Efficient quantum state tomography with auxiliary systems

Quantum state tomography is a fundamental technique in quantum information science for reconstructing the density matrix of an unknown quantum state, thereby providing complete information about the state. It plays a crucial role in quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only complicates experimental design and implementation but also leads to substantial consumption of experimental resources. These limitations significantly hinder the application of state tomography in large-scale quantum systems. To reduce the number of measurement settings and improve sampling efficiency, this study proposes a tomography method based on auxiliary systems. The method can be implemented either through entanglement between the quantum system to be measured and a quantum auxiliary system or through correlations between the quantum system and a probabilistic classical auxiliary system. Measurements performed on the joint system enable more efficient extraction of information about the target quantum state. The method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity scaling as O(d²) for the system of interest with d dimensions, significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for purity estimation based on the proposed circuit, one of which achieves estimation precision at the Heisenberg limit. The effectiveness of the proposed method is validated through detailed theoretical analysis, numerical simulations, and cloud-based quantum computing demonstrations.