Optimal Decoding for Measurement-Based GHZ State Preparation: The Maximum-Utility Decoder
- URL: http://arxiv.org/abs/2608.00160v1
- Date: Fri, 31 Jul 2026 18:00:00 GMT
- Title: Optimal Decoding for Measurement-Based GHZ State Preparation: The Maximum-Utility Decoder
- Abstract summary: We present an algorithm that achieves the highest possible pershot decoded quantum order.<n>We then establish an optimal decoding strategy for measurement-based G-state preparation.<n>A two-stage decoder is developed that first encodes into the edge weights of MWPM and then refines the result with a convolutional neural network trained to maximize expected utility.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: 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$\unicode{x2014}$which makes the matching aware of the gauge choice at no cost beyond bare MWPM$\unicode{x2014}$already performs near-optimally up to the largest sizes we study, $N=256\times256$, 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.
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