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arXiv:2405.07950 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum-like states on complex synchronized networks

Gregory D. Scholes

Published Aug 1, 2024• 19 citations
arXivHTMLPDF

Abstract

Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a ‘QL advantage’ for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions.

DOI
10.1098/rspa.2024.0209

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Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a ‘QL advantage’ for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions.
Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL dits, as higher-dimensional analogs of QL bits. We employ systems of k-regular graphs to construct QL-dits for arbitrary dimensions, where the emergent eigenspectrum of their adjacency matrices defines the QL-state space. The tensor product structure of multi-QL dit systems is realized through the Cartesian product of graphs. Furthermore, we examine the potential computational advantages of employing d-nary QL systems over two-level QL bit systems, particularly in terms of classical resource efficiency. Overall, this study generalizes the paradigm of using synchronized network dynamics for QL information processing to include higher-dimensional QL resources.
Abstract In recent work we have developed a theory for classical systems that mimic the state space of quantum systems. We call the states ‘quantum-like’ (QL) because they show many properties similar to those of a quantum system. Here we study the QL states of a large classical system comprising a network of coupled phase oscillators. In particular, we seek to understand how the nonlinearity of the classical system manifests in the linear representation of the state space. To that end, we study how a classical network is viewed from the perspective of the time evolution of its emergent state. We investigate how mixedness of the QL states diminish or increase as the underlying classical system synchronizes or de-synchronizes respectively. In the limit of a very strongly phase-locked classical network—that is, where couplings between phase oscillators are very large—the state space evolves according to unitary dynamics. When the system is set up so that the classical phase oscillators synchronize with time, we find that the purity measure of the QL states correspondingly increases. Whereas, in cases where the coupling between oscillators is too small to allow the network to synchronize, the classical variables act like a kind of environment that promotes dephasing of initially-prepared superpositions.
Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL-states and QL-gate operations, emerging from the dynamics of complex synchronized networks. The present work builds on this framework to present a simulation of quantum dynamics in spin systems. We focus on the transverse field Ising model, a widely studied Hamiltonian for interacting spin systems. We show how circuits, built using the QL analog of quantum gates, can be used to prepare an arbitrary initial wavepacket and simulate the time evolution of the resulting state for extracting physical observables such as magnetization. Beyond the statevector dynamics, we introduce a set of metrics that characterize how QL-circuit operations transform the underlying graph connections. These metrics quantify the microscopic coupling, phase coherence, and effective inter-block coupling generated by the circuit. These graph-level quantities provide constraints and design targets for potential physical implementations, indicating where interconnection resources, phase control, and effective coupling strength must be engineered in a hardware realization. Overall, this study demonstrates how classical resources can be harnessed to simulate quantum systems and we discuss the advantages and limitations.
We present a quantum-like (QL) model in that contexts (complexes of e.g. mental, social, biological, economic or even political conditions) are represented by complex probability amplitudes. This approach gives the possibility to apply the mathematical quantum formalism to probabilities induced in any domain of science. In our model quantum randomness appears not as irreducible randomness (as it is commonly accepted in conventional quantum mechanics, e.g., by von Neumann and Dirac), but as a consequence of obtaining incomplete information about a system. We pay main attention to the QL description of processing of incomplete information. Our QL model can be useful in cognitive, social and political sciences as well as economics and artificial intelligence. In this paper we consider in a more detail one special application -- QL modeling of brain's functioning. The brain is modeled as a QL-computer.
The main purpose of thispaper is to show that composite quantum-like (QL) systems can closely mimic the separable states of quantum systems, and that suitable physical systems exhibiting these states exist. It is shown that QL graphs can closely emulate states of composite quantum systems, such as coupled two-level systems that display separable linear combinations of states. Examples of classical systems are suggested that show these states. These include multipole moments of waves or networks of phase oscillators. The work indicates that composite QL states can be manifest in complex network structures relevant to quantum biology or engineered into circuits, or even possibly soft matter.
In previous work we proposed a construction of quantum-like bits that could endow a large synchronizing classical system, for example, of oscillators, with quantum-like function that is not compromised by decoherence. In the present paper we investigate further this platform of quantum-like states. First, we discuss a general protocol on how to construct classical synchronizing networks that allow for emergent states. We then study how gates can be implemented on those states. This suggests the possibility of quantum-like information processing on a special class of many-body classical systems. Finally, we show that our approach allows for non-Kolmogorov interference, a feature that separates our model from a classical probabilistic system. This paper aims to explore the mathematical structure of quantum-like resources distilled from classical synchronizing systems and shows how arbitrary gates can be implemented by manipulating many-body correlations.