2026/07/31 by Misha Yutushui, Theo Haas, Simon Trebst
Physics and Astronomy · #quant-ph #cond-mat.dis-nn #cond-mat.stat-mech
7+8 pages, 5+7 figures. Data available at https://doi.org/10.5281/zenodo.21380269
arxiv created 2026/07/31 · arxiv updated 2026/08/04
The meticulous preparation of macroscopic Greenberger-Horne-Zeilinger (GHZ) states provides a foundational resource for quantum technologies such as metrology, cryptography, and fault-tolerant codes. While state-of-the-art measurement-based protocols offer efficient low-depth execution, their performance can be bottlenecked by conventional decoders, such as minimum weight perfect matching (MWPM) or even maximum-likelihood decoding (MLD), which optimize for binary logical recovery and fail to maximize the continuous long-range order characteristic of a GHZ state for two-dimensional geometries. Here we overcome this limitation by framing the decoding problem as minimum Bayesian risk inference, introducing a general paradigm that maximizes the expected utility of the decoded state. Implementing this maximum-utility approach, we construct an algorithm that achieves the highest possible per-shot decoded quantum order and thereby establish an optimal decoding strategy for measurement-based GHZ state preparation. To improve its computational efficiency, we design a scalable two-stage decoder, which first encodes the syndromes into the edge weights of MWPM and then refines the result with a convolutional neural network trained to maximize the expected utility, at a fraction of the cost of the optimal decoder. Remarkably, we find that the first stage alone\unicodex2014which makes the matching aware of the gauge choice at no cost beyond bare MWPM\unicodex2014already performs near-optimally up to the largest sizes we study, N=256×256, closing up to 87% of the gap between the bare-MWPM and optimal decoding thresholds. Generalizing MWPM and MLD, the maximum-utility decoder (MUD) establishes a versatile framework that can be explicitly tailored to the operational demands of specific experiments by redefining the utility function.