Emergent Quantumness in Neural Networks

Abstract

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Towards a Theory of Quantum Gravity from Neural Networks

“…Note that in refs. [ 12 , 13 , 27 ] a functional similar to (14) was obtained, but only in a near equilibrium limit or when the entropy production due to learning is negligible, i.e., . In contrast, in (14) we completed the square and redefined the free energy which allowed us to keep all the terms.…”

“…By following the analysis of refs. [ 12 , 13 , 27 ] we assume that the long time scale dynamics is governed by the principle of stationary entropy production [ 9 ]. The principle states that the path taken by a system is the one for which the entropy production is stationary, subject to whatever constraints are imposed on the system.…”

“…All of the solutions of the Madelung Equations (17) and (18) are also solutions of the Schrodinger equation, but the opposite is not true [ 31 ] and so the system is not exactly quantum. To study a limit when a fully quantum behavior emerges we have to assume that the learning system is described by a grand canonical ensemble of neurons and that the exact number of neurons N is unobservable [ 13 ]. Then a constant shift in the free energy is unobservable, i.e., where the “chemical potential” of neurons (or another Lagrange multiplier which imposes a constraint on the number of neurons) is defined as …”

“…Using (22) the functional (14) can be rewritten as, where the “wave-function” is (See ref. [ 13 ] for details.)…”

“…Moreover, if quantum mechanics does emerge from a statistical theory, for example, due to averaging over some hidden variables [ 10 ], then the hidden variables must be non-local [ 11 ]. In this paper we describe a microscopic theory of neural networks from which the quantum behavior does emerge (for the trainable variables) and the hidden variables (or the non-trainable variables) are non-local [ 12 , 13 ]. In fact, as we shall see, the very notion of locality is also an emergent phenomenon that arises from the learning dynamics of neural networks.…”