Resource-Efficient Emulation of Majorana Zero Mode Braiding on a Superconducting Trijunction

Abstract

Topological superconductivity could host quasiparticles that are key candidates for fault-tolerant quantum computation due to their immunity to noise as they obey non-Abelian exchange statistics. For example, in the case of Majorana Zero Modes (MZM), braiding enables two topologically protected quantum gates. While their direct manipulation in solid-state systems remains experimentally challenging, digital emulation of MZM behavior has provided insight as well as a deeper understanding of controlling these topological quantum systems. This emulation is typically accomplished by mapping the topological and trivial phases of a Majorana system to ferromagnetic and paramagnetic Hamiltonians of a spin-glass model. This approach usually relies on adiabatic evolution of superconducting Hamiltonians, which require circuits with very large depths. In this work, we present a resource-efficient method to emulate MZM braiding in a trijunction geometry using a quantum processor. We introduce direct braiding operators which simulate the evolution more efficiently, reducing the quantum gate overhead. We then further generalize this method to emulate braiding operations in extended trijunction architectures based on Kitaev chains.

Figures (7)

• Figure 1: Pictorial representation of a topological trijunction. The three arms of the device are shown as gray bars. Majorana modes in the trivial phase are depicted as green triangles, while Majorana modes in the topological phase are shown as blue circles. The shaded circles indicate the unpaired, non-local Majorana modes. The purple line represents the coupling between two topological-phase arms, with the remaining arm in the trivial phase.
• Figure 2: Time evolution of the braiding protocol starting at $t=0$, with time increasing counterclockwise around the trijunction. Arms connected by a dashed line are in the topological phase, while the unshaded arm is in the trivial phase. The trijunction sequentially passes through six distinct configurations at discrete time steps, each of duration $\tau$.
• Figure 3: Quantum circuits implementing the trijunction braiding protocol for a single-site trijunction. Panel \ref{['fig:braid-adiabatic']} shows an adiabatic realization of the six-step protocol, while panel \ref{['fig:braid-operator']} shows the corresponding braiding operator based construction. The blocks labeled $|\Psi\rangle_\pm$ and $|\Psi\rangle_\pm^\dagger$ denote the initial and final state preparation respectively for each implementation.
• Figure 4: Sub-steps in a single braiding step of the braiding protocol. Green arrows indicate the motion of the Majorana zero mode (black square shading inside a blue circle), while violet arrows denote the order in which the braiding operators are applied. Green triangles represent Majorana bound states in the normal-phase wire, and blue circles represent Majorana bound states in the topological-phase wire.
• Figure 5: Trade-off between state fidelity and circuit depth for adiabatic simulation of a single-site trijunction. Each bar corresponds to the respective choice of Trotter steps in x-axis. The red bar indicates the minimal setting used in the resource comparison of fig. \ref{['fig:ECRComp']}.