Lifeboat Foundation

Quantum state tomography plays a fundamental role in characterizing and evaluating the quality of quantum states produced by quantum devices. It serves as a crucial element in the advancement of quantum hardware and software, regardless of the underlying physical implementation and potential applications^1,2,3. However, reconstructing the full quantum state becomes prohibitively expensive for large-scale quantum systems that exhibit potential quantum advantages^4,5, as the number of measurements required increases exponentially with system size.

Recent protocols try to solve this challenge through two main steps: efficient parameterization of quantum states and utilization of carefully designed measurement schemes and classical data postprocessing algorithms. For one-dimensional (1D) systems with area law entanglement, the matrix product state (MPS)^{6,7,8,9,10,11,12} provides a compressed representation. It requires only a polynomial number of parameters that can be determined from local or global measurement results. Two iterative algorithms using local measurements, singular value thresholding (SVT)¹³ and maximum likelihood (ML)¹⁴, have been demonstrated in trapped-ion quantum simulators with up to 14 qubits¹⁵. However, SVT is limited to pure states and thus impractical for noisy intermediate-scale quantum (NISQ) systems. Meanwhile, although ML can handle mixed states represented as matrix product operators (MPOs)^16,17, it suffers from inefficient classical data postprocessing.