Variational preparation and characterization of chiral spin liquids in quantum circuits

TL;DR

This work demonstrates how chiral spin liquids can be prepared and characterized on quantum circuits using a variational quantum eigensolver (VQE) framework coupled with a tangent-space excitation Ansatz. By applying this to the Kitaev honeycomb model and an SU(2)_1 chiral spin liquid on a square lattice, the authors show faithful reproduction of ground-state degeneracies and chiral edge modes, and obtain accurate low-energy spectra in both vortex-free and vortex sectors as well as on torus and disk geometries. The approach provides a polynomial-cost pathway to identify topological order and edge physics on near-term quantum devices, using ground-state preparation plus controlled excitations to reveal topological signatures without full state tomography. These results offer a scalable, hardware-friendly route to exploring chiral topological phases and their anyonic excitations in finite systems.

Abstract

Quantum circuits have been shown to be a fertile ground for realizing long-range entangled phases of matter. While various quantum double models with non-chiral topological order have been theoretically investigated and experimentally implemented, the realization and characterization of chiral topological phases have remained less explored. Here we show that chiral topological phases in spin systems, i.e., chiral spin liquids, can be prepared in quantum circuits using the variational quantum eigensolver (VQE) framework. On top of the VQE ground state, signatures of the chiral topological order are revealed using the recently proposed tangent space excitation ansatz for quantum circuits. We show that, both topological ground state degeneracy and the chiral edge mode can be faithfully captured by this approach. We demonstrate our approach using the Kitaev honeycomb model, finding excellent agreement of low-energy excitation spectrum on quantum circuits with exact solution in all topological sectors. Further applying this approach to a non-exactly solvable chiral spin liquid model on square lattice, the results suggest this approach works well even when the topological sectors are not exactly known.

Figures (15)

• Figure 1: Schematic for Kitaev honeycomb model. The lattice is generated by translating a two-site unit cell along $\bf{a}_1$ and $\bf{a}_2$ directions, and contains $A,B$ sublattices. Depending on the orientation, links of the lattice are labeled as $x,y,z$. Six small triangles in each hexagon are illustrated in upper right. The site label for plaquette operator $W_p$ is shown in the yellow hexagon. The non-contractible loop $\Gamma_1$ ($\Gamma_2$) along $\bf{a}_1$ ($\bf{a}_2$) direction is shown in green (red).
• Figure 2: (a) Schematics of HVA for Kitaev honeycomb model. For a $(\phi_1,\phi_2)$ sector, $|\Psi_{0}\rangle$ is a proper initial state with well-defined conserved quantities. The links where $U_{l}^{\alpha}$'s act are shown in the lower panel, highlighted with the corresponding color. In (b) and (c) we show the variational power of HVA for system sizes $(L_1,L_2)=(3,2),(3,3),(4,3)$ with number of sites $N_s=2L_1L_2$. (b) For all three $(\phi_1,\phi_2)$ sectors, comparing with exact ground state, the energy error of optimized variational state decreases with circuit depth, reaching a value close to zero. (c) For these system sizes, the state generated by a circuit with random parameters $\{\bm{\theta}\}$ and depth $D=12,15,18$ can be approached with high precision using optimized circuits with lower depth $D-3$, where a sudden drop in median of infidelity over 100 samples is observed.
• Figure 3: (a) Schematics of one basis state for excitations on quantum circuits, with the detailed implementation illustrated below. (b),(c),(d) and (e) show the low-energy spectrum of a $(L_1,L_2)=(3,3)$ system in zero flux sectors, with $(\phi_1,\phi_2)=(1,1), (-1,1), (1,-1), (-1,-1)$, respectively. The available momenta are indicated by red filled dots in the first Brillouin zone shown on the right. Lines show the spectrum obtained with exact fermion solution, which is labeled by the number of occupied fermion modes $n$ on top of the fermion ground state. Filled symbols in (b)-(e) represent spectrum from our variational ansatz with three ground states, where distinct symbols represent energy levels with different degeneracy. In all sectors, the variational results, including the level degeneracy, agree with exact results over a wide range of energy.
• Figure 4: Variational excitation spectrum in nonzero vortex sectors, in comparison with ED. The system size is $(L_1,L_2)=(3,3)$. We consider creating vortices by acting on the red dots shown in the inset. Three spectra of corresponding $W_p$ configurations are plotted with different colors. The other vortex configurations with $(\langle{W_1\rangle}, \langle{W_2}\rangle)=-1$, $(\langle{W_2\rangle}, \langle{W_4}\rangle)=-1$, $(\langle{W_1}\rangle, \langle{W_3}\rangle)=-1$ and $(\langle{W_3}\rangle, \langle{W_4}\rangle)=-1$ have identical low-energy spectrum as $(\langle{W_2}\rangle, \langle{W_3}\rangle)=-1$.
• Figure 5: Schematics of HVA for the $\mathrm{SU}(2)_1$ CSL on torus geometry. (a) $N_s=16$ site cluster, with torus generated by translation along vectors $\mathbf{t_1}=(4,0), \mathbf{t_2}=(0,4)$. (b) $N_s=15$ site cluster, with torus generated by vectors $\mathbf{t_1}=(-4,1), \mathbf{t_2}=(1,-4)$. In both (a) and (b) links with the same color correspond to gates in one layer.