Efficient Circuit-Based Quantum State Tomography via Sparse Entry Optimization

Many quantum states arising in algorithms and physical systems occupy only a small, structured subset of the exponentially large Hilbert space, yet standard quantum state tomography fails to exploit this structure. We present an efficient circuit-based tomography framework for pure quantum states that are sparse in a computational basis. For an $n$-qubit state supported on $k$ basis elements, the protocol reconstructs all amplitudes using $1 + 2(k-1)$ measurement settings. The method admits both entanglement-assisted and entanglement-free implementations, enabling explicit tradeoffs between two-qubit gate usage and statistical noise. We derive bounds on the required number of CNOT gates from the combinatorial structure of the state support and analyze their effect on reconstruction infidelity. The framework extends naturally to closed-system process tomography and is validated via numerical simulations using Qiskit.