Abstract
The ferromagnetic state of an Ising chain can represent a two-fold degenerate subspace or equivalently a logical qubit which is protected from excitations by an energy gap. We study a a braiding-like exchange operation through the movement of the state in the qubit subspace which resembles that of the localized edge modes in a Kitaev chain. The system consists of two Ising chains in a 1D geometry where the operation is simulated through the adiabatic time evolution of the ground state. The time evolution is implemented via the Suzuki-Trotter expansion on basic single- and two-qubit quantum gates using IBM's Aer QASM simulator. The fidelity of the system is investigated as a function of the evolution and system parameters to obtain optimum efficiency and accuracy for different system sizes. Various aspects of the implementation including the circuit depth, Trotterization error, and quantum gate errors pertaining to the Noisy Intermediate-Scale Quantum (NISQ) hardware are discussed as well. We show that the quantum gate errors, i.e. bit-flip, phase errors, are the dominating factor in determining the fidelity of the system as the Trotter error and the adiabatic condition are less restrictive even for modest values of Trotter time steps. We reach an optimum fidelity $>99\%$ on systems of up to $11$ sites per Ising chain and find that the most efficient implementation of a single braiding-like operation for a fidelity above $90\%$ requires a circuit depth of the order of $\sim 10^{3}$ restricting the individual gate errors to be less than $\sim 10^{-6}$ which is prohibited in current NISQ hardware.
Highlights
The race for developing fault-tolerant quantum computing has only intensified in the last few years [1,2]
While the main issue is still decoherence and local perturbations [4,5], there has been significant progress in two directions: (i) qubits that are engineered to autonomously correct errors with methods such as bosonic codes [6], and (ii) proposed qubits based on non-Abelian anyons that exhibit nontrivial topological statistics, which are immune to decoherence and local perturbations, such as Majorana qubits [7]
This work focuses on the time evolution of a braiding-like operation in the transverse Ising model, the results and the analysis of the dynamics and its implementation on a digital quantum computer are applicable to a wider range of physical phenomena than just the edge modes in a Kitaev model
Summary
The race for developing fault-tolerant quantum computing has only intensified in the last few years [1,2]. While the main issue is still decoherence and local perturbations [4,5], there has been significant progress in two directions: (i) qubits that are engineered to autonomously correct errors with methods such as bosonic codes [6], and (ii) proposed qubits based on non-Abelian anyons that exhibit nontrivial topological statistics, which are immune to decoherence and local perturbations, such as Majorana qubits [7] While these novel qubits are being developed, it is instructive to study their dynamics by simulating them on available noisy intermediate-scale quantum (NISQ) hardware. This work focuses on the time evolution of a braiding-like operation in the transverse Ising model, the results and the analysis of the dynamics and its implementation on a digital quantum computer are applicable to a wider range of physical phenomena than just the edge modes in a Kitaev model. IV, we discuss the system’s expected Suzuki-Trotter error before presenting the results and analysis of the digital simulation and the contributing errors
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
