Robust Chiral Edge Dynamics of a Kitaev Honeycomb on a Trapped Ion Processor

Abstract

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.

Figures (10)

• Figure 1: Model and quantum circuit overview.a, We simulate a 22-qubit Kitaev honeycomb lattice, featuring direction-dependent Ising interactions [see Eq. (\ref{['eq: kitaev']})], an effective field with emergent three-body interactions [see Eq. (\ref{['eq: V']})], and Heisenberg interactions [not shown; see Eq. (\ref{['eq: heis']})]. Flipping a spin in the non-Abelian phase generates an edge excitation (purple bump) which travels clockwise or counterclockwise depending on the sign of the field. b, Quantum circuit schematic for measuring time-dependent spin-spin correlations along the edge. c, Phase diagram of the Kitaev honeycomb in the absence of a magnetic field kitaev2006anyons. Stars indicate points we probe in the set of gapped Abelian phases $A^\alpha$ ($\alpha = x, y, z$) and the phase $B$, which acquires a gap and becomes a non-Abelian spin-liquid phase in the presence of a field.
• Figure 2: Quantum state preparation and temperature.a, Non-Abelian ground state preparation on the H2-1 quantum computer yields plaquette expectations close to the ideal vortex-free limit $\langle W_p\rangle = 1$. b, Percentage of shots with and without vortices, for 300 shots. We use the shots with zero vortices to obtain post-selected results with fewer computational errors. c, Term-resolved energy measurements ($\pm$ standard error) for the non-Abelian ground state. Post-selecting significantly improves agreement with the exact ground state energy. (inset) Effective temperature assignment by matching the measured state energy to the thermal $\langle H\rangle(T)$ curve (see methods). The Hamiltonian parameters are $K^x=K^y=K^z=-1$, $V=0.3$, $h=0.1$, and $J=0$.
• Figure 3: Chiral edge dynamics in the pure Kitaev model $\bm{(J = 0)}$.a, Time evolution of unequal-time spin-spin correlation functions $\langle Z_i(t) Z_\mathrm{C}\rangle$ on the edge in the Abelian phase, following a local excitation at reference site $\mathrm{C}$ (see inset for qubit labels). We observe no chiral response, consistent with a vanishing Chern number and the absence of topologically protected edge states. Parameters are $K^x=K^y=K^z/6=-1/6$, $V=0.3$, $h=0.1$, and $J=0$. b, c, We apply the same protocol to a non-Abelian state with $V>0$ in b, and $V<0$ in c. The directional propagation observed in $\langle Z_{\mathrm{L}}(t) Z_{\mathrm{C}} \rangle$ and $\langle Z_{\mathrm{R}}(t) Z_{\mathrm{C}} \rangle$ reflects the presence of a chiral edge mode and is a dynamical signature of the nonzero Chern number and non-Abelian topological order. Results obtained on quantum hardware, both raw and post-selected, show quantitative agreement with an exact theoretical simulation (dotted line) and a noiseless circuit simulation that includes all algorithmic errors (solid line). We observe a (counter-) clockwise chiral behavior for the ($V<0$) $V>0$ case. Parameters are the same as the Abelian case except that $K^x=K^y=K^z=-1$.
• Figure 4: Chiral dynamics in the Kitaev-Heisenberg model.a, Starting from the ground state of the Kitaev-Heisenberg model with $J = 0.05$, a local spin excitation is applied to site $\mathrm{C}$ and unequal-time correlations are measured, similar to Fig. \ref{['fig:fig3']}. The observed directional propagation is consistent with a nonzero Chern number and the persistence of non-Abelian topological order in the presence of weak Heisenberg interactions. b, For a stronger Heisenberg coupling $J = 0.2$, the chiral dynamics is strongly suppressed. Both panels use $K^{\alpha} = -1 \; (\alpha = x, y, z)$, $V=0.3$, and $h=0.1$.
• Figure 5: Ground state optimization.a, Ground state preparation error $C_\text{GS}$ for the pure Kitaev Hamiltonian ($J=0$) as the variational circuit depth is increased. We consider four ground states, in the Abelian ($K^x/6=K^y/6=K^z=-1$) and non-Abelian ($K^x=K^y=K^z=-1$) phases, and with effective fields $V=\pm 0.3$. (inset) Purple qubits on the boundary denote correction qubits $c_p$ used to correct $W_p=-1$ plaquette operator measurements. b, Ground state optimization for the non-integrable Kitaev-Heisenberg Hamiltonian, with $K^x=K^y=K^z=-1, \ V=0.3$, and $J=0.05, 0.2$.