Simulation of Interaction-Induced Chiral Topological Dynamics on a Digital Quantum Computer

Chiral edge states are highly sought after as paradigmatic topological states relevant to both quantum information processing and dissipationless electron transport. Using superconducting transmon-based quantum computers, we demonstrate chiral topological propagation that is induced by suitably designed interactions, instead of flux or spin-orbit coupling. Also different from conventional 2D realizations, our effective Chern lattice is implemented on a much smaller equivalent 1D spin chain, with sequences of entangling gates encapsulating the required time-reversal breaking. By taking advantage of the quantum nature of the platform, we circumvented difficulties from the limited qubit number and gate fidelity in present-day noisy intermediate-scale quantum era quantum computers, paving the way for the quantum simulation of more sophisticated topological states on very rapidly developing quantum hardware.

Figure 1

Chiral topological propagation from interactions. (a) Chern topology arises from a checkerboard lattice with nonreciprocal couplings carrying effective flux (thick and thin orange arrows). Implemented as a 1D interacting chain, vertical, horizontal, and diagonal couplings become single-boson (red, blue) and two-boson hoppings (pink, orange), respectively. An on-site repulsive interaction is imposed to yield a virtual boundary along the lattice diagonal, along which chiral propagation occurs in the form of oppositely moving two-particle wave packets (black arrows). (b) The Berry curvature $\mathcal{F}$ and (c) its corresponding $\mathbf{d}(\mathbf{k})$-vector skyrmionic texture of our model for $v_1=v_2=1$ and $\varphi =\pi /6$. The total number of chiral edge modes is the Chern number $\int \mathcal{F}{d}^2\mathbf{k}/2\pi =2$. (d) The Wannier polarization representing the center-of-mass translation of a maximally localized wave packet as we thread a flux $k_{+}\to k_{+}+A_y$. It is pumped $C=2$ upon a complete cycle and moves with almost perfect uniformity for most values of the ratios $v_2/v_1$.

Figure 2

Quantum circuit implementation schematics. (a) Schematic of quantum circuit for time evolving an initial state $|{\psi }_0⟩$ according to our Hamiltonian ${\mathcal{H}}_{1\mathrm{D}}$, with readout error mitigation (RO) and postselection (PS) applied on the final measurements. (b) In-principle first-order Trotterization of the $e^{-i{\mathcal{H}}_{1\mathrm{D}}t}$ propagator and the breakdown of a Trotter step in terms of unitaries $u_1$ and $u_2$, as further elaborated in Sec. S2 of Supplemental Material [98]. Pairs of qubits (drawn as double lines) represent each site of the logical Chern lattice. (c) Circuit ansatz for recompilation, comprising layers of entangling CXs and single-qubit rotations, stacked in brickwork pattern. (d) Non-nearest-neighbor CX gates can be implemented using SWAPs, which themselves decompose into nearest-neighbor CXs.

Figure 3

Demonstration of chiral topological propagation on a quantum computer. (a) Evolution of occupancy densities $\rho (x_1,x_2)=⟨{\mu }_{x_1}^{†}{\mu }_{x_1}{\nu }_{x_2}^{†}{\nu }_{x_2}⟩$ as time progresses, with good agreement between exact diagonalization (top row) and hardware data (bottom row), for an initial wave packet at $(x_1,x_2)=(2,4)$. We have normalized the peak of $\rho (x_1,x_2)$ for visual clarity. Hatch-shaded squares indicate the $x_1=x_2$ virtual boundary. (b) 3D visualization of the same chiral propagation, which clearly reveals the peak translating parallel to the $x_1=x_2$ virtual boundary. The sequence of snapshots of measured $\rho (x_1,x_2)$ at $t=0$, 0.75, 1.50, and 2.25 are superimposed (lightest to darkest). (c) Left: monotonic evolution of the center of mass $⟨x_1+x_2⟩$, with localized peak positions (crosses) shifting together with the density profile (blue) and centroid $⟨x_1+x_2⟩={\sum }_{x_1,x_2}(x_1+x_2)\rho (x_1,x_2{)}^4$ (dashed). Exact diagonalization (ED) results agree closely with data from the quantum hardware. Right: the state remains close to the diagonal virtual boundary, as indicated by constant $⟨x_1-x_2⟩=2$ over time. Parameters are $v_1=v_2=1$ and $\varphi =\pi /6$.