Extended quantum models

Hongyu Wang, Yingcheng Li, Yuting Hu +1 more

Abstract We generalize the electric-magnetic (EM) duality in the quantum double (QD) models to the extended QD models of topological orders with gapped boundaries. We also map the extended QD models to the extended Levin-Wen (LW) models with gapped boundaries. To this end, we Fourier-transform and rewrite the extended QD model on a trivalent lattice with a boundary, where the bulk gauge group is a finite group G. Gapped boundary conditions of the model before the transformation are known to be characterized by the subgroups K ⊆ G supplying the boundary degrees of freedom. We find that after the transformation, the boundary conditions are then characterized by the Frobenius algebras AG,K in RepG. An AG,K is the dual space of the quotient of the group algebra of G over that of K , and RepG is the category of the representations of G. The EM duality on the boundary is revealed by mapping the K ’s to AG,K ’s. We also show that our transformed extended QD model can be mapped to an extended LW model on the same lattice via enlarging the Hilbert space of the latter. Moreover, our transformed extended QD model elucidates the phenomenon of anyon splitting in anyon condensation.

Miroslav Svítek

S. Baid, Sáiz, A., Lucas Lamata +5 more

We investigate the Extended Lipkin Model (ELM), whose phase diagram mirrors that of the Interacting Boson Approximation model (IBA). Unlike the standard Lipkin model, the ELM (as the IBA) features both first- and second-order quantum shape phase transitions depending on the model parameters. Our goal is to implement the ELM on a quantum platform, leveraging Machine Learning techniques to identify its quantum phase transitions and critical lines. To achieve this, we offer: i) ground state energy calculations using a variational quantum eigensolver; ii) a detailed formulation for ELM dynamics within quantum computing, facilitating experimental exploration of the IBA phase diagram; and iii) a phase diagram determination using various Machine Learning methods. We successfully replicate the ELM ground-state energy using the Adaptive Derivative-Assembled Pseudo-Trotter ansatz Variational Quantum Eigensolver (ADAPT-VQE) algorithm across the entire phase space. Our framework ensures ELM implementation on quantum platforms with controlled errors. Lastly, our ML predictions yield a meaningful phase diagram for the model. Keywords: Quantum Platforms Nuclear Models ADAPT-VQE Quantum Shape Phase Transitions Interacting Boson Approximation Extended Lipkin Model Machine Learning

Daowen Qiu

Peter P. Rohde

S. Baid, Álvaro Sáiz, Lucas Lamata +5 more

Background: In recent years, the implementation of nuclear physics models in quantum computers has emerged as a promising and novel area of research. Simultaneously, the study of quantum shape phase transitions in nuclear models has gained significant attention. Specifically, the phase diagram of the interacting boson approximation (IBA) has been extensively explored, particularly in connection with large-particle-number-limit considerations. Interestingly, the extended Lipkin model (ELM) serves as a valuable alternative for mimicking the IBA phase diagram and holds the advantage of being more straightforward to implement within a quantum computing platform.Purpose: We explore the ELM and provide: (i) calculations of the ground-state energy using a variational quantum eigensolver; (ii) a comprehensive formulation for implementing the dynamics of the ELM within a quantum computing platform, enabling the experimental exploration of the IBA phase diagram for systems with a small number of particles; and (iii) a determination of the phase diagram of the model using different machine learning (ML) methods. We note that in the ELM, unlike the usual Lipkin model, both first- and second-order quantum shape phase transitions take place depending on the model parameters.Methods: Initially, we employ the adaptive derivative-assembled pseudo-Trotter ansatz variational quantum eigensolver (ADAPT-VQE) to calculate the ground-state energy of the ELM. Next, we introduce the essential formulation and procedures required to implement this model effectively in a quantum computing environment. Finally, we use ML techniques to identify the different phases and critical points of the ELM.Results: We successfully reproduce the ground-state energy of the ELM across the complete phase space of the model using the ADAPT-VQE algorithm. We provide the necessary framework for implementing the ELM in a quantum computing platform, ensuring that the model can be executed with controlled errors. Finally, we obtain meaningful ML predictions that allow us to determine the phase diagram of the model.Conclusions: Our findings offer compelling evidence that the implementation of a nuclear model like the ELM in a quantum computing environment is not only feasible but can also be achieved with manageable error rates. This realization opens the door to detailed experimental investigations of the phase diagram of the ELM (and indirectly of the IBA) in a quantum computer, further advancing our understanding of quantum shape phase transitions and nuclear structure.