Selective and efficient quantum state tomography for multi-qubit systems

The exponential growth with qubit number for the cost (in experimental time and data post-processing) of full quantum state tomography (QST) makes it impractical for large quantum systems. To bypass this bottleneck, we introduce selective and efficient QST (SEEQST), a method for efficiently estimating multiple, targeted elements of an arbitrary N-qubit density matrix. Our approach partitions the density matrix into 2^N disjoint subsets, each containing 2^N elements. We show that any chosen subset can be accurately estimated from just two experimental settings (quantum circuits) using only single-qubit measurements. Crucially, the complexity of estimating any subset remains constant, regardless of the total number of qubits. While designed for selective analysis, SEEQST can also reconstruct the full density matrix using 2^N+1-1 experimental settings, outperforming the standard 3^N scaling. We provide a circuit decomposition for the required measurements, demonstrating a maximum circuit depth that scales logarithmically with N, assuming all-to-all connectivity. The Python code for SEEQST is publicly available at github.com/aniket-ae/SEEQST.