Adaptive truncation

chi_out is the maximum output bond dimension. Set cutoff to trim each bond further using a relative singular-value threshold:

import quimb.tensor as qtn

from portc import apply

H = qtn.MPO_identity(8, phys_dim=2)
psi = qtn.MPS_rand_state(8, bond_dim=4, seed=0)

fixed = apply(H.arrays, psi.arrays, chi_out=32, seed=1)
trimmed = apply(H.arrays, psi.arrays, chi_out=32, cutoff=1e-10, seed=1)

print([a.shape[0] for a in fixed[1:]])
print([a.shape[0] for a in trimmed[1:]])

By default, the sketch size is chi_out. During the right-to-left sweep, the QR factorisation at each site is followed by an SVD of its small triangular factor, and singular values below cutoff * sigma_max are discarded. The output is right-canonical.

OversamplingCopy Anchor Link

Sketching at exactly chi_out lets the error of the randomized range finder dominate. Set oversample to a finite factor of at least 1.0 to sketch at ceil(oversample * chi_out), or to None for the paper's recommendation, max(ceil(1.5 * chi_out), chi_out + 10):

import quimb.tensor as qtn

from portc import apply

H = qtn.MPO_rand(8, bond_dim=4, phys_dim=2, seed=0)
psi = qtn.MPS_rand_state(8, bond_dim=16, seed=0)

out = apply(H.arrays, psi.arrays, chi_out=32, oversample=None, seed=1)

A final left-to-right SVD sweep then truncates the oversampled, right-canonical intermediate to chi_out, discarding singular values below cutoff * sigma_max at each cut. These are the Schmidt values of the intermediate train as truncated so far, not of the random sketch matrix. The result is left-canonical, with the norm on the last site.

Oversampling improves accuracy at a fixed chi_out but costs more: the sweep runs at the larger sketch size, cutoff no longer trims bonds during the sweep, and the final pass adds one SVD per site. With oversample=None, expect a run time several times that of the default.

• cutoff must lie in [0.0, 1.0); 0.0 (the default) disables trimming.
• chi_out remains the upper bound: the cutoff only ever lowers a bond.
• Two-site trains use the exact fallback, which ignores cutoff and oversample, truncates to chi_out and returns right-canonical output.