The approach reformulates QSE as a constrained optimization problem, enabling rigorous statistical error estimates and improving numerical stability through regularization and a Krylov+ expansion to address ill-conditioning.

A large-scale experimental demonstration shows quantum subspace expansion (QSE) combined with informationally complete measurements called classical shadows can accurately recover ground state energies in spin models with up to 80 qubits.

The work provides a viable path to overcoming the measurement overhead of QSE, offering complete statistical error bars and a framework that could generalize to other quantum systems and observables, with future directions including symmetry verification, virtual distillation-inspired methods, and alternative estimators to further reduce sampling costs.

The study confirms that energy mitigation via QSE with classical shadows remains robust across system sizes, though accurately characterizing complex observables such as weight-five observables (e.g., SZY) at larger scales remains challenging due to statistical uncertainties.

Researchers achieved exact diagonalization for a 16-qubit case and showed that Krylov+ estimates lie below the first excited state, indicating substantial overlap with the true ground state and adherence to the variational principle within uncertainties.

Large-scale measurements involve over 32,768 randomized bases per circuit and evaluation of Pauli traces, with total single-qubit Pauli traces reaching up to 5.6×10^13 for 80 qubits, highlighting the scale of the demonstration.