  We introduce AQCtensor, a novel algorithm to produce short-depth quantum
circuits from Matrix Product States (MPS). Our approach is specifically
tailored to the preparation of quantum states generated from the time evolution
of quantum many-body Hamiltonians. This tailored approach has two clear
advantages over previous algorithms that were designed to map a generic MPS to
a quantum circuit. First, we optimize all parameters of a parametric circuit at
once using Approximate Quantum Compiling (AQC) - this is to be contrasted with
other approaches based on locally optimizing a subset of circuit parameters and
"sweeping" across the system. We introduce an optimization scheme to avoid the
so-called ``orthogonality catastrophe" - i.e. the fact that the fidelity of two
arbitrary quantum states decays exponentially with the number of qubits - that
would otherwise render a global optimization of the circuit impractical.
Second, the depth of our parametric circuit is constant in the number of qubits
for a fixed simulation time and fixed error tolerance. This is to be contrasted
with the linear circuit Ansatz used in generic algorithms whose depth scales
linearly in the number of qubits. For simulation problems on 100 qubits, we
show that AQCtensor thus achieves at least an order of magnitude reduction in
the depth of the resulting optimized circuit, as compared with the best generic
MPS to quantum circuit algorithms. We demonstrate our approach on simulation
problems on Heisenberg-like Hamiltonians on up to 100 qubits and find optimized
quantum circuits that have significantly reduced depth as compared to standard
Trotterized circuits.
