Quantum Computation via Sparse Distributed Representation

Abstract: 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.

• The paper's main contribution is the introduction of a classical analog of quantum superposition through sparse distributed representations to efficiently represent an exponential number of states.
• It details a methodology using Q clusters with K binary units, enabling simultaneous state updates and reducing the need for exponential computational resources.
• The approach offers practical benefits for pattern matching and probabilistic inference, suggesting that classical systems can achieve quantum-like speed-up.

Quantum Computation via Sparse Distributed Representation

Introduction

The paper "Quantum Computation via Sparse Distributed Representation" (1707.05660) presents a novel approach to quantum computation by leveraging sparse distributed representations (SDR) to implement classical analogs of quantum superposition. The author challenges the prevailing localist representation assumption in quantum theory (QT) and quantum computing (QC), proposing instead that SDRs offer an effective means to mimic quantum superposition within a classical computational framework. This conceptual shift promises significant computational advantages, particularly in terms of complexity and processing speed.

Sparse Distributed Representations in Quantum Context

In traditional quantum mechanics, each possible state of a quantum system is characterized by a probability amplitude coefficient, typically represented in a disjoint or localist manner. This paper proposes an SDR-based approach, wherein these coefficients are represented via overlapping subsets of representational units. Such representation allows for a classical computation model that echoes the quantum superposition by representing exponential possibilities without the localist constraint of separate state representations.

The SDR model introduced consists of $Q$ clusters, each containing $K$ binary units. Each quantum state's probability amplitude can be represented by activating $Q$ units, one per cluster. Consequently, an exponential number of states, $K^Q$, can be represented distributively while maintaining a fixed number of units. Maxwell's probability distribution over all states maps onto the overlap size of these sparse representations.

Computational Implications

An SDR-based perspective permits simultaneous representation and update of all quantum states, unlike traditional quantum computing requiring separate calculations for each potential state. The author argues that classical computational resources could achieve "quantum speed-up" by iteratively using SDR, where computations on individual units affect all the states their codes participate in. This obviates the exponential resource requirement plaguing current quantum simulations.

A key computational advantage of SDRs lies in their ability to map similar input states to closely overlapping codes, a property termed Similar Input Similar Code (SISC). This allows for efficient probabilistic inference and pattern matching, which are essential for tasks like image retrieval.

Theoretical and Practical Implications

The approach redefines how classical systems can mimic quantum phenomena. By moving from localist to distributed representations, this framework could potentially unravel a classical interpretation for complex quantum properties like superposition and entanglement. The SDR method also suggests a promising route for achieving practical quantum computation on classical hardware without new, non-traditional technology.

Exploratory avenues for future research include integrating this SDR approach with alternative representation models like Geometric Algebra (GA) and further refining the associated algorithms to optimize performance in real-world applications. Moreover, exploring the transition from binary to real-valued units might refine the model's applicability across various cognitive and computational domains.

Conclusion

Gerard J. Rinkus's proposition of implementing quantum computation through sparse distributed representations marks a pivotal shift in quantum computational theory. This paradigm not only challenges established localist depiction standards but also presents a feasible pathway for classical systems to emulate quantum processes efficiently. The framework's promise of computational scalability and enhanced algorithmic efficiency situates it as a noteworthy candidate for future advancements in both AI and quantum computing fields.