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Journal of Experimental and Theoretical Physics

Disentangled Quantum Operation on Majorana Qubits in Qubit Chains

Yu. Makhlin, Stefan Backens, Alexander Shnirman

Published Oct 1, 2019• 1 citation
Publisher

Abstract

Abstract We consider quantum logical gates on Majorana qubits implemented in chain structures of ordinary qubits, spins, or pseudospins. We demonstrate that one can implement a two-qubit operation via local manipulations, using an extra coupler spin in a T-junction geometry, so that this coupler spin remains disentangled from the qubit. Furthermore, we identify a set of symmetry operations, which not only allow us to determine the resulting two-qubit gate, but also to demonstrate robustness of the resulting gate to inaccuracies in the manipulations, known for topological quantum computation.

DOI
10.1134/s1063776119100212

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Abstract We consider quantum logical gates on Majorana qubits implemented in chain structures of ordinary qubits, spins, or pseudospins. We demonstrate that one can implement a two-qubit operation via local manipulations, using an extra coupler spin in a T-junction geometry, so that this coupler spin remains disentangled from the qubit. Furthermore, we identify a set of symmetry operations, which not only allow us to determine the resulting two-qubit gate, but also to demonstrate robustness of the resulting gate to inaccuracies in the manipulations, known for topological quantum computation.
Majorana zero modes can be simulated in structures based on spin or quasi-spin degrees of freedom, e.g., Josephson-qubit chains. Braiding of Majorana degrees of freedom is realized using T-junctions supplied with an auxiliary spin (ancilla). Motivated by prospective experiments, we analyze the braiding in the spin representation, which provides the basis for the analysis of imperfections characteristic to the spin and qubit designs. The result of the braiding operation is straightforwardly found for the initial basis states of the two qubits and the ancilla, up to phase factors. Here, we fix these phase factors and thus describe the complete two-qubit operation. This result is relevant for physical simulation of the Majorana qubits in Josephson-qubit chains and other spin or qubit structures.
We propose and analyze a physical system capable of performing topological quantum computation with Majorana zero modes (MZMs) in a one-dimensional topological superconductor (1DTS). One of the leading methods to realize quantum gates in a 1DTS is to use T junctions, which allows one to maneuver MZMs in such a manner as to achieve braiding. In this paper, we propose a scheme for implementing quantum logical gates in a purely one-dimensional geometry without T junctions, instead replacing it with an auxiliary qubit. This has the additional benefit of introducing a non-Clifford gate, corresponding to one- and two-logical-qubit ZZZ rotations. We first design a topologically protected logical ZZZ gate based entirely on local interactions within the 1DTS. Using an auxiliary qubit coupled to the topological superconductors, we extend the ZZZ gate to non-Clifford single- and multiqubit arbitrary rotations with partial topological protection. Finally, to perform universal quantum computing, we introduce a scheme for performing arbitrary unitary rotations, albeit without topological protection. We develop a formalism based on unitary braids which creates transitions between different topological phases of the 1DTS system. The unitary formalism can be simply converted to an equivalent adiabatic scheme, which we numerically simulate, and we show that high-fidelity operations should be possible with reasonable parameters.
We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum-dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot that is tunnel coupled to two topological superconductors. The effective spin-Majorana exchange facilitates a hybrid cnot gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a cnot gate is implemented that acts on neighboring Majorana qubits and eliminates the necessity of interqubit braiding. In both setups, the spin-Majorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a swap or hybrid swap gate which is sufficient for universal quantum computation without projective measurements.
Among the list of major threats to quantum computation, quantum decoherence poses one of the largest because it generates losses to the environment within a computational system which cannot be recovered via error correction methods. These methods require the assumption that the environmental interaction forces the qubit state into some linear combination of qubit eigenstates. In reality, the environment causes the qubit to enter into a mixed state where the original is no longer recoverable. A promising solution to this problem bases the computational states on the low lying energy excitations within topological materials. The existence of these states is protected by a global parameter within the Hamiltonian which prevents the computational states from coupling locally and decohering. In this paper, the qubit is based on nonlocal, topological Majorana fermions (MF), and the gate operations are generated by swapping or braiding the positions of said MF. The algorithmic calculation for such gate operations is well known, but, the opposite gates-to-braid calculation is currently underdeveloped. Additionally, because one may choose from a number of different possible qubit definitions, the resultant gate operations from calculation to calculation appear different. Here, the calculations for the two- and four-MF cases are recapitulated for the sake of logical flow. This set of gates serves as the foundation for the understanding and construction of the six-MF cases. Using these, a full characterization of the system is made by completely generalizing the list of gates and transformations between possible qubit definitions. A complete description of this system is desirable and will hopefully serve future iterations of topological qubits.
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.
The realization of quantum gates in topological quantum computation still confronts significant challenges in both fundamental and practical aspects. Here, we propose an ancilla-free, deterministic, and topologically protected measurement-based scheme to realize the implementation of Clifford quantum gates on the Majorana qubits. Our scheme is based on a rigorous proof that the single-qubit gate can be performed by leveraging the neighboring Majorana qubit but not disturbing its carried quantum information, enabling an ancilla-free scheme for the topological quantum computing with Majorana qubits. Benefiting from the ancilla-free construction, we show the minimum measurement sequences with four steps to achieve two-qubit Clifford gates by constructing their geometric visualization. The uncertainty of the current strategy can be avoided by manipulating the Majorana modes in their parameter space, as shown in a concrete Majorana platform, correcting the undesired measurement outcomes while maintaining topological protection. Our scheme identifies the minimal operations of measurement-based topological and deterministic Clifford gates and offers an ancilla-free design of topological quantum computation.