In this innovation, the retrieval of stored items from an exponentially large, unsorted database is performed by quantum-inspired resonance using polynomial resources due to quantum-like superposition effect. The model is represented by a modified Madelung equation in which the quantum potential is replaced by different, specifically chosen potentials. As a result, the dynamics attains both quantum and classical properties: it preserves superposition and entanglement of random solutions, while allowing one to measure its state variables using classical methods. Such an optimal combination of characteristics is a perfect match for quantum-inspired information processing.
The formal mathematical difference between quantum and classical mechanics is better pronounced in the Madelung (rather than the Schrodinger) equation. Two factors contribute to this difference: (1) the scale of the system introduced through the Planck constant and (2) the topology of the Madelung equations that includes the feedback (in the form of the quantum potential) from the Liouville equation to the Hamilton-Jacobi equation. Ignoring the scale factor as well as the concrete form of the feedback, this innovation concentrates on preserving the topology while varying the types of the feedback. A general approach to the choice of the feedback was introduced. More specific feedback is linked to the behavioral models of Livings.
It turns out that the eigenvalues of linear parabolic partial differential equations (PDEs) possess similar properties. Consider a linear n-dimensional parabolic PDE subject to boundary conditions. Then the eigenvalues corresponding to each variable form a sequence of monotonously increasing positive numbers. However, each linear combination of these eigenvalues represents another eigenvalue of the solution, and that is the same “combinatorial explosion” shown below.
Due to that property, for each n-string-number label, one can find an excitation force that activates the corresponding eigenvalue. Global (normalization) constraints imposed upon the probability density (in addition to boundary conditions) was achieved by a special form of the excitation force.
This innovation is concerned only with computational capabilities of the proposed model disregarding possible physical interpretations. In particular, the problem of modeling associative memory of exponential capacity using only polynomial resources is addressed. The classic formulation of associative memory is as follows: store a set of patterns in such a way that, when presented with a new pattern, the network responds by producing whichever one of the stored patterns most closely resembles the stored one. The correct response follows training on a set of examples. The best solution provided by Hopfield neural net is the storage capacity proportional to N/logN where N is the dimensionality of the corresponding neural net. The proposed associative memory is based upon quantumlike superposition of solutions to motor dynamics and the resonance between motor and mental dynamics.