The variational quantum eigensolver (VQE) is the flagship algorithm of near-term quantum computing. The pitch: find ground-state energies of molecules using shallow quantum circuits that today's noisy hardware can run. Classical optimization adjusts circuit parameters; the quantum chip evaluates each candidate wavefunction. The resource requirement — circuit depth, number of gates — was believed to scale polynomially with molecular size. This is what made VQE the best hope for quantum advantage in chemistry before fault-tolerant machines arrive.
The scaling is exponential.
Robledo-Moreno et al. (arXiv:2603.13073, March 2026) demonstrate that adaptive VQE — the variant where circuit elements are chosen dynamically based on energy gradients — requires resources that grow exponentially with system size. The predictor is Rényi entropy: the entanglement complexity of the target state determines the circuit depth, and for chemically interesting molecules, that complexity grows exponentially. The correlation between Rényi entropy and required resources fits with R² = 0.99.
The finding is not that VQE fails in practice — that was known from noisy hardware experiments. The finding is that VQE fails in principle. Even with perfect hardware and unlimited classical optimization, the circuit depth required for chemical accuracy on systems beyond toy molecules exceeds what polynomial scaling would allow. The wall is not noise, not optimization landscape, not barren plateaus. It is entanglement structure itself.
This matters because VQE occupied a specific economic niche: the argument for near-term quantum hardware investment. If fault-tolerant quantum computing requires decades of engineering, VQE was the bridge — the algorithm that could deliver value now. Removing that bridge doesn't invalidate quantum computing. It invalidates the timeline. The value is still there, but it sits behind the fault-tolerance barrier that VQE was supposed to circumvent.