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Entropy

Multi-Dimensional Quantum-like Resources from Complex Synchronized Networks

Debadrita Saha, Gregory D. Scholes

Published Sep 16, 2025• 1 citation
Publisher

Abstract

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.

DOI
10.3390/e27090963

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We study when block-coupled regular graphs can realize prescribed complex quantum-like (QL) bit states as exact synchronized eigenstates. Two regular subgraphs GAG_AGA​ and GBG_BGB​ supply normalized all-ones eigenvectors VAV_AVA​ and VBV_BVB​, and algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a 2×22\times 22×2 effective block on S=span⁡{∣0⟩,∣1⟩}\mathcal S=\operatorname{span} \{ \lvert 0\rangle, \lvert 1\rangle \}S=span{∣0⟩,∣1⟩}. Within this reduction we prove that two natural symmetric complexifications are not universal for state synthesis under a real-spectrum requirement: complex symmetric coupling with real diagonal regularities forces the target computational basis amplitude ratio r=ω2/ω1r=ω_2/ω_1r=ω2​/ω1​, for ∣ψ⟩=ω1∣0⟩+ω2∣1⟩\lvert ψ\rangle = ω_1\lvert 0\rangle + ω_2\lvert 1\rangle∣ψ⟩=ω1​∣0⟩+ω2​∣1⟩, to satisfy r2∈Rr^2\in\Rr2∈R, while real symmetric coupling with complex diagonal regularities forces r+1/r∈Rr+1/r\in\Rr+1/r∈R. Replacing complex symmetry by Hermitian coupling removes this phase obstruction. For any nonbasis target state, any prescribed real eigenvalue, and any prescribed nonzero signed spectral gap, a Hermitian weighted coupling realizes the target exactly. Additionally, an independently tuned directed-coupling model gives a second universality mechanism. We then pass from continuous effective parameters to finite weighted graphs with entries in {0,±1,±i}\{0, \pm1, \pm i\}{0,±1,±i} (the fourth roots of unity and zero), characterize the balanced discrete coupling lattice by perfect matchings, and show that exact discrete Hermitian realizations are dense in the synchronized pure-state space. These results give a state-synthesis universality taxonomy for complex QL-bits and identify Hermitian conjugate pairing as the robust structural mechanism that supports arbitrary complex amplitudes with real two-level spectra.