Investigating the Exchange of Ising Chains on a Digital Quantum Computer
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.
Bassel Heiba Elfeky Matthieu C. Dartiailh S. M. Farzaneh Javad Shabani