Correlated Purification for Restoring $N$-Representability in Quantum Simulation

Classical shadow tomography offers a scalable route to estimating properties of quantum states, but the resulting reduced density matrices (RDMs) often violate constraints that ensure they represent $N$-electron states -- known as $N$-representability conditions -- because of statistical and hardware noise. We present a correlated purification framework based on semidefinite programming to restore accuracy to these noisy, unphysical two-electron RDMs. The method performs a bi-objective optimization that minimizes both the many-electron energy and the nuclear norm of the change in the measured 2-RDM. The nuclear norm, often employed in matrix completion, promotes low-rank, physically meaningful corrections to the 2-RDM, while the energy term acts as a regularization term that can improve the purity of the ground state. While the method is particularly effective for the ground state, it can also be applied to excited and non-stationary states by decreasing the weight of the energy relative to the error norm. In an application to fermionic shadow tomography of large hydrogen chains, correlated purification yields substantial reductions in both energy and 2-RDM error, achieving chemical accuracy across dissociation curves. This framework provides a robust strategy for tomography in many-body quantum simulations.