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.
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.
We study when block-coupled regular graphs can realize prescribed complex quantum-like (QL) 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 for state synthesis 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 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.
We study multi-qubit variational quantum states that can be considered as vertex- and edge-weighted graph. These states are constructed as single-layer variational circuits with RX rotations and RZZ entangling gates, corresponding to graphs of arbitrary structure. In general case of quantum graph states of arbitrary structure we derive the geometric measure of entanglement and evaluate quantum correlators. It is shown that these quantities are related to the edge-weight structure around the corresponding vertices in the graph (i.e., edge weights incident to the vertices and vertex weights associated with their closed neighborhoods). In the special case of quantum states representing unweighted graphs, these quantities are related to the degrees of the corresponding vertices in the graph. As an example, we analyze the state associated with the star graph K1,4 using noisy quantum computing on the AerSimulator. The results are in good agreement with theoretical predictions. These findings demonstrate a connection between graph structure and quantum properties, enabling the study of properties of classical graphs via quantum computing.
Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.
The continuous-time quantum walk is a particle evolving by Schr\"odinger's equation in discrete space. Encoding the space as a graph of vertices and edges, the Hamiltonian is proportional to the discrete Laplacian. In some physical systems, however, the Hamiltonian is proportional to the adjacency matrix instead. It is well known that these quantum walks are equivalent when the graph is regular, i.e., when each vertex has the same number of neighbors. If the graph is irregular, however, the quantum walks evolve differently. In this paper, we show that for some irregular graphs, if the particle is initially localized at a certain vertex, the probability distributions of the two quantum walks are identical, even though the amplitudes differ. We analytically prove this for a graph with five vertices and a graph with six vertices. By simulating the walks on all 1 018 689 568 simple, connected, irregular graphs with 11 vertices or less, we found 64 graphs with this notion of equivalence. We also give eight infinite families of graphs supporting these equivalent walks.
This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.
Abstract Braunstein et al have started the study of entanglement properties of the quantum states through graph theoretical approach. Their idea was to start from a simple unweighted graph G and then they have defined the quantum state from the Laplacian of the graph G . A lot of research had already been done using the similar idea. We ask here the opposite one i.e can we generate a graph from the density matrix? To investigate this question, we have constructed a unital map ϕ such that ϕ ( ρ ) = L ρ + ρ , where the quantum state is described by the density operator ρ . The entries of L ρ depends on the entries of the quantum state ρ and the entries are taken in such a way that L ρ satisfies all the properties of the Laplacian. This make possible to design a simple connected weighted graph from the Laplacian L ρ . We show that the constructed unital map ϕ characterize the quantum state with respect to its purity by showing that if the determinant of the matrix ϕ ( ρ ) − I is positive then the quantum state ρ represent a mixed state. Moreover, we study the positive partial transpose (PPT) criterion in terms of the spectrum of the density matrix under investigation and the spectrum of the Laplacian associated with the given density matrix. Furthermore, we derive the inequality between the minimum eigenvalue of the density matrix and the weight of the edges of the connected subgraph of a simple weighted graph to detect the entanglement of d 1 ⨂ d 2 dimensional bipartite quantum states. Lastly, we have illustrated our results with few examples.