Universal Complex Quantum-Like Bits from Hermitian Weighted Graphs

TL;DR

This paper explores how Hermitian weighted graphs can realize complex quantum-like bits with arbitrary states, establishing universality conditions and dense realizations in discrete graph models.

Contribution

It demonstrates that Hermitian couplings remove phase obstructions, enabling universal realization of complex qubit states in weighted graph models.

Findings

Hermitian coupling allows realization of any target state with prescribed eigenvalues.

Discrete graph models with entries in roots of unity are dense in the space of pure states.

A universality taxonomy for complex QL-bits is established based on graph structures.

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 $r=\omega_2/\omega_1$, for $\lvert \psi\rangle = \omega_1\lvert 0\rangle + \omega_2\lvert 1\rangle$, 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.