  Matrix product density operators (MPDOs) are tensor network representations
of locally purified density matrices where each physical degree of freedom is
associated to an environment degree of freedom. MPDOs have interesting
properties for mixed state representations: guaranteed positivity by
construction, efficient conservation of the trace and computation of local
observables. However, they have been challenging to use for noisy quantum
circuit simulation, as the application of noise increases the dimension of the
environment Hilbert space, leading to an exponential growth of bond dimensions.
MPDOs also lack a unique canonical form, due to the freedom in the choice of
basis for the environment Hilbert space, which leads to a vast variation of
bond dimensions.
  In this work, we present a systematic way to reduce the bond dimensions of
MPDOs by disentangling the purified state. We optimize the basis for the
environment Hilbert space by performing density matrix renormalization group
(DMRG)-like sweeps of local 2-qubit basis optimization. Interestingly, we find
that targeting only the disentanglement of the purified state leads to a
reduction of the environment dimension. In other words, a compact MPDO
representation requires a low-entanglement purified state.
  We apply our compression method to the emulation of noisy random quantum
circuits. Our technique allows us to keep bounded bond dimensions, and thus
bounded memory, contrary to previous works on MPDOs, while keeping reasonable
truncation fidelities.
