  We analyze analytically and numerically quantum logic gates in a
one-dimensional spin chain with Heisenberg interaction. Analytic solutions for
basic one-qubit gates and swap gate are obtained for a quantum computer based
on logical qubits. We calculated the errors caused by imperfect pulses which
implement the quantum logic gates. It is numerically demonstrated that the
probability error is proportional to $\epsilon^4$, while the phase error is
proportional to $\epsilon$, where $\epsilon$ is the characteristic deviation
from the perfect pulse duration.
