Analytic solutions for quantum logic gates and modeling pulse errors in a quantum computer with a Heisenberg interaction

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.