A Unified Framework for Sample Complexity of Structured Quantum State Tomography under Noisy Observations

The author develops a unified mathematical framework to quantify the sample complexity of reconstructing structured quantum states. Unlike previous studies that often assume ideal measurement conditions, this framework explicitly incorporates noise at both the state preparation and measurement stages. By using a covering-number analysis, the paper provides a common language to evaluate the intrinsic complexity of diverse state classes, including sparse states, low-rank matrices, and tensor-network models like Matrix Product Operators (MPOs) and Projected Entangled-Pair Operators (PEPOs).

The study derives non-asymptotic recovery guarantees for two primary estimation strategies:

• Noise-aware estimator: By incorporating a calibrated noise model, this estimator achieves recovery guarantees where the sample complexity depends on the state class complexity, measurement ensemble efficiency, and the specific noise levels.
• Noise-unaware estimator: This estimator uses the ideal Born measurement model despite the presence of noise. The analysis reveals that the reconstruction error decomposes into a statistical error term (which improves with more samples) and a deterministic bias term (which remains constant regardless of sample size). This bias represents a fundamental performance floor caused by model mismatch.

As quantum devices scale to hundreds of qubits, traditional tomography becomes exponentially expensive. This framework offers a rigorous way to understand how structural assumptions (like low-rank or tensor-network structure) can be combined with realistic noise models to make tomography feasible. By quantifying the performance gap between noise-aware and noise-unaware methods, the paper provides researchers with a clear theoretical basis for choosing reconstruction strategies in the presence of inevitable hardware imperfections.