Majorana fermions subject to the non-Abelian braid group are believed to be the basic ingredients of future topological quantum computations. In this work, we propose to simulate Majorana fermions of the Kitaev model in electric circuits based on the observation that the circuit Laplacian can be made identical to the Hamiltonian. A set of AC voltages along the chain plays the role of the wave function. We generate an arbitral number of topological segments in a Kitaev chain. A pair of topological edge states emerge at the edges of a topological segment. Its wave function is observable by the position and the phase of a peak in impedance measurement. It is possible to braid any pair of neighboring edge states with the aid of T-junction geometry. By calculating the Berry phase acquired by their eigenfunctions, the braiding is shown to generate one-qubit and two-qubit unitary operations. We explicitly construct Clifford quantum gates based on them. We also present an operator formalism by regarding a topological edge state as a topological soliton intertwining the trivial segment and the topological segment. Our analysis shows that the electric-circuit approach can simulate the Majorana-fermion approach to topological quantum computations.
Abstract Majorana fermions and their generalizations to Zn parafermions are considered promising building blocks of fault-tolerant quantum computers for their ability to encode quantum information nonlocally. In such topological quantum computers, highly robust quantum gates are obtained by braiding pairs of these quasi-particles. However, it is well-known that braiding Majorana fermions or parafermions only leads to a Clifford gate, hindering quantum universality. This paper establishes an exact mapping between Majorana fermions to Z4 parafermions in systems under total parity nonconserving and total parity conserving setting. It is revealed that braiding of Majorana fermions may lead to non-Clifford quantum gates in the 4-dimensional qudit representation spanned by Z4 parafermions, whilst braiding of Z4 parafermions may similarly yield non-Clifford quantum gates in the qubit representation spanned by Majorana fermions. This finding suggests that topologically protected universal quantum computing may be possible with Majorana fermions (Z4 parafermions) by supplementing the usual braiding operations with the braiding of Z4 parafermions (Majorana fermions) that could be formed out of Majorana fermions (Z4 parafermions) via the mapping prescribed here. Finally, the paper discusses how braiding of Majorana fermions or Z4 parafermions could be obtained via a series of parity measurements.
Topological quantum computation by way of braiding of Majorana fermions is not universal quantum computation. There are several attempts to make universal quantum computation by introducing some additional quantum gates or quantum states. However, there is a serious problem in embedding an
M-qubit quantum gate in the
N-qubit system with
N>M. This is an inherent problem to the Majorana system, where quantum gates for logical qubits become nonlocal in terms of physical qubits in general because braiding operations preserve the fermion parity. For instance, the CZ gate could not be embedded in the three-qubit system. We overcome this embedding problem by introducing
2(N+1)-body interactions of Majorana fermions in the
N-qubit system. A universal set of quantum gates is constructed for
N logical qubits, leading to topological-nontopological hybrid universal quantum computation. It would be more robust than conventional universal quantum computation because the quantum gates generated by braiding are topologically protected.