Abstract Fusing Majorana zero modes (MZMs) leads to multiple outcomes, a property unique to non-Abelian anyons. The successful demonstration of this nontrivial fusion rule would be a hallmark for the development of topological quantum computation. Here, we propose that such fusion can be achieved by simply attaching a fermionic mode to a single Majorana zero mode. By modulation of the energy level of the fermionic mode (FM) as well as its coupling with MZM in different sequences, we show that a zero or integer charge pumping can be realized when different fusion loops are chosen. Such fusion loops are intimately related with the nontrivial fusion rule of Majorana modes and are solely determined by the crossings at zero energy in the parameter space. Finally, we demonstrate our proposal in a nanowire-based topological superconductor coupled to a quantum dot. We show that charge pumping is robust for MZMs in the realistic system irrespective of the initial condition of FM state.
Abstract Fusing Majorana zero modes (MZMs) leads to multiple outcomes, a property unique to non-Abelian anyons. The successful demonstration of this nontrivial fusion rule would be a hallmark for the development of topological quantum computation. Here, we propose that such fusion can be achieved by simply attaching a fermionic mode to a single Majorana zero mode. By modulation of the energy level of the fermionic mode (FM) as well as its coupling with MZM in different sequences, we show that a zero or integer charge pumping can be realized when different fusion loops are chosen. Such fusion loops are intimately related with the nontrivial fusion rule of Majorana modes and are solely determined by the crossings at zero energy in the parameter space. Finally, we demonstrate our proposal in a nanowire-based topological superconductor coupled to a quantum dot. We show that charge pumping is robust for MZMs in the realistic system irrespective of the initial condition of FM state.
Abstract We propose and simulate a near-adiabatically coupling probing scheme for nontrivial fusion of a pair of Majorana zero modes (MZMs). The scheme can avoid the complexity of oscillating charge occupation in the probing quantum dot, making thus practical measurements more feasible. We also show how to extract the information of nonadiabatic transition and fermion parity violation caused during moving the MZMs together to fuse, from the initial states prepared with definite fermion parity. All the simulations, including the effective coupling between the fusing MZMs, and their coupling to the probing quantum dot, are based on the lattice model of a Rashba quantum wire in proximity contact with an s -wave superconductor, under the modulation of mini-gate voltage control.
Majorana zero modes, characterized by non-Abelian exchange statistics, have been widely viewed as prime candidates for implementing topological quantum computing. The associated braiding and fusion operations realize topologically protected quantum gates and qubit measurements, thus providing a feasible pathway toward intrinsically fault-tolerant quantum computation. However, the experimental realization of controllable braiding and fusion of Majorana zero modes remains one of the most formidable challenges in the field.In this paper, building upon recent theoretical advances in quantum-dot-assisted schemes for realizing the braiding and fusion of Majorana zero modes, we provide a systematic introduction to the underlying theoretical foundations and detailed operational procedures, and further elucidate the experimental feasibility of the approach.Furthermore, this paper presents two lines of investigation developed within the proposed framework. Firstly, based on the quantum-dot-assisted braiding scheme, the dynamical evolution of Andreev bound states in Majorana nanowires is revealed. Secondly, employing the quantum-dot-assisted fusion scheme, numerical simulations of charge transfer along different fusion paths clarify the nontrivial fusion rules of Majorana zero modes and the corresponding operational procedures.
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.
In this paper we perform real-time simulations for probing the non-Abelian fusion of a pair of Majorana zero modes (MZMs). The nontrivial fusion outcomes can be either a vacuum, or an unpaired fermion, which reflect the underlying non-Abelian statistics. The two possible outcomes can cause different charge variations in the nearby probing quantum dot (QD), while the charge occupation in the dot is detected by a quantum point contact. In particular, we find that gradual fusion and gradual coupling of the MZMs to the QD (in nearly adiabatic switching-on limit) provide a simpler detection scheme than sudden coupling after fusion to infer the coexistence of two fusion outcomes, by measuring the occupation probability of the QD. For the scheme of sudden coupling (after fusion), we propose and analyze continuous weak measurement for the quantum oscillations of the QD occupancy. From the power spectrum of the measurement currents, one can identify the characteristic frequencies and infer thus the coexistence of the fusion outcomes.
We perform simulations for transport and nontrivial fusion of Majorana zero modes in topological superconducting quantum wires. We uncover interesting behaviors of nonadiabatic transition associated with the transport through mini-gate-controlled multiple-segment modulations. Owing to breaking of the initial fermion parity induced by nonadiabatic transitions, a deviation from the statistics of outcomes of nontrivial fusion arises and is analyzed. Moreover, we develop a measurement scheme to infer the amount of fermion parity breaking and nonadiabatic transition probability to excited states, based on the characteristic spectrum of measurement current by a quantum-point-contact detector, by measurement of the charge occupation dynamics in a fusion-outcome-probing quantum dot.
In a recent breakthrough experiment [Nature (London) 614, 445 (2023)], signatures of Majorana zero modes have been observed in tunnel spectroscopy for a minimal Kitaev chain constructed from coupled quantum dots. However, as Ising anyons, Majoranas' most fundamental property of non-Abelian statistics is yet to be detected. Moreover, the minimal Kitaev chain is qualitatively different from topological superconductors in that it supports Majoranas only at a sweet spot. Therefore, it is not obvious whether non-Abelian characteristics such as braiding and fusion can be demonstrated in this platform with a reasonable level of robustness. In this work, we theoretically propose a protocol for detecting the Majorana fusion rules in an artificial Kitaev chain consisting of four quantum dots. In contrast with the previous proposals for semiconductor-superconductor hybrid nanowire platforms, here we do not rely on mesoscopic superconducting islands, which are difficult to implement in quantum dot chains. To show the robustness of the fusion protocol, we discuss the effects of three types of realistic imperfections on the fusion outcomes, e.g., diabatic errors, dephasing errors, and calibration errors. We also propose a fermion parity readout scheme using quantum capacitance. Our work will shed light on future experiments on detecting the non-Abelian properties of Majorana modes in a quantum dot chain.