Synthetic Topological Qubits in Conventional Bilayer Quantum Hall Systems

Abstract

The idea of topological quantum computation is to build powerful and robust quantum computers with certain macroscopic quantum states of matter called topologically ordered states. These systems have degenerate ground states that can be used as robust "topological qubits" to store and process quantum information. However, a topological qubit has not been realized since the proposed systems either require sophisticated topologically ordered states that are not available yet, or require complicated geometries that are too difficult to realize. In this paper, we propose a new experimental setup which can realize topological qubits in a simple bilayer fractional quantum Hall (FQH) system with proper electric gate configurations. Compared to previous works, our proposal is accessible with current experimental techniques and only involves well-established topological states. Our system can realize a large class of topological qubits, generalizing the Majorana zero modes studied in the recent literature to more computationally powerful possibilities. We propose three tunneling and interferometry experiments to detect the existence and non-local topological properties of the topological qubits.

Figures (8)

• Figure 1: (a) Cross-section of proposed device. Top and bottom gates partially deplete the electron fluids, and the fringe fields cause the wave function at the edges to extend further vertically, enhancing the tunneling. Filled circles and crosses indicate the gapless chiral edge states, moving out of or into the page. (b) 3D view. Interlayer backscattering between edge states smoothly connects the layers, allowing for coherent quasiparticle propagation between layers. The red and blue balls connected by a dashed line illustrates an inter-layer path of the quasiparticle.
• Figure 2: A pair of TLJs introduces two non-contractible loops, so the space is topologically equivalent to a torus. The edge states around the outer edge of the sample and around the gates are mapped to edge states around four holes on the surface of the torus.
• Figure 3: Experimental setups to measure zero bias interlayer quasiparticle conductance. (a) The "Andreev reflection" process (see text) in which the exciton current $I_1-I_2$ is reversed across TLJ. (b) Use the edge states of the sample to measure the local quasiparticle density of states, which detect the parafermion zero modes at the ends of the TLJ.(c) Edge theory picture of TLJ in (b). The coordinate $\tilde{x}$ increase from $-L$ to $+L$ following the direction indicated by orange, yellow, red and blue dots. The second layer (blue) is flipped in the right panel. Dotted lines between counterpropagating edge states indicate backscattering, generating an energy gap.
• Figure 4: QPC interferometer setup for detecting the non-Abelian nature of genons. $\Gamma_{1,2}$ are the two QPC's in the first layer, and $\tilde{\Gamma}_{1,2}$ are the two inter-layer QPC's across the TLJs. $L$ and $L'$ indicated by thick dashed lines are the two interference loops.
• Figure 5: The defects (genons) can be viewed as domain walls between two different ways of generating an energy gap in the counterpropagating edge states (double arrows indicate interedge electron tunneling). The zero mode operator $\alpha_{2i-1}$ corresponds to the quasiparticle hopping process shown in the figure, projected into the ground state subspace of the edge theory.