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 $\mathcal S=\operatorname{span} \{ \lvert 0\rangle, \lvert 1\rangle \}$. 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 $\lvert \psi\rangle = \omega_1\lvert 0\rangle + \omega_2\lvert 1\rangle$, to satisfy $r^2\in\R$, while real symmetric coupling with complex diagonal regularities forces $r+1/r\in\R$. 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 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

arXiv:2604.23991 [quant-ph]

(or arXiv:2604.23991v2 [quant-ph] for this version)