Deep Neural Networks as the Semi-classical Limit of Quantum Neural Networks

Abstract

Our work intends to show that: (1) Quantum Neural Networks (QNN) can be mapped onto spinnetworks, with the consequence that the level of analysis of their operation can be carried out on the side of Topological Quantum Field Theories (TQFT); (2) Deep Neural Networks (DNN) are a subcase of QNN, in the sense that they emerge as the semiclassical limit of QNN; (3) a number of Machine Learning (ML) key-concepts can be rephrased by using the terminology of TQFT. Our framework provides as well a working hypothesis for understanding the generalization behavior of DNN, relating it to the topological features of the graphs structures involved. I. INTRODUCTIONA paradoxical result from [1] according to which DNN memorize the training samples by brute force leaves unexplained where the generalization capabilities of DNN come from. This "apparent" paradox, as it has been dubbed by [2], has led to active discussions by many scholars such as [15]. In any case, in our vision, the overall discussion has empirically proved how far the ML community is from building a principled model of DNN and, therefore, understanding their generalization capabilities.So far, ML techniques have been deployed in order to enhance quantum tasks ([16], [17], [18]), while Quantum Computing (QC) has been used to speed up traditional ML algorithms ([19], [20], [21], [22]). We will follow here a different perspective encompassing DNN as a sub-case (semi-classical limit) of QNN. This pathway has been suggested by the analogy with physics. A topical experiment at the base of the quantum revolution around the beginnings of the 20th century pointed out the existence of the photoelectric effect. As it is notorious, the effect has been explained by Albert Einstein resorting to a corpuscular description of the electromagnetic field, namely to the concept of photons as carriers of "quanta" of light. But, actually, the interpretation of this very seminal experiment clashed with a common perspective on quantum physics, widely spread nowadays even in the physicist's community, and that relies on the naive assumption that quantum means microscopic and classical macroscopic. A rather different pathway consists in moving from a quantum theory, with tested semi-classical limit that corresponds to the classical theory, and investigating the varieties of predictions that can then falsify the quantum theory. This approach allows new predictive power and more robust experimental corroboration, and it is the approach we have been following within this paper.

Neurons as hierarchies of quantum reference frames

“…If the QRFs X, Y , and Z perform pure quantum computation without classical intermediate steps, we can treat them in a path-integral formalism [116], and hence as defining a manifold of mappings from I to O. This suggests that information processing in neurons is amenable to a treatment using topological quantum field theory [117] and the emerging theory of topological quantum neural networks [118]. We defer this possibility to future work.…”

Sequential measurements, TQFTs, and TQNNs

“…A framework has been constructed in [17] that associates spin-network states that are colored with irreducible representations of some Lie group G to training and test sample states in (quantum) machine learning. A typical case-study is represented by training and test samples that are images composed of pixels.…”

“…The structure itself of the model shares similarity with partition functions emerging in TQFTs characterizing quantum gravity. This spin-network simulator, as introduced in [127], is a combinatorial TQFT model for computation, which can be extended to the TQNNs, as proposed in [17]. A q-deformation of the spin-network simulator has been investigated as well [128].…”

“…[16]), we have that sequential observations induce TQFTs, in particular, sequential observations of S induce TQFTs with copies of the Hilbert space H S as boundaries. In §6 we establish a link between sequential measurements and topological quantum neural networks (TQNNs), a framework that encode deep neural networks (DNNs) in their semiclassical limit, moving from the relation between TQFT and TQNN constructed in [17].…”

A free energy principle for generic quantum systems

“…Formally, we can think of P A as a weighted sum of "all possible paths" from the boundary state |B(t) to the boundary state |B(t + ∆t) as in Eq. ( 8) [104]; see [105] for discussion. Fig.…”