Deviation Criterion for the Born Rule: A Rigorous Lemma for the Weak Proposition p = f(ŝ · n)

chao wang

Alexia Auffèves, Philippe Grangier

Gennaro De Luca, Vinayak Sharma, Aviral Shrivastava

Quantum Circuit Born Machines (QCBMs) offer a natural approach to generative machine learning by leveraging the Born rule. Recent work has provided a method to classically train QCBMs with Instantaneous Quantum Polynomial (IQP) circuits via the Maximum Mean Discrepancy (MMD) loss. Despite the assumed intractability of sampling from IQP circuits classically, their expectation values can be computed classically, enabling training of these IQP QCBMs. However, quantum machine learning (QML) models have various other challenges, including trainability issues caused by exponential concentration or barren plateaus. While these issues have been explored for parameters sampled from a uniform distribution, little work has been done to rigorously treat the use of arbitrary Gaussian initialization schemes. This work leverages Stein's lemma and Lipschitz concentration bounds for Gaussian random variables to provide an analytical lower bound of the variance of the gradient and a probabilistic concentration bound of the deviation of the gradient from its mean. It discusses strategies to either avoid or encourage exponential concentration, as well as the conditions under which barren plateaus are more likely to occur.

Paul W. Brumer

Travis Bergen

Theorem (Quadratic Projection Measure) + Ansatz (Hilbert Embeddability) + Proposition (Standard Born Rule). Under three structural constraints (basis invariance, projection-integration stability, orientation independence), the projection-induced measure on coherent informational states is forced to be ||^2. The standard quantum-mechanical Born rule emerges as a Proposition under the Hilbert Embeddability Ansatz, with Gleason's theorem providing the converse on the Hilbert representation. Inherits from Paper 0 (axioms A4, A5, A6).

Charles R. Giardina