  Recently, a polynomial time classical algorithm has been found for learning
the shallow representation of a unitary $U$ acting on $n$-qubits, by learning
local inversions and then sewing them back together with ancilla qubits and
SWAP gates. In this work, we bootstrap local inversion learning (LIL) to
optimize quantum circuit depth by learning shallow representations for its
sub-unitaries. We recursively cut circuits and apply the LIL algorithm to
replace sub-circuits with their shallow representations, if it can be found by
the algorithm. If not, we keep cutting until the optimization terminates,
either by finding shallow representations or by reaching constant-depth
sub-circuits. By replacing sub-circuits with their shallow representations, we
hope to obtain some compression of the quantum circuit. Due to the binary
search structure, the optimization algorithm has time complexity logarithmic in
the depth of the original given circuit.
