Nonequilibrium quantum states emerging from flippable spins in random networks

The gist: The eigenstate entanglement of the quantum Ising model depends on the structure of the interaction network, and this dependence originates from the structure of the Hilbert space diagram, which in turn is quantified by its equienergy subgraph.

Introduction to Irregular Interaction Networks

Quantum many-body systems are typically studied assuming translational symmetry in the interaction network Recent experimental advances in various platforms for quantum simulators have enabled the realization of irregular interaction networks, which are intractable to implement with conventional crystal lattices Another hallmark of these advances is the ability to observe the time-dependent behaviour of quantum many-body systems However, the relationship between irregular interaction networks and quantum many-body dynamics remains poorly understood. Here, we investigate the connection between the structure of the interaction network and the eigenstate entanglement of the quantum Ising model by exploring all possible interaction networks up to seven spins. We find that the eigenstate entanglement depends on the structure of the Hilbert space diagram, particularly the structure of the equienergy subgraph. We further reveal a correlation linking the structure of the Hilbert space diagram to the number of unconstrained spin pairs. Our results demonstrate that the minimum eigenstate entanglement of the quantum Ising model is governed by the specific structure of the interaction network.

Eigenstate Entanglement and Hilbert Space Structure

Quantum entanglement is a key quantity for understanding quantum many-body systems The von Neumann entanglement entropy of half the system is a standard measure of entanglement, but for arbitrary network structures, it is often difficult to divide the system fairly into two subsystems due to irregularity. To address this, we use the multipartite (global) entanglement which is defined as Q (rho B) B 1 − 1/N ∑ i=1 tr rho 2 i = 1/N ∑ i=1 2 h i h 1 − tr rho 2 i i = 1/N ∑ i=1 Q (rho B i). We diagonalize the Hamiltonian in equation (1) to obtain the eigenenergies and eigenstates, Hˆ E B = E B. We then compute the eigenstate entanglement using equation (2), Q(E mu E mu). Figure 1 shows the eigenenergies and the corresponding entanglement. The results demonstrate that the profile of eigenstate entanglement varies depending on the network structure. For example, networks 4-23 and 4-41 exhibit lower eigenstate entanglement. This trend of reduced eigenstate entanglement persists as the system size increases up to seven spins.

Structure of the Equienergy Subgraph

We examine the Hilbert space structure of the quantum Ising model by rewriting the Hamiltonian as an Anderson model of localization in Hilbert space. The Hamiltonian in equation (1) can be expressed as Hˆ = sum s s> ⟨s! Hˆ sum s' s'> ⟨s'! = sum s D ss E B s> E B + sum ss'!= s D ss' E B s'> ⟨ s + sum i h'i X B i, (3) where the first term of equation (3) depends on the interaction network. This formulation can be interpreted as a single quantum particle hopping on the N-dimensional hypercube, where the on-site energy is given by U (s) and the hopping energy is given by h i;x. The Hilbert space described by equation (1) forms an N-dimensional hypercube, which can be visualized as a projection of the hypercube into lower dimensions.

Quantifying Network Structure via Circuit Rank

To quantify the Hilbert space structure from the perspective of energy conservation, we investigate the equienergy subgraph of the Hilbert space diagram. We define the equienergy subgraph as the set of vertices for which U (s) = U (s'). The structure of the equienergy subgraph is quantified by the circuit rank, defined as r B e − v + c, (4) where e is the number of edges, v is the number of nodes, and c is the number of connected components. We find that in general, the minimum eigenstate entanglement decreases as the circuit rank of the equienergy subgraph increases. This anticorrrelation becomes stronger as the number of spins in the interaction network increases.

Connection to Unconstrained Spin Pairs

We introduce a structural feature of interaction networks: the degree of each vertex. From the perspective of energy-conserving dynamics, we identify two requirements for hopping in Fock space that are related to the degree of the spins. First, only spins with even degree can flip without changing U (s) because flipping a spin with odd degree always increases or decreases U (s), while flipping an even-degree spin can preserve the energy. Second, to increase the circuit rank of the equienergy subgraph, evendegree spins must not be directly connected. We introduce a quantity to characterize whether a given interaction network satisfies the requirements described above. We define the number of unconstrained spin pairs as L B sum i=1 sum j=i+1 l ij delta Ai,j,0, (5) where l i B (1 + d i) mod 2 ∈ [0, 1] is the flip freedom of the i-th spin. We numerically calculate the level of unconstrained spins L for a given network and examine its relationship with the circuit rank of the equienergy subgraph. We find that the circuit rank of the equienergy subgraph exhibits a strong linear correlation with the level of unconstrained spins L across all network structures up to seven spins.

Dynamical Features and Information Preservation

To address how the low-entanglement eigenstate relates to system’s dynamics, we investigate dynamics from Hilbert space structure. We compute the generalized imbalance I (t) B 1/N sum i=1 psi(0) Z B i psi(t), (6) which quantifies how well the system retains information about its initial state. We perform a Fourier transform of the generalized imbalance to extract the maximum amplitude ˜Imax B max f ˜I (f) and its corresponding frequency fmax B arg max f ˜I (f). When the system is initialized in s c⟩, the maximum amplitude is generally high (˜Imax ≳ 0.1), while the corresponding frequency spans a wide range. These results indicate that the oscillatory behaviour of the generalized imbalance depends on the choice of initial state.

Conclusion

We have shown that the minimum eigenstate entanglement anticorrelates with the circuit rank of the equienergy subgraph of the Hilbert space diagram. We then investigate how the circuit rank is determined by the number of unconstrained spin pairs. We further confirm that the minimum eigenstate entanglement anticorrelates with the number of unconstrained spin pairs L. Finally, we quantify oscillatory time evolution and show that depending on the initial state, the oscillatory behaviour of the generalized imbalance varies.

Limitation and Future Outlook

A limitation of this work is the finite size of the system. As the system size increases, we find that (anti)correlation coefficients in our results approach finite values. We expect that interaction network structure will similarly affect many-body phenomena in the Hubbard model, which we leave for future work. Our work serves not only as a comprehensive catalogue but also as an example of experimentally verifiable predictions connecting network structure and dynamics.

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Generating all possible interaction networks up to seven nodes. Among the generated networks, we remove the unconnected networks and isomorphic networks. For each network structure, we generate the interaction matrix J by assigning J i,j ∈ [+1, -1] to the edges of the network. After the assigning, we check the isomorphism of the networks and removed the isomorphic networks. The total number of network structure are summarized in Table 1.

Table 1 | The number of network structures up to seven vertices. Number of vertices Number of network Number of connected networks 3 4 2 7 4-spin systems |

4 |

11 |

6 |

53

5-spin systems |

34 |

21 |

712

6-spin systems|

156 |

112

Code availability Calculations and visualizations of this work are performed using open-source Python libraries: Matplotlib, NetworkX, NumPy, QuTiP–SciPy. The colour maps of ColorCET are used. All code used to reproduce this work will be available on Zenodo.

1. No-Preamble Network Structure Mapping: The improved AI can map arbitrary interaction networks to a Hilbert space diagram and identify its equienergy subgraph, which is crucial for understanding entanglement.

2. Minimum Entanglement Prediction: The system can predict the minimum eigenstate entanglement of the quantum Ising model based on the structure of the interaction network by correlating it with the circuit rank of the equienergy subgraph.

3. Unconstrained Spin Pair Identification: The AI can determine the number of unconstrained spin pairs by calculating a quantity related to flip freedom, linking this structural feature directly to dynamical properties.

4. Dynamic Feature Prediction: The system can analyze oscillatory behavior in the generalized imbalance via Fourier transform to predict how information is preserved based on the initial state, distinguishing between initializing in "s c> versus s f>."