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.
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BibTeX
@misc{golubev2026extremely,
title = {{Extremely slow scaling of minimal Hamming distance in quantum sampling data}},
author = {P. S. Golubev and I. A. Iakovlev and V. V. Mazurenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.04558},
eprint = {2606.04558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.04558},
}
RIS
TY - UNPB
TI - Extremely slow scaling of minimal Hamming distance in quantum sampling data
AU - P. S. Golubev
AU - I. A. Iakovlev
AU - V. V. Mazurenko
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2606.04558
UR - https://arxiv.org/abs/2606.04558
AB - 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.
ER -