Estimating quantum states from experimental data, typically conducted through quantum state tomography (QST), plays a crucial role in verifying the correctness and evaluating the performance of quantum devices. Nonetheless, executing QST for generic unstructured or low-rank quantum states demands an overwhelming number of state copies, escalating exponentially with the quantity of individual quanta within the system, even with the most optimized measurement configurations. Recent work has shown that for one-dimensional quantum states that can be effectively approximated by matrix product operators (MPOs), a polynomial number of copies of the state suffices for reconstruction. Compared to MPOs in one dimension, projected entangled-pair states (PEPSs) and projected entangled-pair operators (PEPOs), which represent typical low-dimensional structures in two dimensions, are more prevalent as a looped tensor network. However, a formal analysis of the sample complexity required for estimating PEPSs or PEPOs has yet to be established. In this paper, we aim to address this gap by providing theoretical guarantees for the stable recovery of PEPSs and PEPOs. Our analysis primarily focuses on two quantum measurement schemes: (1) informationally complete positive operator valued measures (IC-POVMs), specifically the spherical $t$ designs $(t\ge 3)$, and (2) projective rank-one measurements—in particular, Haar random projective measurements. We first establish stable embeddings for PEPSs (or PEPOs) to ensure that the information contained in the states can be preserved under these two measurement schemes. We then show that a constrained least-squares estimator achieves stable recovery for PEPSs (or PEPOs), with the recovery error bounded when the number of state copies scales linearly under spherical $t$ designs and polynomially under Haar-random projective measurements with respect to the number of qudits. These results provide theoretical support for the reliable use of PEPSs and PEPOs in practical quantum information processing.

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• Quantum information theory