  In recent work Hartmann et al [Phys. Rev. Lett. 102, 057202 (2009)]
demonstrated that the classical simulation of the dynamics of open 1D quantum
systems with matrix product algorithms can often be dramatically improved by
performing time evolution in the Heisenberg picture. For a closed system this
was exemplified by an exact matrix product operator solution of the
time-evolved creation operator of a quadratic fermi chain with a matrix
dimension of just two. In this work we show that this exact solution can be
significantly generalized to include the case of an open quadratic fermi chain
subjected to master equation evolution with Lindblad operators that are linear
in the fermionic operators. Remarkably even in this open system the
time-evolution of operators continues to be described by matrix product
operators with the same fixed dimension as that required by the solution of a
coherent quadratic fermi chain for all times. Through the use of matrix product
algorithms the dynamical behaviour of operators in this non-equilibrium open
quantum system can be computed with a cost that is linear in the system size.
We present some simple numerical examples which highlight how useful this might
be for the more detailed study of open system dynamics. Given that Heisenberg
picture simulations have been demonstrated to offer significant accuracy
improvements for other open systems that are not exactly solvable our work also
provides further insight into how and why this advantage arises.
