  The rapid pace of recent advancements in numerical computation, notably the
rise of GPU and TPU hardware accelerators, have allowed tensor network (TN)
algorithms to scale to even larger quantum simulation problems, and to be
employed more broadly for solving machine learning tasks. The
"quantum-inspired" nature of TNs permits them to be mapped to parametrized
quantum circuits (PQCs), a fact which has inspired recent proposals for
enhancing the performance of TN algorithms using near-term quantum devices, as
well as enabling joint quantum-classical training frameworks which benefit from
the distinct strengths of TN and PQC models. However, the success of any such
methods depends on efficient and accurate methods for approximating TN states
using realistic quantum circuits, something which remains an unresolved
question. In this work, we compare a range of novel and previously-developed
algorithmic protocols for decomposing matrix product states (MPS) of arbitrary
bond dimensions into low-depth quantum circuits consisting of stacked linear
layers of two-qubit unitaries. These protocols are formed from different
combinations of a preexisting analytical decomposition scheme with constrained
optimization of circuit unitaries, and all possess efficient classical
runtimes. Our experimental results reveal one particular protocol, involving
sequential growth and optimization of the quantum circuit, to outperform all
other methods, with even greater benefits seen in the setting of limited
computational resources. Given these promising results, we expect our proposed
decomposition protocol to form a useful ingredient within any joint application
of TNs and PQCs, in turn further unlocking the rich and complementary benefits
of classical and quantum computation.
