Quantum version for the associative learning in large language models

In the TGD framework the model for associative learning, as it is modelled in large language models (LLMs) discussed from TGD point of view, could be generalized to formulate a quantum model for associative learning as it could occur in TGD inspired theory of consciousness.

I have discussed LLMs from TGD point of view already earlier (see this, this, and this). One could also consider the combination of the TGD inspired quantum version of associative learning with the speculative idea of extending a classical computer to a hybrid of classical and quantum computers (see this).

Zero energy ontology from the point of view of LLMs

Zero energy ontology (ZEO) is the first piece of the TGD vision.

1. By holography, spacetime surfaces are analogous to Bohr orbits as basic objects. This means that 3-D structure as 3-surface determines almost deterministically the 4-surface.

The failure of a complete classical determinism is essential. The non-deterministic classical time evolution involves 3-D loci of non-determinism as analogs of 1-D frames of 2-D soap films.

Different Bohr orbits starting from a fixed 3-surface A at the passive boundary of CD would lead to different surfaces B located at the active boundary of CD whose size of CD would increase during the sequence of SSFRs.

At the quantum level, the superpositions of Bohr orbits define zero-energy states in geometric degrees of freedom (“world of classical worlds”, WCW). In fermionic degrees of freedom zero energy states are superpositions of products of fermionic states assignable to the boundaries of CD and to the loci of non-determinism.

The 3-D state at the passive boundary would remain invariant under the sequence of “small” state function reductions (SSFRs). This is the TGD counterpart of the Zeno effect.

The Bohr orbits of a 3-D particle are analogous to random walks A→ B for a particle as a 3-surface. The almost deterministic Bohr orbits A→ B are analogous to the association sequences of language models associated with the many layered neural nets.

The non-deterministic classical time evolution is modellable by a diffusion equation (diffusion) or Schrödinger type equation (dispersion). This process would be the quantum counterpart for the diffusion appearing in LLMs (see this). Whether this process could be seen as analog as a path integral as a sum over a discrete set of paths as Bohr orbits, is an interesting question.

The time reversal of the diffusion/dispersion is used in error correction in LLMs and in ZEO it could correspond to a pair of BSFRs involving a temporary change of the arrow of time. A pair of BSFRs would make it possible for the system to make a fresh start and therefore to learn by trial and error. This is perhaps the most important aspect of conscious intelligence.

On the quantum level, a series of SSFRs corresponds to a subjective time evolution giving rise to a conscious self. It also corresponds to an analog of computation and of mathematical reasoning: the theorem develops step by step as a sequence of SSFRs. In biology this sequence corresponds to biological function and in neuroscience to a behavioral pattern.

Holography =holomorphy hypothesis and learning process

Holography=holomorphy hypothesis allows to reduce classical field equations to purely algebraic conditions (f₁,f₂)=(0,0), where f_i are analytic functions of one hypercomplex and 3 complex coordinates of H=M⁴× CP₂. The solutions are minimal surfaces irrespective of the classical action as long as it is general coordinate invariant and expressible in terms of induced geometry. This means universality of the dynamics and is quantum criticality expressed by the holomorphy. This implies saddle surface property for the spacetime surface meaning that the real parts of f_i do not have minima or maxima in general.

Interestingly, the almost absence of minima meaning a saddlepoint property for most extrema is essential for the success of LLMs, which is in fact not well-understood. In LLMs, the cost function V measuring the size of the teaching error, is minimized in the parameter space by gradient dynamics. If most extrema are saddle points, the process does not get stuck to a local minimum and learning becomes very effective.

Furthermore, in LLMs local flatness of the parameter space is of help since it increases the probability that the gradient dynamics leads to the minimum and also reduces the probability to leave the minimum by a small perturbation.

Could the minimal surface property somehow prevent the sticking in the recent case?

1. It is useful to consider the situation first at the level of a single space-time surface (rather than WCW). At the space-level all points are geometrically saddle points in the geometrical sense by the minimal surface property stating that the trace of the second fundamental form, as an analog of acceleration identifiable as a sum of external curvatures, vanishes. The absece of saddle points in this sense need not have anything to do with the absence of sticking in some dynamics.
2. The quantum learning process would occur in the “world of classical worlds” (WCW) as the space of Bohr orbits rather than at the space-time level. The vacuum functional as an exponent of the classical action is proposed to have by the analog of Langlands duality (see this) also purely number theoretic expression, which would mean computability and enormous simplification (see this).

The Kähler function K, defining vacuum functional as its exponential, is in a central role. Also the degeneracies of the maxima are important. The maxima for the exponential of K”hahler function are thermodynamic analogs for Boltzmann exponents and their degeneracy measured by entropy. One can say that the minimization of energy and maximization of entropy compete.

Note that K is determined only modulo the addition of a real or imaginary part of a holomorphic function of WCW complex coordinates. The Kähler metric of WCW is in complex coordinates of the form

G_MN^*= ∂_M∂_N^*K.

The maxima of vacuum functional exp(K), which correspond to minima of the Kähler function, are of special interest. The Euclidian signature puts strong constraints at the minima of K. Criticality condition means that some second partial derivates of K with respect to the real coordinates vanish.

A good example is the metric of complex plane given by ds²=dzdz^* and has K=zz^* having a minimum at origin. The metric is flat.

It must be however made clear that in the learning the loss function would measure the deviation of B₁ from B₂ and cannot be identified as K. There are two minimization problems involved and it is not clear whether they are consistent.

A model for the learning process

How could the learning process take place?

1. Learning process can be seen mathematically as a construction of a representation for the dynamics of the external world by a subsystem. Associations A₁→ B₁ for the dynamics of the external world serve a teaching material and a representation as for these as associations A→ B in the internal model world is constructed as a model for the dynamics external world.

The external world states A₁→ B₁ would actually correspond to the sensory percepts of the states of the external world and in the learning process the system learns to predict the B₁ as a consequence of A₁.

In the TGD based model for sensory perception (see this and this) as construction of standardized mental images, the feedback loop between sensory organs and magnetic body would make this possible in the same way as in pattern recognition. The deviation of B from B₁ is minimized. This deviation would define the virtual sensory input from the magnetic body to the sensory organ.

Classically A and B (A₁ and B₁) correspond to 3-surfaces at the boundaries of a CD and A (A₁) is fixed in ZEO. At the quantum level, one has zero energy states as superpositions of orbits A→ B (A₁→ B₁).

The parameters characterizing the space-time surfaces, identifiable as the Taylor coefficients of the analytic functions and in the special case of polynomials, define the counterpart of the latent space (see this and this). The coefficients belong to an extension E of rationals and one obtains a hierarchy of extensions having interpretation in terms of evolution (see this, this and this. The coefficients determine almost deterministically the space-time surface as a Bohr orbit.

The failure of non-determinism corresponds to the 3-D loci of non-determinism at the Bohr orbit of A and the discrete variables parametrizing the non-determinism correspond to the parameter space of LLMs.

The space of 3-surfaces at the passive or active boundary of CD would correspond to the latent space as a subspace of the space of features (see this). The cutoff to the degree of the polynomial and to the dimension of the Galois group of the polynomial would induce the analog of dimensional reduction replacing the feature space with a latent space. This cutoff would also reduce the parameter space as the discrete space characterizing the classical non-determinism.The TGD counterpart of the loss landscape (see this) corresponds to a subspace of the parameter space.

A fixed 3-surface at the initial moment at the passive boundary of the CD corresponds to A in the association A→ B. This choice determines the coefficients of the polynomial that define the latent space. The correspondence A→ A₁ could be also learned in the learning process. This correspondence should determine the correspondence B→ B₁. The non-uniqueness of B due to classical non-determinism makes possible many associations.

The construction of a representation means finding non-deterministic space-time surfaces A→B in CD producing an optimal representation for the pair A₁→ B₁, meaning that B is as near as possible B₁. The error function measures the deviation of B from B₁. In LLMs the error function is minimized by a gradient method. The counterpart ot his method in the the case of the construction of conscious association should be understood. The fact that the TGD Universe is fractal is expected to help considerably the construction of conscious associations as representations.

1. The representation could be seen as a simplified version of the original obtained by scaling the size of the cd, either up or down.
2. The reduction of the degree of polynomials used and the algebraic dimension of extension E reduce the complexity. The restriction of an extension of E to E reduces complexity and the hierarchies of extensions of E define complexity hierarchies.
3. Also the hierarchies of analytic maps of (f₁,f₂)→ (g₁(f₁,2₂),g₂(f₁,f₂)) define iteration hierarchies analogous to those associated with fractals and approach to what looks like chaos. One can also “imagine” more complex systems at the level of representation by extending E or performing these iterations.

A hybrid of classical and quantum computer and quantum model for associative learning

One can formulate this picture also in the speculative vision (see this) in which a classical computer becomes a living system as a hybrid of classical and quantum computers.

1. A quantum computation-like process would be associated with classical computation. The classical non-determinism could be maximal in the sense that each tick of the computer clock would involve loci of classical non-determinism making the outputs of the gates non-deterministic.

Classical computation would correspond to the most probable Bohr orbit in the representation of the computation as a zero energy state. If localization in WCW is possible (position measurement in the discrete degrees of freedom of WCW due to non-determinism) this localization could occur at a single Bohr orbit.

The output of a gate would be a superposition of pairs of ordinary bits and OH-O^- qubits. For the OH-O^- qubits, the proton of OH would be transformed to a gravitationally dark proton at the gravitational magnetic body of the Earth or the Sun. This entanglement would be reduced in an SSFR which could, but need not, occur after each clock period.

This would give rise to a computational analog of the associative learning process in which the learning process assigns to the pairs A_{→ B} computations A→ B. Note that classical non-determinism also makes possible the formation of association sequences. See the article A hybrid of classical and quantum computer and quantum model for associative learning.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/01/quantum-version-for-associative.html