Fusion Dynamics of Majorana Zero Modes

TL;DR

The paper addresses how to dynamically fuse Majorana zero modes (MZMs) in realistic platforms and extract the qubit state via charge readouts. It develops time-dependent, many-body simulations for both spinless Kitaev wires and spinful magnet-superconductor hybrid (MSH) structures to implement Z- and X-channel fusions and braiding, connecting these procedures to the Ising anyon fusion rules ${\sigma\times\sigma=\mathbf{1}+\zeta}$. The authors show that fusion outcomes can be read out through local charge differences, with quantized signals emerging in adiabatic limits and robust localization on quantum dots, enabling practical Majorana qubit readout. They further demonstrate that braiding gates (Z, X, $\sqrt{X}$, Hadamard) imprint predictable charge signatures, and that the same physics persists in MSH geometries, supporting scalable Majorana-based quantum operations. Overall, the work provides a comprehensive, dynamical confirmation of Ising anyon fusion in two platform classes and highlights charge-based readout as a viable readout modality for Majorana qubits.

Abstract

Braiding and fusion of Majorana zero modes are key elements of any future topological Majorana-based quantum computer. Here, we investigate the fusion dynamics of Majorana zero modes in the spinless Kitaev model, as well as in a spinful model describing magnet-superconductor hybrid structures. We consider various scenarios allowing us to reproduce the fusion rules of the Ising anyon model. Particular emphasis is given to the charge of the fermion obtained after fusing two Majorana zero modes: as long as it remains on the superconductor, charge quantization is absent. When moving the fermion to a non-superconducting region, such as a quantum dot, nearly-quantized charge can be measured. Our findings confirm for both platforms that fusion dynamics of Majorana zero modes can indeed be used for the readout of Majorana qubits.

Figures (12)

• Figure 1: Schematic of Fusion processes. (a) $\mathcal{F}_Z$ fusion process, which corresponds to the fusion of the two MZMs that make up a Majorana bound state. (b) $\mathcal{F}_{Z_1Z_2}$ fusion process, where in a system with two Majorana bound states (consisting of four MZMs), the two MZMs on the left and the two MZMs on the right are pairwise fused. (c) $\mathcal{F}_X$ fusion process where we fuse the middle two MZMs. The topological charges $a,b,c\in \{1,\zeta\}$ in all cases, with $c$ setting the total topological charge in (b) and (c). In the dynamical simulations, we are interested in determining $a$ or $a$ and $b$.
• Figure 2: Fusion through the $Z$-channel on a Kitaev wire. (a) Graphic of the $\mathcal{F}_Z$ process on a single Kitaev chain. (b), (c) Charge density $\rho_i(t)$ for fusion of two MZMs, initialized at $|0\rangle$ and $|1\rangle$, respectively. (d) The final charge difference $\Delta Q(T)$ between the $|1\rangle$ and $|0\rangle$ states as a function of fusion time $T$ and delay coefficient $\alpha$ of the ramping process. Parameters used for these simulations are $(\tilde{t},\Delta,\mu_{\rm topo}, \mu_{\rm triv},L)=(1,1,-0.4,-5.6,32a_0)$.
• Figure 3: Fusion through the $X$-channel on a Kitaev wire. (a) Graphic of the $\mathcal{F}_X$ process on a single Kitaev chain for two pairs of MZMs. (b), (c) Charge density $\rho_i(t)$ for $X$-channel fusion between two pairs of MZMs, initialized in the even parity sector with $|00\rangle$ and $|11\rangle$, respectively. (d) The final charge difference, $\Delta Q(T)$, between the state $\ket{-}$ and the reference state $\ket{+}$ as function of fusion time $T$ and delay coefficient $\alpha$. Parameters used for these simulations are $(\tilde{t},\Delta,\mu_{\rm topo}, \mu_{\rm triv},L)=(1,1,-0.45,-4.4,56a_0)$.
• Figure 4: Phase readout through measurement of the charge difference in the X-channel. Charge difference, $\Delta Q(\theta,\phi)(T)$, between an arbitrary state $|\psi(\theta,\phi)\rangle_{\rm logic}$ and reference state $\ket{+}_{\rm logic}$ as a function of $\theta \in [0,\pi]$ and $\phi \in [0,2\pi]$.
• Figure 5: Fusion through the $Z$-channel after braids. (a) Definition of site labeling on the triple T-junction. (b) Graphic of $\mathcal{F}_{Z_1Z_2}$ fusion on a triple T-junction. (c, f, i) Braiding world lines for a $Z$, $\sqrt{X}$ and $X$ gate. (d,e), (g,h), (j,k) The local charge density, $\rho_i(t)$, along the triple T-junction, is plotted after a $Z$, $\sqrt{X}$, $X$ gate, respectively, where we fuse via the $\mathcal{F}_{Z_1Z_2}$ channel. We initialize at $|00\rangle$ for (d, g, j) and $|11\rangle$ for (e, h, k). $(\tilde{t},\Delta,\mu_{\rm topo}, \mu_{\rm triv},L_{\rm leg})=(1,1,-0.75,-7.2,12a_0)$. Note that the time $t=0$ is directly after the braid.