It looks like the paper specifically claims to create an optimized quantum algorithm for SR pair finding which is most time consuming part the Schnorr fatoring algorithm. It starts with a classical computational lattice problem, defines the optimization problem in a specific form (3), and then maps it to a Hamiltonian to represent a lattice matrix and a PauliZ matrix ((1, 0), (0, -1)) (4) and if we have a Hamiltonian, we can create a QC.
The proof of the reduced memory complexity is linked in section 31 but it is really long winded however, it seems the classical Schnorr lattice reduction problem for factoring integers has a space complexity of O((logN)^(α)/α*loglogN)) which according to the author, since the paper can map SR pair finding (the time consuming piece of Schnorr's algorithm) to a Hamiltonian, we then have a quantum factoring algorithm of the same complexity.
The proof of the reduced memory complexity is linked in section 31 but it is really long winded however, it seems the classical Schnorr lattice reduction problem for factoring integers has a space complexity of O((logN)^(α)/α*loglogN)) which according to the author, since the paper can map SR pair finding (the time consuming piece of Schnorr's algorithm) to a Hamiltonian, we then have a quantum factoring algorithm of the same complexity.
Seems pretty handwavy to me.