Universal Complex Quantum-Like Bits from Hermitian Weighted Graphs

• URL: http://arxiv.org/abs/2604.23991v1
• Date: Mon, 27 Apr 2026 03:09:53 GMT
• Title: Universal Complex Quantum-Like Bits from Hermitian Weighted Graphs
• Abstract summary: Block-coupled regular graphs can realize prescribed complex quantum-like bit states as exact synchronized eigenstates.<n>We show that exact discrete Hermitian realizations are dense in the synchronized pure-state space.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study when block-coupled regular graphs can realize prescribed complex quantum-like bit states as exact synchronized eigenstates. Two regular subgraphs $G_A$ and $G_B$ supply normalized all-ones eigenvectors $V_A$ and $V_B$, and algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a $2\times 2$ effective block on $\mathcal S=\operatorname{span} \{ \lvert 0\rangle, \lvert 1\rangle \}$. Within this reduction we prove that two natural symmetric complexifications are not universal under a real-spectrum requirement: complex symmetric coupling with real diagonal regularities forces the target computational basis amplitude ratio $r=ω_2/ω_1$, for $\lvert ψ\rangle = ω_1\lvert 0\rangle + ω_2\lvert 1\rangle$, to satisfy $r^2\in\mathbb{R}$, while real symmetric coupling with complex diagonal regularities forces $r+1/r\in\mathbb{R}$. 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 $\{0, \pm1, \pm i\}$ (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 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.

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