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Topological Quantum Computing on a Conventional Quantum Computer

Xiao Xiao, James K. Freericks, Alexander F. Kemper

Published Jun 9, 2020• 3 citations
OpenAlex

Abstract

Majorana fermions are self-adjoint fermionic particles that are believed to exist as the elementary excitations in nanoengineered devices with superconductors and ferromagnets. They can be employed in topological quantum computation and can aid the solution of a class of spin models created by Kitaev. In both cases, the braiding of Majorana fermions plays a critical role in these applications. We explicitly construct quantum circuits for Majorana braiding and run them on IBM quantum computers to prepare the ground states of strongly correlated Kitaev-inspired models. The entanglement entropy for these models is then measured to determine quantum phase transitions. We show how maintaining particle-hole symmetry is critical to carrying out this work.

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We study the possibility to realize Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work. To achieve this we first apply a uniform conical magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p+ip topological superconductor of spinons. We then study possible vortex binding in such system to a topologically trivial spot in the ground state. We consider two cases in the system. One is a vacancy and the other is a fully polarized spin. We show that in both cases, the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform conical magnetic field. The distribution and asymptotic behavior of these Majorana zero modes is studied. The Majorana zero modes in both cases decay exponentially in space, and are robust against local perturbations and other Majorana zero modes far away, which makes them promissing candidate for braiding in topological quantum computing.

•Mar 4, 2021•0 citations
Majorana fermions are self-adjoint fermionic particles that are believed to exist as the elementary excitations in nanoengineered devices with superconductors and ferromagnets. They can be employed in topological quantum computation and can aid the solution of a class of spin models created by Kitaev. In both cases, the braiding of Majorana fermions plays a critical role in these applications. We explicitly construct quantum circuits for Majorana braiding and run them on IBM quantum computers to prepare the ground states of strongly correlated Kitaev-inspired models. The entanglement entropy for these models is then measured to determine quantum phase transitions. We show how maintaining particle-hole symmetry is critical to carrying out this work.
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.
Topological protection is employed in fault-tolerant error correction and in developing quantum algorithms with topological qubits. But, topological protection intrinsic to models being simulated , also robustly protects calculations, even on NISQ hardware. We leverage it by simulating Kitaev-inspired models on IBM quantum computers and accurately determining their phase diagrams. This requires constructing conventional quantum circuits for Majorana braiding to prepare the ground states of Kitaev-inspired models. The entanglement entropy is then measured to calculate the quantum phase boundaries. We show how maintaining particle-hole symmetry when sampling through the Brillouin zone is critical to obtaining high accuracy. This work illustrates how topological protection intrinsic to a quantum model can be employed to perform robust calculations on NISQ hardware, when one measures the appropriate protected quantum properties. It opens the door for further simulation of topological quantum models on quantum hardware available today.
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. We generate an arbitral number of segments in a Kitaev chain which are in the topological phase. A pair of Majorana-like edge states emerge at the edges of a topological sector, which is observed by impedance peaks. When the Kitaev chain contains NNN topological sectors, the degeneracy of the zero-admittance states is 2N2^{N}2N. A set of AC voltages along the chain plays a role of the wave function. An \textit{LC} resonator attached to each Majorana-like edge state registers the phase and the absolute value of the voltage. One qubit consists of the in-phase and opposite-phase states made of two \textit{LC} resonators installed in one topological sector. 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. The present scheme may simulate topological quantum computations based on electric circuits.
We study the possibility to realize a Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work. To achieve this we first apply a uniform [111] magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p + ip topological superconductor of spinons. We then study possible vortex binding in such system to a topologically trivial spot in the ground state. We consider two cases in the system: one is a vacancy and the other is a fully polarized spin. We show that in both cases, the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform [111] magnetic field. The distribution and asymptotic behavior of these Majorana zero modes are studied. The Majorana zero modes in both cases decay exponentially in space, and are robust against local perturbations and other Majorana zero modes far away, which makes them promising candidates for braiding in topological quantum computing.
The chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain 2D topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of a quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, the Majorana zero mode in the bulk of the corresponding topological matter, is known to be applicable in topological quantum computations. Here we show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes and propose a platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can use quantum coherent chiral Majorana fermions to implement the Hadamard gate and the phase gate, and the junction conductance yields a natural readout for the qubit state.
We study the possibility to realize Majorana zero mode that is robust and may be easily manipulated for braiding in quantum computing in the ground state of the Kitaev model in this work. To achieve this we first apply a uniform conical magnetic field to the gapless Kitaev model and turn the Kitaev model to an effective p+ip topological superconductor of spinons. We then study possible vortex binding in such system to a topologically trivial spot in the ground state. We consider two cases in the system. One is a vacancy and the other is a fully polarized spin. We show that in both cases, the system binds a vortex with the defect and a robust Majorana zero mode in the ground state at a weak uniform conical magnetic field. The distribution and asymptotic behavior of these Majorana zero modes is studied. The Majorana zero modes in both cases decay exponentially in space, and are robust against local perturbations and other Majorana zero modes far away, which makes them promissing candidate for braiding in topological quantum computing.