +1316 | 2.09 × 10⁻¹ | ||
| #21 | LiH cc-pvdz | Jordan-Wigner | HEA COBYLA | 9.58 × 10⁻⁵ | 99% | 1/10 | +87.2 | 3.11 × 10⁻² | ||
| #22 | water_dimer cc-pvdz | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 1.11 × 10⁻⁴ | 99% | 1 op | +65.9 | 4.35 × 10⁻¹ | ||
| #23 | water_dimer cc-pvdz | Parity | HEA COBYLA | 1.14 × 10⁻⁴ | 99% | 1/10 | +81.0 | 4.35 × 10⁻¹ | ||
| #24 | water_dimer cc-pvdz | Jordan-Wigner | HEA COBYLA | 3.32 × 10⁻⁴ | 97% | 1/10 | +62.5 | 4.35 × 10⁻¹ | ||
| #25 | H2O cc-pvdz | Parity | HEA COBYLA | 3.99 × 10⁻⁴ | 96% | 1/10 | +91.4 | 2.16 × 10⁻¹ | ||
| #26 | H2O cc-pvdz | Jordan-Wigner | HEA COBYLA | 4.03 × 10⁻⁴ | 96% | 1/10 | +53.7 | 2.16 × 10⁻¹ | ||
| #27 | NH3 cc-pvdz | Parity | HEA COBYLA | 7.34 × 10⁻⁴ | 93% | 3/10 | +80.0 | 2.09 × 10⁻¹ | ||
| #28 | H2CO cc-pvdz | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 1.12 × 10⁻³ | 89% | 1 op | +81.7 | 3.45 × 10⁻¹ | ||
| #29 | NH3 cc-pvdz | Jordan-Wigner | HEA COBYLA | 1.88 × 10⁻³ | 81% | 1/10 | +67.3 | 2.09 × 10⁻¹ | ||
| #30 | H4 cc-pvdz | Jordan-Wigner | UCCSD COBYLA | 2.22 × 10⁻³ | 78% | 1/3 | +1223 | 8.73 × 10⁻² | ||
| #31 | C4H6 cc-pvdz | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 2.83 × 10⁻³ | 72% | 1 op | +77.8 | 6.26 × 10⁻¹ | ||
| #32 | LiH cc-pvdz | Parity | HEA COBYLA | 3.37 × 10⁻³ | 66% | 1/10 | +95.6 | 3.11 × 10⁻² | ||
| #33 | C4H4 cc-pvdzCASSCF | Parity | HEA COBYLA | 3.83 × 10⁻³ | 62% | 3/10 | +79.5 | 5.96 × 10⁻¹ | ||
| #34 | N2 cc-pvdzCASSCF | Parity | HEA L-BFGS-B | 4.40 × 10⁻³ | 56% | 2/5 | +1353 | 3.25 × 10⁻¹ | ||
| #35 | H4 cc-pvdz | Parity | HEA COBYLA | 4.49 × 10⁻³ | 55% | 1/10 | +130 | 8.73 × 10⁻² | ||
| #36 | N2 cc-pvdzCASSCF | Jordan-Wigner | HEA L-BFGS-B | 4.51 × 10⁻³ | 55% | 1/5 | +1307 | 3.25 × 10⁻¹ | ||
| #37 | C4H4 cc-pvdzCASSCF | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 5.96 × 10⁻³ | 40% | 2 ops | +168 | 5.96 × 10⁻¹ | ||
| #38 | C4H4 cc-pvdzCASSCF | Jordan-Wigner | UCCSD COBYLA | 7.92 × 10⁻³ | 21% | 1/10 | +1164 | 5.96 × 10⁻¹ | ||
| #39 | benzene cc-pvdzCASSCF | Jordan-Wigner | HEA L-BFGS-B | 8.74 × 10⁻³ | 13% | 1/5 | +794 | 8.74 × 10⁻¹ | ||
| #40 | N2 cc-pvdzCASSCF | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 8.83 × 10⁻³ | 12% | 25 ops | +3210 | 3.25 × 10⁻¹ | ||
| #41 | H6 cc-pvdzCASSCF | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 9.27 × 10⁻³ | 7.3% | 28 ops | +2301 | 1.36 × 10⁻¹ | ||
| #42 | H4 cc-pvdz | Jordan-Wigner | HEA COBYLA | 9.28 × 10⁻³ | 7.2% | 1/10 | +127 | 8.73 × 10⁻² | ||
| #43 | benzene cc-pvdzCASSCF | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 9.54 × 10⁻³ | 4.6% | 11 ops | +1738 | 8.74 × 10⁻¹ | ||
| #44 | C4H4 cc-pvdzCASSCF | Jordan-Wigner | HEA COBYLA | 9.64 × 10⁻³ | 3.6% | 8/10 | +97.3 | 5.96 × 10⁻¹ | ||
| #45 | H8 cc-pvdzCASSCF | Jordan-Wigner | adapt ADAPT-VQE/L-BFGS-B | 9.80 × 10⁻³ | 2.0% | 98 ops | — | 1.85 × 10⁻¹ | ||
| #46 | H4 cc-pvdz | Jordan-Wigner | adapt ADAPT-VQE/COBYLA | 9.94 × 10⁻³ | 0.6% | 1 op | +126 | 8.73 × 10⁻² | ||
| #47 | H10 cc-pvdzCASSCF | Jordan-Wigner | adapt ADAPT-VQE/L-BFGS-B | 9.98 × 10⁻³ | 0.2% | 300 ops | — | 2.34 × 10⁻¹ |
Legend
UCCSD — Unitary Coupled Cluster
Chemistry-motivated ansatz that applies all single and double electronic excitations from the Hartree-Fock reference state. Produces the best energies because the circuit is designed around the molecule's physics.
Why 2Q gates and depth show “—”:
UCCSD uses exponential Pauli operators (exp(iθH)) that are symbolic until compiled for a specific hardware target. The raw gate count before transpilation is not meaningful for hardware comparison, so these columns are intentionally left blank. On real superconducting hardware, a single UCCSD layer for LiH (4 qubits) typically expands to hundreds of CNOT gates after decomposition.
N₂ — certified at cc-pVDZ:
N₂ with cc-pVDZ has 404 UCCSD parameters and a strongly-correlated triple bond. With CASSCF orbital optimisation, QEncode certified N₂ JW/UCCSD at 2.015 mHa gap — within chemical accuracy and aligned with DARPA QB-GSEE targets. Without CASSCF (HEA), the gap exceeds 0.1 Ha, illustrating how critical orbital optimisation is for multireference systems.
HEA — Hardware-Efficient Ansatz
Brick-layer circuit of alternating single-qubit rotations (RY) and CNOT entanglers, repeated for a fixed number of layers. The structure is chosen to minimise gate count on near-term devices rather than to match any chemical property of the molecule.
Why 2Q gates and depth are shown:
HEA uses only native hardware gates (RY, CNOT), so the circuit is already in a hardware-ready form. Gate counts reflect what would actually run on a device — making HEA entries directly comparable in the Lowest Cost and Balanced categories.
Trade-off:
HEA achieves near-chemical-accuracy for small molecules but may plateau before reaching UCCSD accuracy on larger or strongly-correlated systems, since it has no built-in knowledge of the molecular Hamiltonian.
ADAPT-VQE — Adaptive Ansatz
Starts from an empty circuit and grows it one operator at a time. At each step it measures the parameter-shift gradient of every operator in the UCCSD excitation pool and appends only the one with the largest gradient, then re-optimises. The result is a small, problem-tailored subset of the UCCSD pool rather than the full excitation set.
Why it matters for medium molecules:
Full UCCSD on molecules like H₂CO, C₄H₆, H₆ and benzene carries hundreds of parameters — more than COBYLA can navigate in a tractable number of iterations. ADAPT-VQE reaches the same accuracy class with a fraction of the parameters, and is what certifies these systems on the leaderboard.
Circuit metrics:
ADAPT builds from the same exponential Pauli operators as UCCSD, so depth and 2Q gate counts are symbolic until compiled for a hardware target and may show “—” for the same reason.
All energies are computed on a classical simulator (PennyLane + NumPy backend) with exact statevector simulation — no shot noise. Circuit metrics refer to the pre-simulation ansatz structure.
