Exponential Scaling Barriers for Variational Quantum Eigensolvers

The Variational Quantum Eigensolver (VQE) is widely regarded as a promising algorithm for calculating ground states of quantum systems that are intractable for classical computers. This promise is typically motivated by the hope of mitigating the exponential growth of Hilbert space with system size. Here we scrutinize how the computational cost of adaptive VQE scales with the size of the target system. We demonstrate that the Rényi entropy derived from classical simulations predicts the required number of adaptive iterations of VQE with high accuracy ($R^2 \approx 0.99$). We validate this on a benchmarking set of more than 20 different molecules with active spaces ranging from four to ten orbitals. For these molecules, we find an exponential scaling of the number of adaptive iterations, and in turn, of the circuit depth with the system size. We therefore conclude that it is unlikely that VQE in its current form is able to simulate large molecular systems with high fidelity without exponential resource requirements.

💡 Research Summary

The paper conducts a systematic investigation of how the computational cost of adaptive Variational Quantum Eigensolvers (ADAPT‑VQE) scales with the size and complexity of target molecular systems. Using classical CASSCF calculations, the authors extract the configuration‑interaction (CI) coefficients and construct a probability distribution over electronic configurations. From this distribution they compute the Rényi entropy hα, which serves as a quantitative metric of multi‑reference character and overall problem complexity. By scanning the Rényi order α, they identify α*≈0.25 as the value that maximizes the correlation between hα and the logarithm of the number of ADAPT iterations (n_ADAPT) required to reach chemical accuracy (ε_chem≈1 kcal/mol). Across a benchmark set of 21 molecules with active spaces ranging from four to ten orbitals, the relationship log n_ADAPT ≈ a · h* + b holds with an exceptional coefficient of determination (R²≈0.99).

The study evaluates three operator pools—Qubit‑ADAPT, QEB‑ADAPT, and CEO‑ADAPT—combined with the TETRIS extension that allows simultaneous addition of non‑overlapping operators. Despite these algorithmic variations, the exponential scaling persists: both n_ADAPT and the resulting circuit depth (measured by CNOT count) grow exponentially with the Rényi entropy and, consequently, with the size of the active space. Extrapolations to chemically relevant but classically challenging systems such as Cr₂,