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arXiv:2205.10114 · Physical Review A

Scalable quantum control and non-Abelian anyon creation in the Kitaev honeycomb model

Omar Raii, Florian Mintert, Daniel Klaus Burgarth

Published Dec 2, 2022• 6 citations
arXivHTMLPDF

Abstract

The Kitaev honeycomb model is a system allowing for experimentally realizable quantum computation with topological protection of quantum information. Practical implementation of quantum information processing typically relies on adiabatic, i.e., slow dynamics. Here we show that the restriction to adiabatic dynamics can be overcome with optimal control theory, enabled by an extension of the fermionization of the Kitaev honeycomb model to the time-dependent case. Moreover, we present a quantum control method that is applicable to large lattice models due to subexponential scaling.

DOI
10.1103/physreva.106.062401

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The Kitaev honeycomb model is a system allowing for experimentally realizable quantum computation with topological protection of quantum information. Practical implementation of quantum information processing typically relies on adiabatic, i.e., slow dynamics. Here we show that the restriction to adiabatic dynamics can be overcome with optimal control theory, enabled by an extension of the fermionization of the Kitaev honeycomb model to the time-dependent case. Moreover, we present a quantum control method that is applicable to large lattice models due to subexponential scaling.
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Quantum computation provides a unique opportunity to explore new regimes of physical systems through the creation of non-trivial quantum states far outside of the classical limit. However, such computation is remarkably sensitive to noise and undergoes rapid dephasing in most cases. One potential solution to these prosaic concerns is to encode and process the information using topological manipulations of so-called anyons, particles in two dimensions with non-Abelian statistics. Unfortunately, practical implementation of such a topological system remains far from complete, both in terms of physical methods but also in terms of connecting the underlying topological field theory with a specific physical model, including the imperfections expected in any realistic device. Here we develop a complete picture of such topological quantum computation using a variation of the Kitaev honeycomb Hamiltonian as the basis for our approach. We show the robustness of this system against noise, confirm the non-Abelian statistics of the quasi-particles to be Ising anyons, and develop new techniques for turning topological information into measurable spin quantities.
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 consider a two-dimensional spin system that exhibits Abelian anyonic excitations. Manipulations of these excitations enable the construction of a quantum computational model. While the one-qubit gates are performed dynamically, the model offers the advantage of having a two-qubit gate that is of topological nature. The transport and braiding of anyons on the lattice can be performed adiabatically enjoying the robust characteristics of geometrical evolutions. The same control procedures can be used when dealing with non-Abelian anyons. A possible implementation of the manipulations with optical lattices is developed.
We analyze the effect of local spin operators in the Kitaev model on the honeycomb lattice. We show, in perturbation around the isolated-dimer limit, that they create Abelian anyons together with fermionic excitations which are likely to play a role in experiments. We derive the explicit form of the operators creating and moving Abelian anyons without creating fermions and show that it involves multispin operations. Finally, the important experimental constraints stemming from our results are discussed.
In this chapter, we consider Kitaev's honeycomb lattice model (Kitaev, 2006). This is an analytically tractable spin model that gives rise to quasiparticles with Abelian as well as non-Abelian statistics. Some of its properties are similar to the fractional quantum Hall effect, which has been studied experimentally in great detail even though it evades exact analytical treatment (Moore and Read, 1991). Due to its simplicity, the honeycomb lattice model is likely to be the first topological spin model to be realised in the laboratory, e.g., with optical lattice technology (Micheli et al. , 2006). Understanding its properties can facilitate its physical realisation and can provide a useful insight into the mechanisms underlining topological insulators and the fractional quantum Hall effect. The honeycomb lattice model comprises interacting spin-½ particles arranged on the sites of a honeycomb lattice. It is remarkable that such a simple model can support a rich variety of topological behaviours. For certain values of its couplings, Abelian anyons emerge that behave like the toric code anyons. For another coupling regime, non-Abelian anyons emerge that correspond to the Ising anyons. The latter are manifested as vortex-like configurations of the original spin model that can effectively be described by Majorana fermions. These are fermionic fields that are antiparticles of themselves. They were first introduced in the context of high-energy physics (Majorana, 1937) and become increasingly important in the analysis of solid state phenomena (Wilczek, 2009).