Abstract We establish a framework for constructing robust quantum-like states based on k -regular bipartite graphs, aiming to realize quantum-inspired information processing through classical oscillatory networks. We first introduce the k -regular bipartite graph and its mathematical representation, and analyze its intrinsic spectral properties. In contrast to ordinary k -regular non-bipartite graphs, the intrinsic bipartite symmetry, enhanced spectral gap, and global delocalization mechanism confer superior robustness to the emergent states. We verify that the emergent states of these bipartite graphs satisfy the axiomatic requirements of quantum-like states, and show that the global delocalization property ensures stable operation under structural noise. Based on the axiomatic requirements of quantum-like states, we further realize basic quantum-inspired gate operations in the classical network framework. Our numerical simulations reveal that the k -regular bipartite graph system exhibits enhanced coherence and stability under noise interference compared with non-bipartite graph systems. This research not only broadens the application boundaries of quantum-inspired computing but also provides a new feasible path for the practical implementation of classical simulation of quantum information processing.
Abstract We establish a framework for constructing robust quantum-like states based on k -regular bipartite graphs, aiming to realize quantum-inspired information processing through classical oscillatory networks. We first introduce the k -regular bipartite graph and its mathematical representation, and analyze its intrinsic spectral properties. In contrast to ordinary k -regular non-bipartite graphs, the intrinsic bipartite symmetry, enhanced spectral gap, and global delocalization mechanism confer superior robustness to the emergent states. We verify that the emergent states of these bipartite graphs satisfy the axiomatic requirements of quantum-like states, and show that the global delocalization property ensures stable operation under structural noise. Based on the axiomatic requirements of quantum-like states, we further realize basic quantum-inspired gate operations in the classical network framework. Our numerical simulations reveal that the k -regular bipartite graph system exhibits enhanced coherence and stability under noise interference compared with non-bipartite graph systems. This research not only broadens the application boundaries of quantum-inspired computing but also provides a new feasible path for the practical implementation of classical simulation of quantum information processing.
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.
The present work is motivated by the need for robust, large-scale coherent states that can play possible roles as quantum resources. A challenge is that large, complex systems tend to be fragile. However, emergent phenomena in classical systems tend to become more robust with scale. Do these classical systems inspire ways to think about robust quantum networks? This question is studied by characterizing the complex quantum states produced by mapping interactions between a set of qubits from structure in graphs. We focus on maps based on k-regular random graphs where many edges were randomly deleted. We ask how many edge deletions can be tolerated. Surprisingly, it was found that the emergent coherent state characteristic of these graphs was robust to a substantial number of edge deletions. The analysis considers the possible role of the expander property of k-regular random graphs.
Graph states are entangled states that are essential for quantum information processing. As experimental advances enable the realization of large-scale graph states, efficient fidelity estimation methods are crucial for assessing their robustness against noise. However, calculations of exact fidelity become intractable for large systems due to the exponential growth in the number of stabilizers. In this work, we show that the fidelity between any ideal graph state and its noisy counterpart under IID Pauli noise can be mapped to the partition function of a classical spin system, enabling efficient computation via statistical mechanical techniques. Using this approach, we analyze the fidelity for regular graph states under depolarizing noise and uncover the emergence of phase transitions in fidelity between the pure-state regime and the noise-dominated regime. Specifically, in 2D, phase transitions occur only when the degree satisfies d ≥ 6 , while in 3D they already appear at d ≥ 5 . However, for graph states with excessively high degree, such as fully connected graphs, the phase transition disappears. Robustness of graph states against noise is thus determined by their connectivity and spatial dimensionality. Graph states with lower degree and/or dimensionality, which exhibit a smooth crossover, demonstrate greater robustness, while highly connected or higher-dimensional graph states are more fragile. Extreme connectivity, as the fully connected graph state possesses, restores robustness. Furthermore, we show that the fidelity can be rewritten in the form of the partition function of a constraint-percolation problem. Within this picture, we discuss the qualitative difference between 2D regular graph states with d = 6 and d = 5 regarding the presence or absence of a phase transition, as well as the suppressed critical behavior of fully connected graph states.
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.
Spectral clustering is one of the most robust methods of an unsupervised machine learning field. Compared with the well-known k-means, spectral clustering uses the spectral properties of the Laplacian matrix to project data to a low-dimensional clustering with higher efficiency. However, for a sparse network with a substantial degree of heterogeneity, standard spectral clustering often overfits noise in the periphery of a sparse and stochastic graph. In our work, we combined the quantum version of the k-means algorithm with regularized spectral clustering and proposed a new quantum machine learning algorithm. It is more like a combination of classical and quantum algorithms. Compared with the established quantum k-means algorithm, our method has obvious computational advantages in non-convex or nested structures. At this stage, the research of quantum algorithms mainly stays in the simulation of classical computers. Our method does not need to expend a lot of effort to preprocess the data at the initial quantum state stage. Under the assumption of quantum data encoding, we theoretically demonstrate the feasibility of the quantum regularized spectral clustering algorithm. Then combine Q-means algorithm complexity advantages to complete the clustering task. This work aims to pave the path and help realize quantum spectral clustering algorithms on real quantum computers in the future and help create other versions of quantum machine learning algorithms. We provide simulations and data examples to illustrate these results.
In previous work we proposed a construction of quantum-like bits that could endow a large synchronizing classical system, for example, of oscillators, with quantum-like function that is not compromised by decoherence. In the present paper we investigate further this platform of quantum-like states. First, we discuss a general protocol on how to construct classical synchronizing networks that allow for emergent states. We then study how gates can be implemented on those states. This suggests the possibility of quantum-like information processing on a special class of many-body classical systems. Finally, we show that our approach allows for non-Kolmogorov interference, a feature that separates our model from a classical probabilistic system. This paper aims to explore the mathematical structure of quantum-like resources distilled from classical synchronizing systems and shows how arbitrary gates can be implemented by manipulating many-body correlations.