Accuracy versus bond dimension
This tutorial applies a random MPO to a random MPS with apply and measures how
the error of the compressed product falls as the output bond dimension chi_out
grows. As a yardstick it uses quimb's contract-then-truncate, which computes
the exact product and truncates it with SVDs: close to the best a train of that
bond dimension can do.
import matplotlib.pyplot as plt
import numpy as np
import quimb.tensor as qtn
from src_method import applyThe problem
A 16-site MPO with bond dimension 4 applied to an MPS with bond dimension 16: the exact product has bond dimension 64, which is still small enough to form and use as the reference.
n_sites = 16
H = qtn.MPO_rand(n_sites, bond_dim=4, phys_dim=2, seed=0)
psi = qtn.MPS_rand_state(n_sites, bond_dim=16, phys_dim=2, seed=1)
exact = H.apply(psi, compress=False)
norm = exact.norm()
print(f"exact bond dimension: {exact.max_bond()}")exact bond dimension: 64Sweeping chi_out
apply takes and returns plain lists of site arrays, so .arrays unwraps the
quimb networks and qtn.MatrixProductState wraps the result back up. A fixed
seed keeps the random sketches, and the figure, reproducible.
chis = [4, 8, 12, 16, 24, 32, 48, 64]
src_errors, svd_errors = [], []
for chi in chis:
phi = qtn.MatrixProductState(apply(H.arrays, psi.arrays, chi_out=chi, seed=2))
src_errors.append(phi.distance(exact) / norm)
truncated = exact.copy()
truncated.compress(max_bond=chi)
svd_errors.append(truncated.distance(exact) / norm)
for chi, e_src, e_svd in zip(chis, src_errors, svd_errors):
print(f"chi_out={chi:3d} SRC {e_src:.2e} SVD {e_svd:.2e}")chi_out= 4 SRC 9.38e-01 SVD 7.92e-01
chi_out= 8 SRC 7.44e-01 SVD 4.91e-01
chi_out= 12 SRC 5.99e-01 SVD 3.30e-01
chi_out= 16 SRC 4.56e-01 SVD 2.27e-01
chi_out= 24 SRC 2.83e-01 SVD 1.09e-01
chi_out= 32 SRC 1.48e-01 SVD 5.02e-02
chi_out= 48 SRC 3.61e-02 SVD 9.76e-03
chi_out= 64 SRC 0.00e+00 SVD 0.00e+00fig, ax = plt.subplots(figsize=(6, 4))
ax.semilogy(chis, np.maximum(src_errors, 1e-16), "o-", label="SRC (apply)")
ax.semilogy(chis, np.maximum(svd_errors, 1e-16), "s--", label="exact + SVD truncation")
ax.set_xlabel("chi_out")
ax.set_ylabel("relative error")
ax.legend()
fig.tight_layout()
Both curves fall together and reach rounding error once chi_out matches the
exact bond dimension of 64. SRC sits somewhat above the SVD truncation: it never
forms the exact product, and its randomized sketch is not optimal. In exchange,
its cost grows with chi_out**2 times the input bond dimensions instead of with
the cube of the exact bond dimension, which is what makes it pay off once the
exact product no longer fits.