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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.