  Quantum-logic-based ${}^{27}\textrm{Al}^+$ optical clock has been
demonstrated in several schemes as there are different choices of the auxiliary
ion species. In this paper, we present the first detailed evaluation of the
systematic shift and the total uncertainty of an ${}^{27}\textrm{Al}^+$ optical
clock sympathetically cooled by a ${}^{40}\textrm{Ca}^+$ ion. The total
systematic uncertainty of the ${}^{40}\textrm{Ca}^+ - {}^{27}\textrm{Al}^+$
quantum logic clock has been estimated to be $7.9 \times 10^{-18}$, which was
mainly limited by the uncertainty of the quadratic Zeeman shift. By comparing
the frequency of two counter-propagating clock beams on the same ion, we
measured the frequency stability to be $3.7 \times 10^{-14} /\sqrt{\tau}$.
