Unfamiliar terms? Jump to the Key Terms section at the end.

In a previous article in this series, I used Bell and CHSH tests to establish something narrow but important: that a real superconducting quantum processor behaves quantum mechanically at all. Entanglement survived contact with hardware. Non-classical correlations were measurable.

That work answered a necessary question, but not a useful one.

Once quantum behaviour is established, the more pertinent question is not what can be done next, but what dominates when that behaviour is used operationally.

Variational Quantum Eigensolvers (VQE) are an obvious place to look for the answer. They are the canonical near-term workload: hybrid, noisy, and built almost entirely around repeated expectation estimation. If there are structural limits to near-term quantum optimisation, they should appear here first.

This article is not an attempt to extract the best possible energy from a molecule. It is a deliberately constrained stress test of what dominates when a theoretically sound quantum workflow is executed under real conditions.

The problem is intentionally minimal.

The Hamiltonian is a sum of six Pauli terms (II, IZ, ZI, ZZ, XX, YY) with coefficients fixed by molecular geometry. For H₂ at equilibrium bond length, these coefficients are well known from standard quantum chemistry. The exact ground-state energy is −1.915 Hartree.

This is not an approximation. It is the full electronic Hamiltonian after qubit mapping.

Nothing here is tuned for performance. There are no adaptive ansätze, no term grouping, no error mitigation, and no extended optimisation runs. Those constraints are deliberate. The goal is not to demonstrate what VQE could achieve under ideal engineering, but to observe what dominates when it is run as-is on real hardware.

Before touching hardware, I ran the same Hamiltonian and ansatz on an exact statevector simulator. The result converges cleanly to the known ground-state energy to numerical precision.

This matters because it establishes a baseline. Algorithmically, there is no problem to solve. Any deviation that follows is not a flaw in the formulation.

Before hardware noise enters the picture, finite sampling already changes the character of the problem.

Running the same workflow under a finite shot budget produces apparently strong results very quickly. Best-of-run energies dip impressively during optimisation.

Those results do not survive validation.

Re-evaluating the best-found parameter settings at higher shot counts exposes the bias. The optimiser is chasing statistical fluctuation rather than a stable minimum. The validated energy is higher, but consistent.

At 2,000 shots per expectation evaluation, sampling error is approximately ±0.013 Hartree. Chemical accuracy, the threshold where quantum chemistry predictions become useful, is roughly ±0.0016 Hartree. The optimiser is not finding minima within that tolerance. It is fitting noise.

This is the first structural lesson. Without explicit validation, VQE results are systematically misleading under sampling noise. Best-of-run values are selection artefacts, not estimates. This effect appears before hardware noise becomes relevant.

The most important hardware effect is not noise, but representation.

A two-qubit circuit does not remain a two-qubit object. After transpilation, the runtime represents the circuit at full device width, in this case across a 156-qubit backend, even though only two physical qubits carry operations.

This is not an error. The runtime schedules, validates, and prices jobs based on the full device representation because observables, layouts, and execution metadata must be consistent across the target backend. Cost follows representation rather than logical intent.

This is a reminder that near-term quantum workloads are shaped as much by orchestration and execution models as by circuit physics.

The hardware experiment mirrors the constrained setup used earlier.

The result is straightforward. On hardware, the optimisation trajectory descends rapidly at first, moving from approximately −0.4 Hartree to around −1.7 Hartree over the first ten iterations, before flattening.

The shape of the curve matters more than the endpoint. Early descent exists, but progress slows quickly as sampling variance and noise dominate. The final measured energy of −1.680 Hartree corresponds to a 12.3% error relative to the exact value, roughly 150 times chemical accuracy.

This outcome is expected under the imposed constraints. Finite shots, no error mitigation, a shallow ansatz, and conservative optimisation are sufficient to explain it. Validation confirms the system reached its effective limit rather than producing a statistical outlier.

By this point, energy values matter less than counts.

Each shot produces a single measurement outcome. Each expectation estimate requires thousands of shots to reduce variance. Each optimisation step requires at least one expectation estimate. The measurement cost compounds multiplicatively.

This is why VQE cost scales with iteration count rather than circuit depth (see article header image). Even trivial circuits become expensive when they must be measured repeatedly, one evaluation at a time, through a real service interface.

Across simulator and hardware runs, the same pattern appears.

Dominated:

Did not dominate:

These are accounting facts rather than subtle performance effects.

This experiment also makes clear which interventions would matter.

What would help:

What would not:

These are not algorithmic breakthroughs. They are economic ones.

VQE behaves as advertised. It descends, stabilises, and produces physically meaningful estimates on real hardware.

What it does not do is escape the economics of measurement and orchestration. Any near-term optimisation workflow built on repeated expectation estimation inherits these constraints. Improvements in ansatz design or optimiser choice matter, but they are second-order compared to how often expectation values must be measured and how that work is scheduled.

Claims that ignore the time and cost of repeated expectation evaluation are incomplete.

The progression from Bell tests to VQE is not a march toward advantage. It is a diagnostic sequence.

Bell tests establish that entanglement survives contact with hardware. VQE reveals what dominates when that behaviour is used operationally. Today, that constraint is the cost of repeated measurement and orchestration overhead.

Repeated measurement and orchestration time are the limiting factors. Optimisation efforts that do not account for this constraint are misdirected, regardless of algorithmic sophistication.

This article is part of the Quantum Optimisation Before Quantum Advantage series. Code and supporting artefacts are available in the accompanying repository.