Generation of high-fidelity controlled-NOT logic gates by coupled superconducting qubits

Abstract

Building on the previous results of the Weyl chamber steering method, we demonstrate how to generate high-fidelity controlled-NOT (CNOT) gates by direct application of certain physically relevant Hamiltonians with fixed coupling constants containing Rabi terms. Such Hamiltonians are often used to describe two superconducting qubits driven by local rf pulses. It is found that in order to achieve 100% fidelity in a system with capacitive coupling of strength $g$, one Rabi term suffices. We give the exact values of the physical parameters needed to implement such CNOT gates. The gate time and all possible Rabi frequencies are found to be $t=\ensuremath{\pi}∕(2g)$ and ${\ensuremath{\Omega}}_{1}∕g=\sqrt{64{n}^{2}\ensuremath{-}1},n=1,2,3,\dots{}$. Generation of a perfect CNOT gate in a system with inductive coupling, characterized by additional constant $k$, requires the presence of both Rabi terms. The gate time is again $t=\ensuremath{\pi}∕(2g)$, but now there is an infinite number of solutions, each of which is valid in a certain range of $k$ and is characterized by a pair of integers $(n,m)$, $\frac{{\ensuremath{\Omega}}_{1,2}}{g}=\sqrt{16{n}^{2}\ensuremath{-}{(\frac{k\ensuremath{-}1}{2})}^{2}}\ifmmode\pm\else\textpm\fi{}\sqrt{16{m}^{2}\ensuremath{-}{(\frac{k+1}{2})}^{2}}$. We distinguish two cases, depending on the sign of the coupling constant: (i) the antiferromagnetic case $(k\ensuremath{\geqslant}0)$ with $n\ensuremath{\geqslant}m=0,1,2,\dots{}$ and (ii) the ferromagnetic case $(k\ensuremath{\leqslant}0)$ with $n>m=0,1,2,\dots{}$. We conclude with consideration of fidelity degradation by switching to resonance. Simulation of time evolution based on the fourth-order Magnus expansion reveals characteristics of the gate similar to those found in the exact case, with slightly shorter gate time and shifted values of the Rabi frequencies.