14. Using quimb within jax, flax and optax

quimb is designed (using autoray) to handle many different array backends, including jax. If you put jax arrays in your tensors, then quimb will dispatch all operations to jax functions, and moreover tensor network algorithms can then be traced through in order to compute gradients, and/or jit-compiled.

While quimb has its own optimizer interface (TNOptimizer) which uses jax or other libraries within it to compute the gradients, it is also possible to instead use quimb within other optimization frameworks. Here we demonstrate this with:

• flax - a library for designing machine learning ‘models’, which is also compatible with netket for example, and

• optax - an optimization library for jax which can itself be jit-compiled.

The resulting computation is then entirely jit-compiled, with quimb merely orchestrating the initial computational graph.

Here’ll we do a simple 1D MERA optimization on the Heisenberg model:

import quimb.tensor as qtn
from quimb.experimental.merabuilder import MERA

# our ansatz and hamiltonian
L = 16
psi = MERA.rand(L, D=8, seed=42, cyclic=False)
ham = qtn.ham_1d_heis(L)

psi.draw(
    color=['UNI', 'ISO'],
    fix={psi.site_ind(i): (i, 0) for i in range(L)},
)

As with TNOptimizer, we need a loss_fn which takes a tensor network and returns a scalar quantity to minimize. Often we also need a norm_fn, which first maps the tensor network into a constrained space (for example, with all unitary tensors):

def norm_fn(psi):
    # parametrize our tensors as isometric/unitary
    return psi.isometrize(method="cayley")

def loss_fn(psi):
    # compute the total energy, here quimb handles constructing
    # and contracting all the appropriate lightcones
    return psi.compute_local_expectation(ham)

# our initial energy:
loss_fn(norm_fn(psi))

-0.015578916803187806

Then we are ready to construct our ‘model’ using flax (something very similar can also be done for e.g. haiku models):

import jax
import flax.linen as nn
import optax

class CustomModule(nn.Module):

    def setup(self):
        # strip out the initial raw arrays
        params, skeleton = qtn.pack(psi)
        # save the stripped 'skeleton' tn for use later
        self.skeleton = skeleton

        # assign each array as a parameter to optimize
        self.params = {
            i: self.param(f'param_{i}', lambda _: data)
            for i, data in params.items()
        }

    def __call__(self):
        psi = qtn.unpack(self.params, self.skeleton)
        return loss_fn(norm_fn(psi))

Next we define a single, jit-compiled, optimization step:

# initialize our model and loss/gradient function
model = CustomModule()
params = model.init(jax.random.PRNGKey(42))
loss_grad_fn = jax.value_and_grad(model.apply)

# initialize our optimizer
tx = optax.adabelief(learning_rate=0.01)
opt_state = tx.init(params)

@jax.jit
def step(params, opt_state):
    # our step: compute the loss and gradient, and update the optimizer
    loss, grads = loss_grad_fn(params)
    updates, opt_state = tx.update(grads, opt_state, params)
    params = optax.apply_updates(params, updates)
    return params, opt_state, loss

And now we are ready to optimize!

import tqdm

its = 1_000
pbar = tqdm.tqdm(range(its))

for _ in pbar:
    params, opt_state, loss_val = step(params, opt_state)
    pbar.set_description(f"{loss_val}")

-6.904621124267578: 100%|██████████| 1000/1000 [00:33<00:00, 29.88it/s] 

Finally if we want to insert the optimized raw parameters back into a tensor network then we can do so with:

mera_opt = psi.copy()

# resinsert the raw, optimized arrays
for i, t in mera_opt.tensor_map.items():
    t.modify(data=params['params'][f'param_{i}'].__array__())

# then we want the constrained form
mera_opt = norm_fn(mera_opt)

Then we can check the energy outside of jax:

loss_fn(mera_opt)

-6.90447020410412

and that the state is still unitary and thus normalized:

mera_opt.H @ mera_opt

1.0000000242457867

14.1. Notes

• quimb also registers all Tensor and TensorNetwork classes with jax’s pytree system, so they can be used as directly input and output to functions such as jax.grad and jax.jit.

• jax by default converts everything to single precision (thus the deviation from 1.0 above), see here about disabling this behavior