Extremely slow scaling of minimal Hamming distance in quantum sampling data

Abstract

Quantum data can be obtained from a diverse range of sources, including direct measurements from noisy quantum processors, cold-atom simulators, and classical approximations such as variational neural-network states. However, our ability to characterize these systems is fundamentally limited, as the available measurement data are often sparse compared to the exponentially large Hilbert space of the system. To address this, we propose using the expected minimal Hamming distance calculated for a set of unique bitstrings as a robust metric revealing a distinct power-law behaviour. Crucially, we have established a direct connection between the empirical power-law parameters and the intrinsic dimension of the quantum measurement outcomes. Through various examples of real experiments and simulations, we show that the power-law parameters reliably characterize the underlying geometric structure of measurement outcomes and identify quantum phase transitions from limited quantum information, without the need for accumulating extensive statistics or explicitly calculating physical observables. This enables the analysis of completely different quantum experiments within a single framework. Furthermore, we demonstrate the versatility of our approach on classical high-dimensional data, showing that the power-law parameters can successfully discriminate different classes of handwritten digits from the MNIST dataset.