Universal Complex Quantum-Like Bits from Hermitian Weighted Graphs

Abstract

We study when block-coupled regular graphs can realize prescribed complex quantum-like (QL) 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 $\mathcal{S}=\mathrm{span}\{|0⟩,|1⟩\}$. 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 $r={\omega }_2/{\omega }_1$, for $|\psi ⟩={\omega }_1|0⟩+{\omega }_2|1⟩$, 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 $\{0,\pm 1,\pm i\}$ (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 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.

38 pages, 1 figure, 1 table. Substantially revised and expanded: rewritten introduction; added pseudo-Hermitian and exceptional-point analysis, full-spectrum isolation results, explicit finite graph-size bounds, and a worked balanced example