  One of the first proposals for the use of quantum computers was the
simulation of quantum systems. Over the past three decades, great strides have
been made in the development of algorithms for simulating closed quantum
systems and the more complex open quantum systems. In this tutorial, we
introduce the methods used in the simulation of single qubit Markovian open
quantum systems. It combines various existing notations into a common framework
that can be extended to more complex open system simulation problems. The only
currently available algorithm for the digital simulation of single qubit open
quantum systems is discussed in detail. A modification to the implementation of
the simpler channels is made that removes the need for classical random
sampling, thus making the modified algorithm a strictly quantum algorithm. The
modified algorithm makes use of quantum forking to implement the simpler
channels that approximate the total channel. This circumvents the need for
quantum circuits with a large number of C-NOT gates.
