Approximate/compressed contraction

In this example we use cotengra to find a ‘compressed’ contraction tree (as detailed in 2206.07044) for a \(10\times10\times10\) tensor network representation of the partition function of the 3D classical ising model. We then use quimb to actually perform the compressed contraction.

%config InlineBackend.figure_formats = ['svg']
import cotengra as ctg
import quimb.tensor as qtn

First we get the tensor network from quimb, though note we could also specify it as usual via inputs, output and size_dict.

tn = qtn.TN3D_classical_ising_partition_function(
    10, 10, 10, beta=0.3
)

Then we set-up our compressed contraction tree finder:

chi = 16
copt = ctg.ReusableHyperCompressedOptimizer(
    chi,
    max_repeats=256,
    minimize='combo-compressed', 
    progbar=True,
    # # save paths to disk:
    # directory=True  
)

quimb knows about cotengra and will directly return the ContractionTreeCompressed generated during the search:

tree = tn.contraction_tree(copt)

log2[SIZE]: 21.00 log10[FLOPs]: 10.89: 100%|████████████████████████| 256/256 [00:15<00:00, 16.02it/s]

We can perform various bits of introspection on both the compressed contraction tree and optimizer now before we actually perform the contraction.

copt.last_opt.plot_trials()
../_images/ae3563da3e7100c0d9554f1682505a4d2511878af8e1cf92a3eb02fad15284a7.svg
(<Figure size 800x300 with 1 Axes>, <Axes: xlabel='TRIAL', ylabel='SCORE'>)

We can see that the best tree was generated with the Span algorithm, so we can visualize the actual spanning tree like so:

tree.plot_span(
    node_max_size=1, 
    edge_max_width=1, 
    edge_color='order', 
    node_color='order',
    edge_colormap='Spectral_r',
    node_colormap='Spectral_r',
    colorbars=False,
)
../_images/2f6c8fde7b82a1994bae2f283063c13d14f45dd8cf683bfd71ae1fd6c675ee4d.svg
(<Figure size 500x500 with 1 Axes>, <Axes: >)

Here we optionally convert to GPU arrays to make the contraction (much) quicker:

def to_backend(x):
    import torch

    return torch.tensor(x, dtype=torch.float32, device='cuda')

tn.apply_to_arrays(to_backend)

Now we actually perform the contraction, this requires a lot of machinery that only quimb can provide currently, so we just need to supply either the raw path or optimizer to the following method:

tn.contract_compressed_(
    optimize=tree.get_path(),  # or optimize=copt
    max_bond=chi,
    equalize_norms=1.0,
    progbar=True,
)

log2[SIZE]: 0.00/21.00: 100%|███████████████████████████████████████| 999/999 [00:11<00:00, 90.42it/s]
TensorNetwork3D(tensors=1, indices=0, Lx=10, Ly=10, Lz=10, max_bond=1)
Tensor(shape=(), inds=[], tags={I9,0,9, X9, Y0, Z9, I9,1,9, Y1, I9,2,9, Y2, I9,3,9, Y3, I9,4,9, Y4, I9,5,9, Y5, I9,6,9, Y6, I9,7,9, Y7, I9,8,9, Y8, I9,9,9, Y9, I9,0,8, Z8, I9,1,8, I9,2,8, I9,3,8, I9,4,8, I9,5,8, I9,6,8, I9,7,8, I9,8,8, I9,9,8, I9,0,7, Z7, I9,1,7, I9,2,7, I9,3,7, I9,4,7, I9,5,7, I9,6,7, I9,7,7, I9,8,7, I9,9,7, I9,0,6, Z6, I9,1,6, I9,2,6, I9,3,6, I9,4,6, I9,5,6, I9,6,6, I9,7,6, I9,8,6, I9,9,6, I9,0,5, Z5, I9,1,5, I9,2,5, I9,3,5, I9,4,5, I9,5,5, I9,6,5, I9,7,5, I9,8,5, I9,9,5, I9,0,4, Z4, I9,1,4, I9,2,4, I9,3,4, I9,4,4, I9,5,4, I9,6,4, I9,7,4, I9,8,4, I9,9,4, I9,0,3, Z3, I9,1,3, I9,2,3, I9,3,3, I9,4,3, I9,5,3, I9,6,3, I9,7,3, I9,8,3, I9,9,3, I9,0,2, Z2, I9,1,2, I9,2,2, I9,3,2, I9,4,2, I9,5,2, I9,6,2, I9,7,2, I9,8,2, I9,9,2, I9,0,1, Z1, I9,1,1, I9,2,1, I9,3,1, I9,4,1, I9,5,1, I9,6,1, I9,7,1, I9,8,1, I9,9,1, I9,0,0, Z0, I9,1,0, I9,2,0, I9,3,0, I9,4,0, I9,5,0, I9,6,0, I9,7,0, I9,8,0, I9,9,0, I8,0,9, X8, I8,1,9, I8,2,9, I8,3,9, I8,4,9, I8,5,9, I8,6,9, I8,7,9, I8,8,9, I8,9,9, 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I1,4,4, I1,5,4, I1,6,4, I1,7,4, I1,8,4, I1,9,4, I1,0,3, I1,1,3, I1,2,3, I1,3,3, I1,4,3, I1,5,3, I1,6,3, I1,7,3, I1,8,3, I1,9,3, I1,0,2, I1,1,2, I1,2,2, I1,3,2, I1,4,2, I1,5,2, I1,6,2, I1,7,2, I1,8,2, I1,9,2, I1,0,1, I1,1,1, I1,2,1, I1,3,1, I1,4,1, I1,5,1, I1,6,1, I1,7,1, I1,8,1, I1,9,1, I1,0,0, I1,1,0, I1,2,0, I1,3,0, I1,4,0, I1,5,0, I1,6,0, I1,7,0, I1,8,0, I1,9,0, I0,0,9, X0, I0,1,9, I0,2,9, I0,3,9, I0,4,9, I0,5,9, I0,6,9, I0,7,9, I0,8,9, I0,9,9, I0,0,8, I0,1,8, I0,2,8, I0,3,8, I0,4,8, I0,5,8, I0,6,8, I0,7,8, I0,8,8, I0,9,8, I0,0,7, I0,1,7, I0,2,7, I0,3,7, I0,4,7, I0,5,7, I0,6,7, I0,7,7, I0,8,7, I0,9,7, I0,0,6, I0,1,6, I0,2,6, I0,3,6, I0,4,6, I0,5,6, I0,6,6, I0,7,6, I0,8,6, I0,9,6, I0,0,5, I0,1,5, I0,2,5, I0,3,5, I0,4,5, I0,5,5, I0,6,5, I0,7,5, I0,8,5, I0,9,5, I0,0,4, I0,1,4, I0,2,4, I0,3,4, I0,4,4, I0,5,4, I0,6,4, I0,7,4, I0,8,4, I0,9,4, I0,0,3, I0,1,3, I0,2,3, I0,3,3, I0,4,3, I0,5,3, I0,6,3, I0,7,3, I0,8,3, I0,9,3, I0,0,2, I0,1,2, I0,2,2, I0,3,2, I0,4,2, I0,5,2, I0,6,2, I0,7,2, I0,8,2, I0,9,2, I0,0,1, I0,1,1, I0,2,1, I0,3,1, I0,4,1, I0,5,1, I0,6,1, I0,7,1, I0,8,1, I0,9,1, I0,0,0, I0,1,0, I0,2,0, I0,3,0, I0,4,0, I0,5,0, I0,6,0, I0,7,0, I0,8,0, I0,9,0}),backend=torch, dtype=torch.float32, data=tensor(1., device='cuda:0')

Note becuase its a large contraction, we perform in inplace and accumulate the norms of all intermediate tensors into tn.exponent.

mantissa, exponent = (tn.contract(), tn.exponent)
mantissa, exponent

(tensor(1., device='cuda:0'), tensor(379.5602, device='cuda:0'))

So our final partition function is \(Z=1\times10^{379.5602}\).

If we just change the entries of the tensor network (and not the actual geometry) then we can reuse the same path or optimizer. For example here we just change the inverse temperature of the model, \(\beta\):

tn = qtn.TN3D_classical_ising_partition_function(
    10, 10, 10, beta=0.25
)
tn.apply_to_arrays(to_backend)

tn.contract_compressed_(
    optimize=copt,  # here we demonstrate the cache of the optimizer
    max_bond=chi,
    equalize_norms=1.0,
    progbar=True,
)

log2[SIZE]: 0.00/21.00: 100%|███████████████████████████████████████| 999/999 [00:10<00:00, 91.62it/s]
TensorNetwork3D(tensors=1, indices=0, Lx=10, Ly=10, Lz=10, max_bond=1)
Tensor(shape=(), inds=[], tags={I9,0,9, X9, Y0, Z9, I9,1,9, Y1, I9,2,9, Y2, I9,3,9, Y3, I9,4,9, Y4, I9,5,9, Y5, I9,6,9, Y6, I9,7,9, Y7, I9,8,9, Y8, I9,9,9, Y9, I9,0,8, Z8, I9,1,8, I9,2,8, I9,3,8, I9,4,8, I9,5,8, I9,6,8, I9,7,8, I9,8,8, I9,9,8, I9,0,7, Z7, I9,1,7, I9,2,7, I9,3,7, I9,4,7, I9,5,7, I9,6,7, I9,7,7, I9,8,7, I9,9,7, I9,0,6, Z6, I9,1,6, I9,2,6, I9,3,6, I9,4,6, I9,5,6, I9,6,6, I9,7,6, I9,8,6, I9,9,6, I9,0,5, Z5, I9,1,5, I9,2,5, I9,3,5, I9,4,5, I9,5,5, I9,6,5, I9,7,5, I9,8,5, I9,9,5, I9,0,4, Z4, I9,1,4, I9,2,4, I9,3,4, I9,4,4, I9,5,4, I9,6,4, I9,7,4, I9,8,4, I9,9,4, I9,0,3, Z3, I9,1,3, I9,2,3, I9,3,3, I9,4,3, I9,5,3, I9,6,3, I9,7,3, I9,8,3, I9,9,3, I9,0,2, Z2, I9,1,2, I9,2,2, I9,3,2, I9,4,2, I9,5,2, I9,6,2, I9,7,2, I9,8,2, I9,9,2, I9,0,1, Z1, I9,1,1, I9,2,1, I9,3,1, I9,4,1, I9,5,1, I9,6,1, I9,7,1, I9,8,1, I9,9,1, I9,0,0, Z0, I9,1,0, I9,2,0, I9,3,0, I9,4,0, I9,5,0, I9,6,0, I9,7,0, I9,8,0, I9,9,0, I8,0,9, X8, I8,1,9, I8,2,9, I8,3,9, I8,4,9, I8,5,9, I8,6,9, I8,7,9, I8,8,9, I8,9,9, 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I5,2,1, I5,3,1, I5,4,1, I5,5,1, I5,6,1, I5,7,1, I5,8,1, I5,9,1, I5,0,0, I5,1,0, I5,2,0, I5,3,0, I5,4,0, I5,5,0, I5,6,0, I5,7,0, I5,8,0, I5,9,0, I4,0,9, X4, I4,1,9, I4,2,9, I4,3,9, I4,4,9, I4,5,9, I4,6,9, I4,7,9, I4,8,9, I4,9,9, I4,0,8, I4,1,8, I4,2,8, I4,3,8, I4,4,8, I4,5,8, I4,6,8, I4,7,8, I4,8,8, I4,9,8, I4,0,7, I4,1,7, I4,2,7, I4,3,7, I4,4,7, I4,5,7, I4,6,7, I4,7,7, I4,8,7, I4,9,7, I4,0,6, I4,1,6, I4,2,6, I4,3,6, I4,4,6, I4,5,6, I4,6,6, I4,7,6, I4,8,6, I4,9,6, I4,0,5, I4,1,5, I4,2,5, I4,3,5, I4,4,5, I4,5,5, I4,6,5, I4,7,5, I4,8,5, I4,9,5, I4,0,4, I4,1,4, I4,2,4, I4,3,4, I4,4,4, I4,5,4, I4,6,4, I4,7,4, I4,8,4, I4,9,4, I4,0,3, I4,1,3, I4,2,3, I4,3,3, I4,4,3, I4,5,3, I4,6,3, I4,7,3, I4,8,3, I4,9,3, I4,0,2, I4,1,2, I4,2,2, I4,3,2, I4,4,2, I4,5,2, I4,6,2, I4,7,2, I4,8,2, I4,9,2, I4,0,1, I4,1,1, I4,2,1, I4,3,1, I4,4,1, I4,5,1, I4,6,1, I4,7,1, I4,8,1, I4,9,1, I4,0,0, I4,1,0, I4,2,0, I4,3,0, I4,4,0, I4,5,0, I4,6,0, I4,7,0, I4,8,0, I4,9,0, I3,0,9, X3, I3,1,9, I3,2,9, I3,3,9, I3,4,9, I3,5,9, I3,6,9, I3,7,9, I3,8,9, I3,9,9, I3,0,8, I3,1,8, I3,2,8, I3,3,8, I3,4,8, I3,5,8, I3,6,8, I3,7,8, I3,8,8, I3,9,8, I3,0,7, I3,1,7, I3,2,7, I3,3,7, I3,4,7, I3,5,7, I3,6,7, I3,7,7, I3,8,7, I3,9,7, I3,0,6, I3,1,6, I3,2,6, I3,3,6, I3,4,6, I3,5,6, I3,6,6, I3,7,6, I3,8,6, I3,9,6, I3,0,5, I3,1,5, I3,2,5, I3,3,5, I3,4,5, I3,5,5, I3,6,5, I3,7,5, I3,8,5, I3,9,5, I3,0,4, I3,1,4, I3,2,4, I3,3,4, I3,4,4, I3,5,4, I3,6,4, I3,7,4, I3,8,4, I3,9,4, I3,0,3, I3,1,3, I3,2,3, I3,3,3, I3,4,3, I3,5,3, I3,6,3, I3,7,3, I3,8,3, I3,9,3, I3,0,2, I3,1,2, I3,2,2, I3,3,2, I3,4,2, I3,5,2, I3,6,2, I3,7,2, I3,8,2, I3,9,2, I3,0,1, I3,1,1, I3,2,1, I3,3,1, I3,4,1, I3,5,1, I3,6,1, I3,7,1, I3,8,1, I3,9,1, I3,0,0, I3,1,0, I3,2,0, I3,3,0, I3,4,0, I3,5,0, I3,6,0, I3,7,0, I3,8,0, I3,9,0, I2,0,9, X2, I2,1,9, I2,2,9, I2,3,9, I2,4,9, I2,5,9, I2,6,9, I2,7,9, I2,8,9, I2,9,9, I2,0,8, I2,1,8, I2,2,8, I2,3,8, I2,4,8, I2,5,8, I2,6,8, I2,7,8, I2,8,8, I2,9,8, I2,0,7, I2,1,7, I2,2,7, I2,3,7, I2,4,7, I2,5,7, I2,6,7, I2,7,7, I2,8,7, I2,9,7, I2,0,6, I2,1,6, I2,2,6, I2,3,6, I2,4,6, I2,5,6, I2,6,6, I2,7,6, I2,8,6, I2,9,6, I2,0,5, I2,1,5, I2,2,5, I2,3,5, I2,4,5, I2,5,5, I2,6,5, I2,7,5, I2,8,5, I2,9,5, I2,0,4, I2,1,4, I2,2,4, I2,3,4, I2,4,4, I2,5,4, I2,6,4, I2,7,4, I2,8,4, I2,9,4, I2,0,3, I2,1,3, I2,2,3, I2,3,3, I2,4,3, I2,5,3, I2,6,3, I2,7,3, I2,8,3, I2,9,3, I2,0,2, I2,1,2, I2,2,2, I2,3,2, I2,4,2, I2,5,2, I2,6,2, I2,7,2, I2,8,2, I2,9,2, I2,0,1, I2,1,1, I2,2,1, I2,3,1, I2,4,1, I2,5,1, I2,6,1, I2,7,1, I2,8,1, I2,9,1, I2,0,0, I2,1,0, I2,2,0, I2,3,0, I2,4,0, I2,5,0, I2,6,0, I2,7,0, I2,8,0, I2,9,0, I1,0,9, X1, I1,1,9, I1,2,9, I1,3,9, I1,4,9, I1,5,9, I1,6,9, I1,7,9, I1,8,9, I1,9,9, I1,0,8, I1,1,8, I1,2,8, I1,3,8, I1,4,8, I1,5,8, I1,6,8, I1,7,8, I1,8,8, I1,9,8, I1,0,7, I1,1,7, I1,2,7, I1,3,7, I1,4,7, I1,5,7, I1,6,7, I1,7,7, I1,8,7, I1,9,7, I1,0,6, I1,1,6, I1,2,6, I1,3,6, I1,4,6, I1,5,6, I1,6,6, I1,7,6, I1,8,6, I1,9,6, I1,0,5, I1,1,5, I1,2,5, I1,3,5, I1,4,5, I1,5,5, I1,6,5, I1,7,5, I1,8,5, I1,9,5, I1,0,4, I1,1,4, I1,2,4, I1,3,4, I1,4,4, I1,5,4, I1,6,4, I1,7,4, I1,8,4, I1,9,4, I1,0,3, I1,1,3, I1,2,3, I1,3,3, I1,4,3, I1,5,3, I1,6,3, I1,7,3, I1,8,3, I1,9,3, I1,0,2, I1,1,2, I1,2,2, I1,3,2, I1,4,2, I1,5,2, I1,6,2, I1,7,2, I1,8,2, I1,9,2, I1,0,1, I1,1,1, I1,2,1, I1,3,1, I1,4,1, I1,5,1, I1,6,1, I1,7,1, I1,8,1, I1,9,1, I1,0,0, I1,1,0, I1,2,0, I1,3,0, I1,4,0, I1,5,0, I1,6,0, I1,7,0, I1,8,0, I1,9,0, I0,0,9, X0, I0,1,9, I0,2,9, I0,3,9, I0,4,9, I0,5,9, I0,6,9, I0,7,9, I0,8,9, I0,9,9, I0,0,8, I0,1,8, I0,2,8, I0,3,8, I0,4,8, I0,5,8, I0,6,8, I0,7,8, I0,8,8, I0,9,8, I0,0,7, I0,1,7, I0,2,7, I0,3,7, I0,4,7, I0,5,7, I0,6,7, I0,7,7, I0,8,7, I0,9,7, I0,0,6, I0,1,6, I0,2,6, I0,3,6, I0,4,6, I0,5,6, I0,6,6, I0,7,6, I0,8,6, I0,9,6, I0,0,5, I0,1,5, I0,2,5, I0,3,5, I0,4,5, I0,5,5, I0,6,5, I0,7,5, I0,8,5, I0,9,5, I0,0,4, I0,1,4, I0,2,4, I0,3,4, I0,4,4, I0,5,4, I0,6,4, I0,7,4, I0,8,4, I0,9,4, I0,0,3, I0,1,3, I0,2,3, I0,3,3, I0,4,3, I0,5,3, I0,6,3, I0,7,3, I0,8,3, I0,9,3, I0,0,2, I0,1,2, I0,2,2, I0,3,2, I0,4,2, I0,5,2, I0,6,2, I0,7,2, I0,8,2, I0,9,2, I0,0,1, I0,1,1, I0,2,1, I0,3,1, I0,4,1, I0,5,1, I0,6,1, I0,7,1, I0,8,1, I0,9,1, I0,0,0, I0,1,0, I0,2,0, I0,3,0, I0,4,0, I0,5,0, I0,6,0, I0,7,0, I0,8,0, I0,9,0}),backend=torch, dtype=torch.float32, data=tensor(1., device='cuda:0')
mantissa, exponent = (tn.contract(), tn.exponent)
mantissa, exponent

(tensor(1., device='cuda:0'), tensor(344.7334, device='cuda:0'))