The Threshold Theorem in Watts: Fault Tolerance as a Question About Objective Probability
Abstract
In 2011 Hagar & Sergioli proposed a new interpretation of objective probability in deterministic physics. On it, the probability of a physical state supervenes on the resources, energy over time, required to realize it from a given state, relative to the resources available (arXiv:1101.3521). The motivation for that paper was an objective alternative to QBism, the view that sees quantum probabilities as subjective degrees of belief. Here I apply this interpretation to a more practical subject matter: fault-tolerant quantum computing (FTQC). Under the resource-bounded measure Hagar & Sergioli proposed, the threshold theorem becomes a claim about classification: it asserts that error-corrected logical states belong to the class of relatively cheaply realizable states, whose probability remains near 1 as the machine grows. The theorem originally derived this claim from a resource inventory that was partial, and left out four resources consumed by error correction: calibration of a drifting device, decoding within the correction cycle, coherence as a finite time budget, and entropy flush through fresh ancillas. These entered the original derivation at zero price. Here I translate the feasibility of FTQC into a measurable quantity, watts per decade of suppressed logical error (a decade, in the engineer's usage, being one factor of ten in the error rate). I then show that the published record already contains its first data points for this translation, and I state the two measurements that would settle the question empirically. The three-decade debate on FTQC, conducted so far as an exchange about noise-model assumptions, turns out to be, under this interpretation, a quantitative dispute about a single object: the resource-bounded objective probability of the target logical states.
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BibTeX
@misc{hagar2026threshold,
title = {{The Threshold Theorem in Watts: Fault Tolerance as a Question About Objective Probability}},
author = {Amit Hagar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.22276},
eprint = {2607.22276},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.22276},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Threshold Theorem in Watts: Fault Tolerance as a Question About Objective Probability
AU - Amit Hagar
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2607.22276
UR - https://arxiv.org/abs/2607.22276
AB - In 2011 Hagar & Sergioli proposed a new interpretation of objective probability in deterministic physics. On it, the probability of a physical state supervenes on the resources, energy over time, required to realize it from a given state, relative to the resources available (arXiv:1101.3521). The motivation for that paper was an objective alternative to QBism, the view that sees quantum probabilities as subjective degrees of belief. Here I apply this interpretation to a more practical subject matter: fault-tolerant quantum computing (FTQC). Under the resource-bounded measure Hagar & Sergioli proposed, the threshold theorem becomes a claim about classification: it asserts that error-corrected logical states belong to the class of relatively cheaply realizable states, whose probability remains near 1 as the machine grows. The theorem originally derived this claim from a resource inventory that was partial, and left out four resources consumed by error correction: calibration of a drifting device, decoding within the correction cycle, coherence as a finite time budget, and entropy flush through fresh ancillas. These entered the original derivation at zero price. Here I translate the feasibility of FTQC into a measurable quantity, watts per decade of suppressed logical error (a decade, in the engineer's usage, being one factor of ten in the error rate). I then show that the published record already contains its first data points for this translation, and I state the two measurements that would settle the question empirically. The three-decade debate on FTQC, conducted so far as an exchange about noise-model assumptions, turns out to be, under this interpretation, a quantitative dispute about a single object: the resource-bounded objective probability of the target logical states.
ER -