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.
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.
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.
Quantum spin liquids, a highly topologically entangled, dynamically correlated state where quantum fluctuations preclude any long-range ordering down to absolute zero. In the search for a topologically robust qubit, the scientific community has been in continuous hunt for real quantum spin liquid systems. Alexei Kitaev in his exactly solvable model for a spin-1/2 two-dimensional honeycomb lattice, presented a system that hosts a topologically protected state (Majorana zero-modes). Under an applied external field, the Kitaev spin liquids turn into a topologically non-trivial chiral spin-liquid state with non-abelian anionic excitations, which is crucial for quantum computing. Earlier theoretical predictions advocated that Kitaev physics can be realized in spin-orbit-coupled Mott insulators such as honeycomb irradiates and ruthenates. However, the experimental findings continuously challenge the theoretical aspects, indicating the presence of non-Kitaev interactions in real materials, where the dimensionality, disorder (vacancy), chemical composition, generalized spin-S, and external perturbations (pressure, magnetic field, temperature) can actively tune the Kitaev interactions and the ground state excitations. In this review article, a comprehensive discussion is included with an updated literature survey in the context of the potential of Raman spectroscopy as a probe for Kitaev quantum spin liquids.
Kitaev's honeycomb model is a paradigmatic exactly solvable system hosting a quantum spin liquid with non-Abelian anyons and topologically protected edge modes, offering a platform for fault-tolerant quantum computation. However, real candidate Kitaev materials invariably include complex secondary interactions that obscure the realization of spin-liquid behavior and demand novel quantum computational approaches for efficient simulation. Here we report quantum simulations of a 22-site Kitaev honeycomb lattice on a trapped-ion quantum processor, without and with non-integrable Heisenberg interactions that are present in real materials. We develop efficient quantum circuits for ground-state preparation, achieving high accuracy with energy errors equivalent to an effective temperature of 0.2 (in units of the Kitaev interactions), consistent with the experimentally relevant spin-liquid regime. Starting from these states, we apply controlled perturbations and measure time-dependent spin correlations along the system's edge. In the non-Abelian phase, we observe chiral edge dynamics consistent with a non-zero Chern number, a hallmark of topological order, which vanishes upon transition to the Abelian toric code phase. Extending to the non-integrable Kitaev-Heisenberg model, we find that weak Heisenberg interactions preserve chiral edge dynamics, while stronger couplings suppress them, signaling the breakdown of topological protection. Our work demonstrates a viable route for probing dynamical signatures of topological order in quantum spin liquids using programmable quantum hardware, opening new pathways for quantum simulation of strongly correlated materials.
We propose an efficient circuit structure of variational quantum circuit Ans\"atze used for the variational quantum eigensolver (VQE) algorithm in calculating gapped topological phases on the currently feasible noisy intermediate-scale quantum computers. An efficient circuit Ansatz should include two layers: the initialization layer and the variational layer. In the initialization layer, a fixed-depth circuit state with a compatible entanglement structure to the target topological phase is constructed. The circuit state is further adjusted subsequently to capture the details of the local correlations, which is dictated with the Hamiltonian, in the parametrized variational layer. Based on this strategy, we design a circuit Ansatz to investigate the symmetry-protected topological Haldane phase in a nonexactly solvable alternating spin-21 Heisenberg chain by VQE calculations. Main characterizations of the Haldane phase, including the long-ranged string order, the fourfold nearly degenerate ground states associated with four different localized edge-mode patterns for the system with open boundaries, and the twofold degeneracy of the entanglement spectrum, are all observed for the optimized shallow circuit state with only one-depth variational layer both in numerical simulations and on real quantum computers. We further demonstrate that the computational capacity (i.e., expressibility) of this quantum circuit Ansatz is determined not by the system size but only by the intrinsic correlation length of the system, thus implying that the scalable VQE calculation is possible.
We study the finite-temperature nature of a quantum phase transition between an Abelian and a non-Abelian topological phase in an exactly solvable model of the chiral spin liquid of Yao and Kivelson [Phys. Rev. Lett. 99, 247203 (2007)]. As it is exactly solvable, this model can serve as a testbed for developing better measures for describing topological quantum phase transitions. We use the global flux and entanglement entropy to characterize this phase transition, and we discuss to what extent the existence of a topologically ordered ground state with non-Abelian excitations is revealed at finite temperature.
We describe a coupled-chain construction for chiral spin liquids in two-dimensional spin systems. Starting from a one-dimensional zigzag spin chain and imposing SU(2) symmetry in the framework of non-Abelian bosonization, we first show that our approach faithfully describes the low-energy physics of an exactly solvable model with a three-spin interaction. Generalizing the construction to the two-dimensional case, we obtain a theory that incorporates the universal properties of the chiral spin liquid predicted by Kalmeyer and Laughlin: charge-neutral edge states, gapped spin-1/2 bulk excitations, and ground-state degeneracy on the torus signaling the topological order of this quantum state. In addition, we show that the chiral spin liquid phase is more easily stabilized in frustrated lattices containing corner-sharing triangles, such as the extended kagome lattice, than in the triangular lattice. Our field-theoretical approach invites generalizations to more exotic chiral spin liquids and may be used to assess the existence of the chiral spin liquid as the ground state of specific lattice systems.