Non-perturbative Floquet engineering of the toric-code Hamiltonian and its ground state

TL;DR

This work demonstrates a nonperturbative Floquet-engineering framework to realize Kitaev's toric-code Hamiltonian through a hybrid continuous-digital protocol that constructs clean four-spin plaquette interactions in a scalable way. By numerically optimizing single-plaquette driving and exploiting commutativity among plaquettes, it achieves an effective Hamiltonian Hw with dominant four-spin terms and negligible higher-order contamination, enabling ground-state preparation and topological-qubit operations. The authors validate the approach with ground-state preparation achieving near-ideal energies and entanglement signatures, and confirm anyon braiding statistics in a 5×4 lattice, while also presenting a minimal nine-qubit device to realize a topological crossover via Floquet adiabatic ramp. The scheme naturally extends to larger lattices and Z2 lattice gauge theory with matter, and opens paths toward hole-based topological quantum computation, albeit with scaling challenges for long-range logical operators in bigger systems.

Abstract

We theoretically propose a quantum simulation scheme for the toric-code Hamiltonian, the paradigmatic model of a quantum spin liquid, based on time-periodic driving. We develop a hybrid continuous-digital strategy that exploits the commutativity of different terms in the target Hamiltonian. It allows one to realize the required four-body interactions in a nonperturbative way, attaining strong coupling and the suppression of undesired processes. In addition, we design an optimal protocol for preparing the topologically ordered ground states with high fidelity. A proof-of-principle implementation of a topological device and its use to simulate the topological phase transition are also discussed. The proposed scheme finds natural implementation in architectures of superconducting qubits with tuneable couplings.

Figures (9)

• Figure 1: Sketch of the spin lattice and of the quantum simulation approach, which will be detailed in the following. Square symbols indicate spins, solid lines represent tuneable pairwise couplings, and wavy purple lines represent couplings that are driven in a given time substep $\tau$, with associated angular frequency $\omega=2\pi/\tau$. In each substep $\tau$, different subsystems are decoupled from each other and driven to produce effective four-spin interactions (light-blue squares). At the end of the sequence (which is periodic with period $T=4\tau$), a complete toric-code Hamiltonian is achieved with high accuracy. The proposed scheme further allows one to prepare its ground state(s) and to study anyonic quasiparticles ('$\mathrm{e}$' and '$\mathrm{m}$').
• Figure 2: (a) Sketch of the driving scheme for realizing a four-spin term in a single four-spin plaquette, involving oscillating fields at angular frequency $\omega$ and $2\omega$; (b) Components (larger than $10^{-7} \omega$ in magnitude) of the effective plaquette Hamiltonian, confirming the achievement of a clean four-spin interaction, in units of $\omega$.
• Figure 3: (a) Two-by-two toric-code Hamiltonian with mixed boundary conditions. (b) Time evolution of the probability of remaining in the initial state $\ket{G}$, where $\ket{G}$ is a ground state of the toric code Hamiltonian shown in (a), for the undriven (dashed) and driven (solid) system in contact with thermal baths at temperature $k_BT=\omega_\alpha/15$, for different system-bath coupling values $\lambda$ and for $\omega_\alpha=2\cdot 10^3\omega$. (c) Sensitivity of the single plaquette protocol to errors in the control amplitudes. Different colors correspond to different maximal error amplitude $\eta_{\mathrm{max}}$. The bars represent the operator with maximal magnitude at a given weight (number of non-identity operators in the tensor product), which is indicated on the top of the bar.
• Figure 4: (a) Lattice connectivity, equivalent to a distorted square lattice. Some triangular cells appear when including links (dashed lines) needed to implement mixed boundaries. (b) Trotter sequence to realize the Hamiltonian on the whole lattice. In each of the four steps of duration $\tau$, a group $G_k$ of disconnected plaquettes is driven (blue colour), while couplings connecting such plaquettes are turned off. The full lattice Hamiltonian progressively builds up (light blue).
• Figure 5: (a) Weight vs magnitude $|c_\ell|$ of the operators composing the effective Hamiltonian for a 5-by-5 system. The desired four-spin and two-spin boundary terms (exemplified in the inset) are at least two orders of magnitude larger than undesired terms. The red colour of markers has non-unit opacity, such that more colour-intense markers appear as a result of overlapping markers. (b) Floquet heating dynamics, quantified via $Q(n)$ of Eq. \ref{['eq:Qheat']}, as a function of the number of Floquet periods $n$, for different system sizes with mixed boundary conditions as depicted in the inset.