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Physical review. B./Physical review. B

Manipulating Majorana qubit states without braiding

Fei-Lei Xiong, Hon-Lam Lai, Wei-Min Zhang

Published Nov 10, 2021• 4 citations
Publisher

Abstract

We propose a scheme to manipulate the Majorana qubit states other than braiding. By utilizing electron transport through Majorana zero modes and tuning the magnetic flux in the setup, the aim of Majorana qubit state manipulation can be fulfilled. The principal part in the setup is an Aharonov-Bohm (AB) interferometer consisting of two topological superconducting chains (TSCs) connected to two leads. We obtain the exact master equation as well as its solution and study the general properties of the real-time dynamics of the Majorana qubit states. The parity of the Majorana qubit state can be almost perfectly polarized by tuning the bias and can also be flipped by switching the sign of the cross couplings between the TSCs and the leads. Moreover, the qubit coherence shows a phase rigidity due to the intrinsic particle-hole symmetry of the Majorana AB interferometer, which is different from that in the double-quantum-dot AB interferometers without topological properties.

DOI
10.1103/physrevb.104.205417

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We propose a scheme to manipulate the Majorana qubit states other than braiding. By utilizing electron transport through Majorana zero modes and tuning the magnetic flux in the setup, the aim of Majorana qubit state manipulation can be fulfilled. The principal part in the setup is an Aharonov-Bohm (AB) interferometer consisting of two topological superconducting chains (TSCs) connected to two leads. We obtain the exact master equation as well as its solution and study the general properties of the real-time dynamics of the Majorana qubit states. The parity of the Majorana qubit state can be almost perfectly polarized by tuning the bias and can also be flipped by switching the sign of the cross couplings between the TSCs and the leads. Moreover, the qubit coherence shows a phase rigidity due to the intrinsic particle-hole symmetry of the Majorana AB interferometer, which is different from that in the double-quantum-dot AB interferometers without topological properties.
We propose an Aharonov-Bohm interferometer consisted of two topological superconducting chains (TSCs) to generate coherence of Majorana qubits, each qubit is made of two Majorana zero modes (MZMs) with the definite fermion parity. We obtain the generalized exact master equation as well as its solution and study the real-time dynamics of the MZM qubit states under various operations. We demonstrate that by tuning the magnetic flux, the decoherence rates can be modified significantly, and dissipationless MZMs can be generated. By applying the bias voltage to the leads, one can manipulate MZM qubit coherence and generate a nearly pure superposition state of Majorana qubit. Moreover, parity flipping between MZM qubits with different fermion parities can be realized by controlling the coupling between the leads and the TSCs through gate voltages.
We study the decoherence dynamics of Majorana qubit braiding operations in a T-junction topological superconducting chain (TSC) system, in which the braiding is performed by controlling the electron-chemical potentials of the TSCs and the couplings between them. By solving rigorously the Majorana qubit dynamics, we show how the Majorana qubit coherence is generated through Bogoliubon correlations formed by exchanging Majorana zero modes (MZMs) between two TSCs in braiding operations. We also demonstrate with the exact master equation how MZMs and also the Bogoliubon correlations dissipate due to charge fluctuations of the controlling gates at both the zero and finite temperatures. As a result, Majorana qubit coherence and also the fermion parity conservation cannot be immune from local perturbations during braiding operations.
Majorana fermions are long-sought exotic particles that are their own antiparticles. Here we propose to utilize superconducting circuits to construct two superconducting-qubit arrays where Majorana modes can occur. A so-called Majorana qubit is encoded by using the unpaired Majorana modes, which emerge at the left and right ends of the chain in the Majorana-fermion representation. We also show this Majorana qubit in the spin representation and its advantage, over a single superconducting qubit, regarding quantum coherence. Moreover, we propose to use four superconducting qubits as the smallest system to demonstrate the braiding of Majorana modes and show how the states before and after braiding Majoranas can be discriminated.
Majorana qubits offer a promising way to store and manipulate quantum information by encoding it into the state of Majorana zero modes. As the information is stored in a topological property of the system, local noise cannot lead to decoherence. Manipulation of the information is achieved by braiding the zero modes. The measurement, however, is challenging as the information is well hidden and thus inherently hard to access. Here, we discuss a setup for measuring the state of a Majorana qubit by employing standard tools of microwave engineering. The basic physical effect that we employ is the fact that a voltage-biased Josephson junction hosting Majorana fermions allows photons to be emitted and absorbed at half the Josephson frequency. We show that in the dispersive regime, our setup allows us to perform a quantum nondemolition measurement and to reach the quantum limit. An appealing feature of our setup is that the interaction of the Majorana qubit with the measurement device can be turned on and off at will by changing the dc bias of the junction.
Constructing a universal operation scheme for Majorana qubits remains a central issue for the topological quantum computation. We study the Landau-Zener-St\"uckelberg interference in a Majorana qubit and show that this interference can be used to achieve controllable operations. The Majorana qubit consists of an rf superconducting quantum interference device (SQUID) with a topological nanowire Josephson junction which hosts Majorana bound states. In the SQUID, a magnetic flux pulse can drive the quantum evolution of the Majorana qubit. The qubit experiences two Landau-Zener transitions when the amplitude of the pulse is tuned around the superconducting flux quanta h/2eh/2eh/2e. The Landau-Zener-St\"uckelberg interference between the two transitions rotates the Majorana qubit, with the angle controlled by the time scale of the pulse. This rotation operation implements a high-speed single-qubit gate on the Majorana qubit, which is a necessary ingredient for the topological quantum computation.
We analyze the control of Majorana zero-energy states by mapping the fermionic system onto a chain of Ising spins. Although the topological protection is lost for the Ising system, the mapping provides additional insight into the nature of the quantum states. By controlling the local magnetic field, one can separate the Ising chain into ferromagnetic and paramagnetic phases, corresponding to topological and nontopological sections of the fermionic system. In this paper we propose (topologically nonprotected) protocols performing the braiding operation, and in fact also more general rotations. We first consider a TTT-junction geometry, but we also propose a protocol for a purely one-dimensional (1D) system. Both setups rely on an extra spin-12\frac{1}{2}21​ coupler. By including the extra spin in the TTT-junction geometry, we overcome limitations due to the 1D character of the Jordan--Wigner transformation. In the 1D geometry the coupler, which controls one of the Ising links, should be manipulated once the ferromagnetic (topological) section of the chain is moved far away. We also propose experimental implementations of our scheme. One is based on a chain of flux qubits which allows for all needed control fields. We also describe how to translate our scheme for the 1D setup to a chain of superconducting wires hosting each a pair of Majorana edge states.