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.
We study when block-coupled regular graphs can realize prescribed complex quantum-like bit states as exact synchronized eigenstates. Two regular subgraphs GA and GB supply normalized all-ones eigenvectors VA and VB, and algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a 2×2 effective block on S=span{∣0⟩,∣1⟩}. 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=ω2/ω1, for ∣ψ⟩=ω1∣0⟩+ω2∣1⟩, to satisfy r2∈R, while real symmetric coupling with complex diagonal regularities forces r+1/r∈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,±1,±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.
Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL dits, as higher-dimensional analogs of QL bits. We employ systems of k-regular graphs to construct QL-dits for arbitrary dimensions, where the emergent eigenspectrum of their adjacency matrices defines the QL-state space. The tensor product structure of multi-QL dit systems is realized through the Cartesian product of graphs. Furthermore, we examine the potential computational advantages of employing d-nary QL systems over two-level QL bit systems, particularly in terms of classical resource efficiency. Overall, this study generalizes the paradigm of using synchronized network dynamics for QL information processing to include higher-dimensional QL resources.
We define a weighted graph associated with the standard realification of a qubit Hamiltonian, whose vertices correspond to the four real coordinate channels of the doubled state space and whose edge weights are determined by the magnitudes of infinitesimal couplings in the realified Schrödinger equation. We prove that this graph is connected if and only if the Hamiltonian possesses nonzero off-diagonal coupling in the basis in which the graph is constructed, derive the complete weighted-Laplacian spectrum in closed form, and obtain an explicit expression for its Fiedler value. We propose a two-qubit extension supported by structural argument and numerical evidence. The graph depends on the chosen computational basis; whether an equivalent basis-independent construction exists remains an open question.
Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a ‘QL advantage’ for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions.
Given a graph with Hermitian adjacency matrix H, perfect state transfer occurs from vertex a to vertex b if the (b,a)-entry of the unitary matrix exp(−iHt) has unit magnitude for some time t. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are ±i), the oriented 3-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.
Overview This work develops a finite-orbit extension of reflected spectral-jet methods for degenerate minimax problems with symmetry-related active spectral branches. The main objective is to move beyond a single reflected pair of branches. A finite symmetry group acts simultaneously on normal coordinates and active branch labels, producing a finite orbit of competing spectral branches. The paper introduces group-averaged spectral jets, centered orbit deviations, primitive quadratic-isolation criteria, representation-theoretic tests for low-order branch splitting, and analytic descriptions of multi-branch coactivity geometry. The abstract framework is realized in: exact non-diagonal C3- and D3-symmetric Hermitian models; a fixed-scenario rank-two Hermitian compression problem; a three-location single-fault Pauli-history model; and the full eight-label independent three-bit quantum error-history process. The full-process QEC realization exhibits a particularly sharp phenomenon: truncating the history space to zero- and single-fault events can reverse the local optimization geometry of the same quantum code. Finite-orbit quadratic isolation Let G be a finite group acting on a normal space N and on a finite transitive branch orbit Ω. For active branches ℓω, define the centered branch deviation relative to the group-averaged jet JG: dω=ℓω−JG. If the orbit average has uniform quadratic normal concavity, Dn2JG⪯−2μz2I, while every centered branch deviation vanishes through second order at the group-fixed manifold and has uniformly controlled cubic remainder, then maxω∈Ωℓω(z,p,s,n)≤JG(z,p,s,0)−2μz2∣n∣2 throughout a sufficiently small normal tube. This generalizes reflected-pair isolation to a finite active orbit. The paper also identifies a representation-theoretic sufficient condition for the disappearance of low-order centered splitting: HomG(VΩ0,Symk(N))={0}, where VΩ0 is the zero-sum branch-label representation. Multi-branch coactivity For analytic optimized branches W1,…,Wm, normalized branch-difference maps are used to define continued coactivity sets through a degenerate perturbation point. If k+1 branches are active and the normalized difference map has rank k, then the corresponding regular coactivity set is locally a codimension-k real-analytic manifold. Thus: two active branches generate a codimension-one transition hypersurface; three active branches generate a codimension-two triple-coactivity locus; and higher active multiplicities generate higher-codimension coactivity strata. Near a regular coactivity point, the active envelope is locally equivalent to the standard chamber fan max{0,y1,…,yk}. The paper also distinguishes these regular strata from singular orbit-fixed junctions, where centered branch splitting begins quadratically or cubically and the normalized difference Jacobian vanishes. Hermitian realizations An exact C3/D3-symmetric non-diagonal Hermitian prototype realizes: a proper group-fixed manifold; three distinct simple spectral branches; cubic orbit splitting; exact quadratic normal isolation; three dominance chambers; and an isolated triple-coactivity junction. A second realization starts from a fixed 4×4 Hermitian scenario and its C3 unitary orbit. Compression to a moving rank-two graph subspace produces the exact jet JG(z,ν)=1−32z2∣ν∣2+32z3∣ν∣2Re(ω−jν)+2726z4∣ν∣4+O(z5), together with an exact quadratic-isolation inequality. Three-location finite-history QEC realization The single-fault model uses the four history labels {I,X0,X1,X2} on three physical qubits. For the repetition code C=span{∣000⟩,∣111⟩}, the Knill–Laflamme scalar-compression conditions hold exactly in this restricted history sector, so the environment-record leakage vanishes. A Fourier-syndrome graph deformation produces three cyclic marginal-leakage branches with exact compressed products Pj(z,q)=1+q1[32zRe(ω−jν)σz+3qσx],q=z2∣ν∣2. The centered splitting is quadratic rather than cubic. The resulting upper envelope has six dominance sectors and a singular triple-coactivity junction. This gives a concrete QEC realization of the fact that finite C3 symmetry alone does not eliminate quadratic orbit splitting. Full eight-label three-bit process The complete independent three-bit history process is indexed by h∈{0,1}3⟹Wh=X0h0X1h1X2h2. For an arbitrary rank-two code, the complete complementary-register leakage admits the exact Bloch reduction L8(V)=max∣u∣=1∑s=0(u⋅ks)DpPsDp1, where ks is the Bloch vector of the traceless part of the compressed history product V†WsV. At the repetition code, the full process has an exact logical-complement obstruction: L8(C)=8∏j=02εj(1−εj). For equal independent bit-flip probabilities (εj=ε), let τ=[ε(1−ε)]3/2. Then L8(C)=8τ. For the Fourier-syndrome graph family, the nontrivial history products split into: a weight-one C3 orbit; a weight-two C3 orbit; and the triple-event logical-complement singleton. In the intrinsic graph coordinate w, the complete leakage satisfies L8(w)=8τ−16τ∣w∣2+o(∣w∣2)(w→0). Consequently, the repetition code is a strict local maximum of the complete eight-label leakage throughout the two-real-dimensional Fourier-syndrome graph chart. Sector-truncation reversal The same code and the same graph deformation have opposite local effects in the truncated and complete history models. In the four-label single-fault sector, Lsingle(C)=0, and every nontrivial sufficiently small graph deformation produces positive leakage. In the full eight-label process, L8(C)=8τ, and the same deformation decreases leakage quadratically. Thus the paper proves an exact sector-truncation reversal: local protection in the truncated process⟷local leakage maximality in the full process The restoration of double- and triple-event histories does not merely change the leakage quantitatively. It changes the local minimax geometry itself. Variational consequence The finite-orbit isolation mechanism enters hard-normal minimax selection through the effective lower functional Qeff(α,β,n)=H(a0,t)+β+21⟨Qα,α⟩+c∣n∣2,β≥0. This yields compactness, localization, and accumulation constraints for exact and near minimizers. Scope The paper does not claim: a finite-parameter closed formula for the complete eight-label leakage away from the local graph regime; a full-Grassmannian minimax theorem for the three-bit process; a complete classification of all minimizing or zero-leakage codes; a fourth-order classification of the interaction between the weight-one and weight-two history orbits; or a treatment of semisimple active spectral clusters. The active spectral branches in the abstract framework are assumed to be locally simple. Matrix-valued spectral jets associated with multiple eigenvalue clusters remain a separate extension. Reproducibility The accompanying standalone research bundle contains: the manuscript PDF and LaTeX source; theorem-facing proof notes for the single-fault and full eight-label QEC models; a deterministic SymPy verification script; a machine-readable JSON verification report; a tested Python/SymPy environment specification; a portable SHA-256 manifest verifier; a complete SHA-256 manifest; and package-status, audit, and change-log files. The verifier checks exact covariance identities, orbit averages, centered jet cancellations, characteristic polynomials, Hermitian compression formulas, Pauli-product compressions, leakage coefficients, logical-complement constants, and intrinsic graph-coordinate expansions. Compactness, analytic continuation, uniform branch localization, trace-norm asymptotics, variational liminf arguments, and recovery constructions are supplied analytically in the manuscript and proof notes rather than claimed as fully machine-verified. All proof-critical scripts reject optimized Python mode, and no floating-point comparison enters a PASS decision. Related work This paper extends the reflected-pair framework developed in: Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions, Version v0.2r1, DOI: 10.5281/zenodo.21636022. Its QEC applications are also related to the finite-history minimax line developed in: Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax, Version v0.3r2, DOI: 10.5281/zenodo.21598820.
This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: rgen=2ν(G×), where ν(G×) is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network Kp,q, the exact rank is rankKθ=min{p+1,q+1,ord(eiθ)}, with infinite multiplicative order interpreted in the natural way. The K2,2 example therefore exhibits a rank stratification (4→3→2) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with rstr=4,rsym=3. Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3