  Quantum superposition says 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., its probability of being observed, 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 small subset
of an overall population of units, and the subsets can overlap. Specifically, I
consider an SDR model in which the overall population consists of Q WTA
clusters, each with K binary units. Each coefficient is represented by a set of
Q units, one per cluster. Thus, K^Q coefficients can be represented with KQ
units. Thus, the particular world state, X, whose coefficient's representation,
R(X), is the set of Q units active at time t has the max probability and the
probability of every other state, Y_i, at time t, is measured by R(Y_i)'s
intersection with R(X). Thus, R(X) simultaneously represents both the
particular state, X, and the 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 meets the criterion of quantum computing. Such an algorithm has
already been described: it achieves this "quantum speed-up" without esoteric
hardware, and in fact, on a single-processor, classical (Von Neumann) computer.
