Public rankings include only certified entries with official trust filtering (2026-07-16).
Entry counts can differ by molecule when some configurations are not yet certified. All runs use the cc-pVDZ basis set.
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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 | CCSD(T) corr. | Status | |
|---|---|---|---|---|---|---|---|
H2 cc-pvdz | Parity | UCCSD | 0 | 3.46 × 10⁻² | |||
| #2 | H2 cc-pvdz | Parity | HEA | 2.22 × 10⁻¹⁶ | 3.46 × 10⁻² | ||
| #3 | HF cc-pvdz | Parity | UCCSD | 1.42 × 10⁻¹⁴ | 2.11 × 10⁻¹ | ||
| #4 | HF cc-pvdz | Parity | HEA | 2.84 × 10⁻¹⁴ | 2.11 × 10⁻¹ | ||
| #5 | HF cc-pvdz | Bravyi-Kitaev | HEA | 3.91 × 10⁻¹² | 2.11 × 10⁻¹ | ||
| #6 | HF cc-pvdz | Bravyi-Kitaev | UCCSD | 1.44 × 10⁻¹¹ | 2.11 × 10⁻¹ | ||
| #7 | H2 cc-pvdz | Bravyi-Kitaev | UCCSD | 7.92 × 10⁻¹¹ | 3.46 × 10⁻² | ||
| #8 | H2 cc-pvdz | Bravyi-Kitaev | HEA | 1.89 × 10⁻¹⁰ | 3.46 × 10⁻² | ||
| #9 | H2 cc-pvdz | Jordan-Wigner | UCCSD | 1.05 × 10⁻⁹ | 3.46 × 10⁻² | ||
| #10 | HF cc-pvdz | Jordan-Wigner | UCCSD | 1.83 × 10⁻⁹ | 2.11 × 10⁻¹ | ||
| #11 | HF cc-pvdz | Jordan-Wigner | HEA | 4.76 × 10⁻⁹ | 2.11 × 10⁻¹ | ||
| #12 | BeH2 cc-pvdz | Parity | HEA | 1.19 × 10⁻⁸ | 6.90 × 10⁻² | ||
| #13 | H2 cc-pvdz | Jordan-Wigner | HEA | 1.36 × 10⁻⁸ | 3.46 × 10⁻² | ||
| #14 | BeH2 cc-pvdz | Jordan-Wigner | HEA | 5.43 × 10⁻⁸ | 6.90 × 10⁻² | ||
| #15 | H2O cc-pvdz | Jordan-Wigner | UCCSD | 1.44 × 10⁻⁷ | 2.16 × 10⁻¹ | ||
| #16 | water_dimer cc-pvdz | Jordan-Wigner | UCCSD | 1.93 × 10⁻⁶ | 4.35 × 10⁻¹ | ||
| #17 | LiH cc-pvdz | Jordan-Wigner | UCCSD | 2.86 × 10⁻⁶ | 3.11 × 10⁻² | ||
| #18 | BeH2 cc-pvdz | Jordan-Wigner | UCCSD | 6.65 × 10⁻⁶ | 6.90 × 10⁻² | ||
| #19 | NH3 cc-pvdz | Jordan-Wigner | UCCSD | 3.24 × 10⁻⁵ | 2.09 × 10⁻¹ | ||
| #20 | LiH cc-pvdz | Jordan-Wigner | HEA | 9.58 × 10⁻⁵ | 3.11 × 10⁻² | ||
| #21 | water_dimer cc-pvdz | Jordan-Wigner | adapt | 1.11 × 10⁻⁴ | 4.35 × 10⁻¹ | ||
| #22 | water_dimer cc-pvdz | Parity | HEA | 1.56 × 10⁻⁴ | 4.35 × 10⁻¹ | ||
| #23 | H2O cc-pvdz | Parity | HEA | 2.96 × 10⁻⁴ | 2.16 × 10⁻¹ | ||
| #24 | water_dimer cc-pvdz | Jordan-Wigner | HEA | 3.32 × 10⁻⁴ | 4.35 × 10⁻¹ | ||
| #25 | H2O cc-pvdz | Jordan-Wigner | HEA | 4.03 × 10⁻⁴ | 2.16 × 10⁻¹ | ||
| #26 | H2CO cc-pvdz | Jordan-Wigner | adapt | 1.12 × 10⁻³ | 3.45 × 10⁻¹ | ||
| #27 | NH3 cc-pvdz | Jordan-Wigner | HEA | 1.88 × 10⁻³ | 2.09 × 10⁻¹ | ||
| #28 | H4 cc-pvdz | Jordan-Wigner | UCCSD | 2.22 × 10⁻³ | 8.73 × 10⁻² | ||
| #29 | BeH2 cc-pvdz | Parity | UCCSD | 2.22 × 10⁻³ | 6.90 × 10⁻² | ||
| #30 | C4H6 cc-pvdz | Jordan-Wigner | adapt | 2.83 × 10⁻³ | 6.26 × 10⁻¹ | ||
| #31 | N2 cc-pvdzCASSCF | Jordan-Wigner | HEA | 4.51 × 10⁻³ | 3.25 × 10⁻¹ | ||
| #32 | LiH cc-pvdz | Parity | HEA | 5.18 × 10⁻³ | 3.11 × 10⁻² | ||
| #33 | H4 cc-pvdz | Parity | HEA | 5.62 × 10⁻³ | 8.73 × 10⁻² | ||
| #34 | C4H4 cc-pvdzCASSCF | Jordan-Wigner | adapt | 5.96 × 10⁻³ | 5.96 × 10⁻¹ | ||
| #35 | C4H4 cc-pvdzCASSCF | Parity | HEA | 6.11 × 10⁻³ | 5.96 × 10⁻¹ | ||
| #36 | NH3 cc-pvdz | Parity | HEA | 6.91 × 10⁻³ | 2.09 × 10⁻¹ | ||
| #37 | C4H4 cc-pvdzCASSCF | Jordan-Wigner | UCCSD | 7.92 × 10⁻³ | 5.96 × 10⁻¹ | ||
| #38 | benzene cc-pvdzCASSCF | Jordan-Wigner | HEA | 8.74 × 10⁻³ | 8.74 × 10⁻¹ | ||
| #39 | N2 cc-pvdzCASSCF | Jordan-Wigner | adapt | 8.83 × 10⁻³ | 3.25 × 10⁻¹ | ||
| #40 | H6 cc-pvdzCASSCF | Jordan-Wigner | adapt | 9.27 × 10⁻³ | 1.36 × 10⁻¹ | ||
| #41 | H4 cc-pvdz | Jordan-Wigner | HEA | 9.28 × 10⁻³ | 8.73 × 10⁻² | ||
| #42 | N2 cc-pvdzCASSCF | Parity | HEA | 9.50 × 10⁻³ | 3.25 × 10⁻¹ | ||
| #43 | benzene cc-pvdzCASSCF | Jordan-Wigner | adapt | 9.54 × 10⁻³ | 8.74 × 10⁻¹ | ||
| #44 | C4H4 cc-pvdzCASSCF | Jordan-Wigner | HEA | 9.64 × 10⁻³ | 5.96 × 10⁻¹ | ||
| #45 | H8 cc-pvdzCASSCF | Jordan-Wigner | adapt | 9.80 × 10⁻³ | 1.85 × 10⁻¹ | ||
| #46 | H4 cc-pvdz | Jordan-Wigner | adapt | 9.94 × 10⁻³ | 8.73 × 10⁻² | ||
| #47 | H10 cc-pvdzCASSCF | Jordan-Wigner | adapt | 9.98 × 10⁻³ | 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.
