Local overlapping quantum tomography with semidefinite programming

Many quantum computing systems work based on local interactions, making the local reduced density matrices (RDMs) of the system particularly important. While full-state tomography scales exponentially with system size, local overlapping tomography provides an efficient way to construct local RDMs using a number of product measurements that remains independent of system size. However, the reconstructed local RDMs are often nonphysical and inconsistent due to shot noise, requiring a higher number of measurements to accurately estimate observables. In this work, we introduce semidefinite programming (SDP) as a post- processing step for local RDMs obtained through local overlapping tomography, effectively mitigating the effects of shot noise and ensuring physical consistency. To illustrate the power of this method, we apply it to the estimation of ground state energy and ground state preparation for local Hamiltonians. We also integrate SDP with a variational method, algorithmic cooling, which heuristically constructs quantum circuits to minimize the energy of the Hamiltonian. Instead of updating the variational circuits with local RDMs from naive tomography, we use the local RDMs optimized under SDP constraints. Numerical simulations, including for frustrated and frustration-free Hamiltonians, highlight the advantages of the local overlapping tomography with SDP in terms of accuracy and resource efficiency, making it a valuable tool for practical quantum computations.