Molecular ground states: VQE against exact diagonalisation

The problem

Find the ground-state energy of a molecule. Everything a chemist wants downstream depends on it: reaction barriers, binding affinities, catalyst design, spectra. The threshold that matters is chemical accuracy, 1 kcal/mol or 0.0016 Hartree, because that is roughly the precision at which a computed energy predicts a reaction rate correctly. Above it the number is interesting, below it the number is useful.

Our instance is H2 in the STO-3G basis at its equilibrium bond length of 0.735 angstrom, mapped to two qubits by the parity transformation with two-qubit reduction. It is the standard textbook molecule and its Hamiltonian is published, so every figure here can be checked against the literature rather than taken on trust.

The classical baseline

Exact diagonalisation of the qubit Hamiltonian. Like the enumeration used on the finance page, this is not a heuristic: it returns the true ground state, so VQE is scored against the correct answer.

The finding that matters

The ansatz reaches the exact ground state perfectly. Given noiseless expectation values, VQE returns -1.137306 Hartree, matching diagonalisation to six decimal places and comfortably inside chemical accuracy. The quantum circuit is not the limitation.

With 4,096 shots per Pauli term it misses chemical accuracy, by roughly a factor of two, and the miss varies between runs. The entire error is statistical noise in the energy estimates that drive the optimiser. The optimiser is being steered by a slightly wrong number at every step and settles near, but not at, the minimum.

This is the real constraint on VQE, and it is rarely the one discussed. Precision costs shots, shot count scales as roughly 1/error squared, and every shot is wall-clock time on hardware with a queue. Reaching chemical accuracy reliably on this two-qubit molecule needs substantially more sampling than 4,096 shots per term. On a molecule with hundreds of Hamiltonian terms, that multiplies.

The second run landing at +0.0018 rather than +0.0031 Hartree is the same phenomenon. We report both because reporting only the better one would misrepresent the method's reliability.

Where classical methods actually stop

Exact diagonalisation stores a matrix of dimension 2^n by 2^n. At 20 qubits that is about 16 terabytes in dense form, and at 30 qubits roughly 17 exabytes. On that arithmetic alone the classical wall looks close.

It is not, and this is where most quantum chemistry marketing overstates the case. Nobody runs dense diagonalisation on large Hamiltonians. Sparse iterative eigensolvers never form the matrix, density matrix renormalisation group methods handle strongly correlated systems with hundreds of orbitals, and coupled cluster is the workhorse of production computational chemistry at sizes far past anything a quantum computer can address. Classical quantum chemistry is a mature, highly optimised field.

The honest statement is narrower and still meaningful: for strongly correlated systems, where coupled cluster degrades and the wavefunction cannot be compressed, the classical cost does grow exponentially. Transition-metal catalysts and multi-reference systems are the standard examples. Those are the cases where a fault-tolerant quantum computer would offer something classical methods cannot, and they are also a small fraction of the calculations a pharmaceutical company runs in a day.

What would have to change

An industrially relevant molecule needs hundreds to thousands of logical qubits, and the Hamiltonian term count grows as roughly the fourth power of the orbital count, so the sampling cost we just measured on a two-qubit molecule grows accordingly. Error correction is not optional at that scale: the circuits are far too deep for today's error rates.

Of the six sectors benchmarked on this site, this is the one the published roadmaps actually move. Quantinuum targets roughly 100 logical qubits with Sol in 2027, and IBM Starling and Quantinuum Apollo target hundreds by 2029. There is a serious literature on quantum chemistry in the 25 to 100 logical qubit range, so those milestones land squarely on molecules that matter. Contrast that with routing, which would need 40,000 logical qubits for a 200-stop round, or scheduling at 268,425 for one factory instance. Hundreds of logical qubits is transformative here and irrelevant there.

One caveat on Sol specifically. Its logical qubits come from the iceberg code, which is distance 2 and therefore detects errors and discards the run rather than correcting them. Postselection acceptance falls off with circuit depth, so Sol suits shallow chemistry circuits far better than deep ones. Apollo and Starling, with genuine correction, are the milestones for the harder molecules.

The practical guidance for an R&D director in 2026 is that quantum chemistry is the right domain to watch and the wrong domain to budget for as a production tool today. The numbers above are why. They are also why this page is more useful than a case study: it tells you what to monitor, which is logical qubit counts, code distance and the shot cost of expectation values, not headline physical qubit numbers.