Quantum Computation via Sparse Distributed Representation

Abstract

Quantum superposition states that any physical system simultaneously exists in all of its possible states, the number of which is exponential in the number of entities composing the system. The strength of presence of each possible state in the superposition-i.e., the probability with which it would be observed if measured-is represented by its probability amplitude coefficient. The assumption that these coefficients must be represented physically disjointly from each other, i.e., localistically, is nearly universal in the quantum theory/computing literature. Alternatively, these coefficients can be represented using sparse distributed representations (SDR), wherein each coefficient is represented by a small subset of an overall population of representational units and the subsets can overlap. Specifically, I consider an SDR model in which the overall population consists of Q clusters, each having K binary units, so that each coefficient is represented by a set of Q units, one per cluster. Thus, K Q coefficients can be represented with KQ units. We can then consider the particular world state, X, whose coefficient's representation, R(X), is the set of Q units active at time t to have the maximal probability and the probabilities of all other states, Y, to correspond to the size of the intersection of R(Y) and R(X). Thus, R(X) simultaneously serves both as the representation of the particular state, X, and as a probability distribution over all states. Thus, set intersection may be used to classically implement quantum superposition. If algorithms exist for which the time it takes to store (learn) new representations and to find the closest-matching stored representation (probabilistic inference) remains constant as additional representations are stored, this would meet the criterion of quantum computing. Such algorithms, based on SDR, have already been described. They achieve this "quantum speed-up" with no new esoteric technology, and in fact, on a single-processor, classical (Von Neumann) computer.

A Theory of Sequence Indexing and Working Memory in Recurrent Neural Networks

“…Such models include holographic reduced representations (HRR; Plate, 1991Plate, , 2003 and hyperdimensional computing (HDC) (Gayler, 1998;Kanerva, 2009), and will be referred to here by the umbrella term vector symbolic architectures (VSA; see Gayler, 2003; section 4.1.1). VSA models have been shown to be able to solve challenging tasks of cognitive reasoning (Rinkus, 2012;Kleyko & Osipov, 2014;Gayler, 2003). VSA principles have been recently incorporated into standard neural networks for advanced machine learning tasks (Eliasmith et al, 2012), inductive reasoning (Rasmussen & Eliasmith, 2011), and processing of temporal structure (Graves, Wayne, & Danihelka, 2014;Graves et al, 2016;Danihelka, Wayne, Uria, Kalchbrenner, & Graves, 2016).…”

Transparallel mind: classical computing with quantum power

“…Fairly recently, ideas have emerged-in both cognitive science and computer science-that certain distributed representations allow for information processing that is mathematically analogous to quantum computing (Aerts et al 2009), or might even allow for classical computing with quantum power (Rinkus 2012). Returning in these approaches is that good candidates for that are so-called holographic reduced representations (Plate 1991).…”

“…The idea that the brain represents information in a reduced, distributed, fashion has been around for a while (Hinton 1990). Fairly recently, ideas have emerged -in both cognitive science and computer science -that certain distributed representations allow for information processing that is mathematically analogous to quantum computing (Aerts et al 2009), or might even allow for classical computing with quantum power (Rinkus 2012).…”

Estimation of Vectors Similarity by Their Randomized Binary Projections

“…Some approaches and methods require specialized algorithms for storage and handling of vector data in a certain format (representation). Sparse [28,29] (with a small share of nonzero components) binary vectors are used, for example, in associative-projective neural networks [30,31] and in the efficient binary version [32][33][34] of distributed associative memory [10,11,[35][36][37][38].…”