Construction and Rigorous Analysis of Quantum-Like States

TL;DR

This work formalizes the construction of quantum-like bits from classical, near-regular graphs by combining two regular subgraphs with an $l$-regular bipartite connector. It proves the existence of Hadamard-like basis states and shows how to realize arbitrary single-qubit-like states $|\psi\rangle=a|+\rangle+b|-\rangle$ by tuning graph regularities via symmetric detuning $\Delta$ or via asymmetric directed coupling with $\Delta_C$, including switching rules to cover all amplitude regimes. The authors connect these quantum-like states to classical random walks on the underlying graphs, analyze the role of the spectral gap, and explore extensions to real-valued, complex-weighted, and multi-QL-bit systems, highlighting potential pathways toward gates and higher-qubit architectures. The framework offers a rigorous, graph-theoretic mechanism for encoding and manipulating quantum-like information in classical networks, with implications for robust information storage and computation in synchronized-network systems and for understanding spectral-graph constraints on quantum-like dynamics.

Abstract

Extending upon the observations of the emergence of quantum-like states from classical complex synchronized networks, this work adds mathematical rigor to the analysis of single Quantum-Like (QL) bits constructed by eigenvectors of the adjacency matrices of such networks. First, we rigorously show that symmetric construction of such networks (regular undirected/symmetric bipartite graph $G_C$ connecting two regular undirected subgraphs $G_A,\,G_B$) leads to an equal superposition of the $|+\rangle, |-\rangle$ Hadamard states (with basis $|0\rangle,\,|1\rangle$ set from eigenvectors of the subgraphs), and provide an analysis of sufficient conditions on the network for construction of such states. Second, we prove two methods to construct any arbitrary single qubit state $|ψ\rangle = a|0\rangle + b|1\rangle,\, |a|^2+|b|^2=1$ and provide a switching lemma for the boundaries of both methods. The first method of construction is by detuning the regularities of the two subgraphs and the second is by asymmetrically constructing the bipartite connection matrix $C$ by allowing it to be directed, and then detuning those regularities. While the intuition is derived from the motivation of using complex synchronized networks for quantum information storage and computations, the proofs for constructing eigenvectors that interact in a quantum-like fashion only require the structure of the graph embedded in the adjacency matrix. Practically, this means that synchronization is not important to creating quantum-like bits, only that the edge weights are generally unit or close to unit and that the subgraphs are regular. As such, the results on combinations of random k-regular graphs (precisely Erdős-Rényi graphs) may be independently interesting.

Figures (8)

• Figure 1: Regular graph and adjacency spectrum. Each subgraph has n/2 nodes. m_subgraph is set to None, indicating no edges deleted for a total of 678 edges $\approx$ 2*(#edges in a k-regular subgraph of size $\frac{n}{2}$)+Pr(connect)*(#possible (undirected) connecting edges) = $2*(k(\frac{n}{2})/2)+0.1*(\frac{n}{2})^2 = 20*30+0.1*30^2 = 690$. Note that the two emergent eigenvectors are approximately $20\pm 3 = k\pm l$ for $k$-regular subgraphs $G_A$, $G_B$ and approximately $l$-regular connecting graph $G_C$ (randomly adding edges uniformly adds approximately $l \coloneqq \frac{\mathrm{\#edges~added}}{\mathrm{v}}*2$ edges to each node). See \ref{['lem:basic_comp_eigvecs']} for more details.
• Figure 2: Basic visualization of the bounds of $\Delta$ applied to $\lvert\psi\rangle$. The x-axis is $a$, the y-axis is $b$, and the z-axis is $\Delta$. The cylinder is defined by the circle (extended to 3D) $a^2+b^2=1$ and is the set of possible points under the definition of $\lvert\psi\rangle$ in \ref{['eqn:arb_qubit_def']}. By plotting it against $\Delta$, we can see what values $\lvert\psi\rangle$ are possible given the equation for $\Delta$, \ref{['eqn:delta_tuning_eqn']}, as shown by the four lines running up the cylinder. See text after \ref{['thm:arb_state_construction']} for more.
• Figure 3: Full visualization of the bounds of $\Delta$ applied to $\lvert\psi\rangle$. The x-axis is $a$, the y-axis is $b$, and the z-axis is $\Delta$. Refer to \ref{['fig:basic_visualization']} for introduction. Added to this plot are six planes. The vertical four planes are defined by the four variations of $a$ in \ref{['eqn:ab_dfn_by_delta']} ($\pm \otimes \pm$) and demonstrate the possible $a$ values given tunable parameter $\Delta$. As analyzed in \ref{['eqn:delta_int_constraints']}, $\Delta$ is also constrained to be a rational number. For simplicity, the plot shows only positive integer values of $\Delta$ as xy-planes perpendicular to the z-axis. Feasible values for $\lvert\psi\rangle$ are therefore constrained to intersections of the xy-planes with the feasible vertical lines.
• Figure 4: Basic visualization of the feasible values of $\Delta_C$ (green line, $\to \infty$ as $x\to y$) and $\Delta_C^{-1}$ (grey line, $\to \infty$ as $x \to -y$) applied to $\lvert\psi\rangle$. The x-axis is $a$, the y-axis is $b$, and the z-axis is $\Delta_C \; (\Delta_C^{-1})$. The cylinder is defined by the circle (extended to 3D) $a^2+b^2=1$ and is the set of possible points under the definition of $\lvert\psi\rangle$ in \ref{['eqn:arb_qubit_def']}. By plotting it against $\Delta_C \; (\Delta_C^{-1})$, we can see what values $\lvert\psi\rangle$ are possible given the equation for $\Delta_C \; (\Delta_C^{-1})$, \ref{['eqn:delta_c_tuning_eqn']}, as shown by the four lines running up the cylinder. See text after \ref{['lem:asym_coup_eigvecs']} for more.
• Figure 5: Full visualization of the feasible values of $\Delta_C$ (green line, $\to \infty$ as $x\to y$) and $\Delta_C^{-1}$ (grey line, $\to \infty$ as $x \to -y$) applied to $\lvert\psi\rangle$. The x-axis is $a$, the y-axis is $b$, and the z-axis is $\Delta$. Refer to \ref{['fig:basic_visualization_delta_c']} for introduction. Added to this plot are six planes. The four vertical planes are defined by the four variations of $a$ in \ref{['eqn:ab_dfn_by_delta_c']} ($\pm \otimes \pm$) and demonstrate the possible $a$ values given tunable parameter $\Delta_C \; (\Delta_C^{-1})$. As analyzed in \ref{['eqn:delta_c_int_constraints']}, $\Delta_C$ is also constrained to be a rational number. For simplicity, the plot shows only positive integer values of $\Delta_C$ as xy-planes perpendicular to the z-axis. Feasible values for $\lvert\psi\rangle$ are therefore constrained to intersections of the xy-planes with the feasible vertical lines.

Theorems & Definitions (6)

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