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How to optimize a QML model using JAX and JAXopt

Josh Izaac

Josh Izaac

Maria Schuld

Maria Schuld

Published: January 17, 2024. Last updated: October 06, 2024.

Once you have set up a quantum machine learning model, data to train with, and cost function to minimize as an objective, the next step is to perform the optimization. That is, setting up a classical optimization loop to find a minimal value of your cost function.

In this example, we’ll show you how to use JAX, an autodifferentiable machine learning framework, and JAXopt, a suite of JAX-compatible gradient-based optimizers, to optimize a PennyLane quantum machine learning model.

demos/_static/demonstration_assets/How_to_optimize_QML_model_using_JAX_and_JAXopt/socialthumbnail_large_How_to_optimize_QML_model_using_JAX_and_JAXopt_2024-01-16.png

Set up your model, data, and cost

Here, we will create a simple QML model for our optimization. In particular:

  • We will embed our data through a series of rotation gates.

  • We will then have an ansatz of trainable rotation gates with parameters weights; it is these values we will train to minimize our cost function.

  • We will train the QML model on data, a (5, 4) array, and optimize the model to match target predictions given by target.

import pennylane as qml
import jax
from jax import numpy as jnp
import jaxopt

jax.config.update("jax_platform_name", "cpu")

n_wires = 5
data = jnp.sin(jnp.mgrid[-2:2:0.2].reshape(n_wires, -1)) ** 3
targets = jnp.array([-0.2, 0.4, 0.35, 0.2])

dev = qml.device("default.qubit", wires=n_wires)

@qml.qnode(dev)
def circuit(data, weights):
    """Quantum circuit ansatz"""

    # data embedding
    for i in range(n_wires):
        # data[i] will be of shape (4,); we are
        # taking advantage of operation vectorization here
        qml.RY(data[i], wires=i)

    # trainable ansatz
    for i in range(n_wires):
        qml.RX(weights[i, 0], wires=i)
        qml.RY(weights[i, 1], wires=i)
        qml.RX(weights[i, 2], wires=i)
        qml.CNOT(wires=[i, (i + 1) % n_wires])

    # we use a sum of local Z's as an observable since a
    # local Z would only be affected by params on that qubit.
    return qml.expval(qml.sum(*[qml.PauliZ(i) for i in range(n_wires)]))

def my_model(data, weights, bias):
    return circuit(data, weights) + bias

We will define a simple cost function that computes the overlap between model output and target data, and just-in-time (JIT) compile it:

@jax.jit
def loss_fn(params, data, targets):
    predictions = my_model(data, params["weights"], params["bias"])
    loss = jnp.sum((targets - predictions) ** 2 / len(data))
    return loss

Note that the model above is just an example for demonstration – there are important considerations that must be taken into account when performing QML research, including methods for data embedding, circuit architecture, and cost function, in order to build models that may have use. This is still an active area of research; see our demonstrations for details.

Initialize your parameters

Now, we can generate our trainable parameters weights and bias that will be used to train our QML model.

weights = jnp.ones([n_wires, 3])
bias = jnp.array(0.)
params = {"weights": weights, "bias": bias}

Plugging the trainable parameters, data, and target labels into our cost function, we can see the current loss as well as the parameter gradients:

print(loss_fn(params, data, targets))

print(jax.grad(loss_fn)(params, data, targets))

0.2923263
{'bias': Array(-0.7543211, dtype=float32, weak_type=True), 'weights': Array([[-1.9507737e-01,  5.2854680e-02, -4.8925218e-01],
       [-1.9968897e-02, -5.3287193e-02,  9.2290491e-02],
       [-2.7175546e-03, -9.6470118e-05, -4.7957897e-03],
       [-6.3544415e-02,  3.6111102e-02, -2.0519719e-01],
       [-9.0263695e-02,  1.6375934e-01, -5.6426287e-01]], dtype=float32)}

Create the optimizer

We can now use JAXopt to create a gradient descent optimizer, and train our circuit.

To do so, we first create a function that returns the loss value and the gradient value during training; this allows us to track and print out the loss during training within JAXopt’s internal optimization loop.

def loss_and_grad(params, data, targets, print_training, i):
    loss_val, grad_val = jax.value_and_grad(loss_fn)(params, data, targets)

    def print_fn():
        jax.debug.print("Step: {i}  Loss: {loss_val}", i=i, loss_val=loss_val)

    # if print_training=True, print the loss every 5 steps
    jax.lax.cond((jnp.mod(i, 5) == 0) & print_training, print_fn, lambda: None)

    return loss_val, grad_val

Note that we use a couple of JAX specific functions here:

  • jax.lax.cond instead of a Python if statement

  • jax.debug.print instead of a Python print function

These JAX compatible functions are needed because JAXopt will automatically JIT compile the optimizer update step.

opt = jaxopt.GradientDescent(loss_and_grad, stepsize=0.3, value_and_grad=True)
opt_state = opt.init_state(params)

for i in range(100):
    params, opt_state = opt.update(params, opt_state, data, targets, True, i)

Step: 0  Loss: 0.29232630133628845
Step: 5  Loss: 0.08928854763507843
Step: 10  Loss: 0.0715944766998291
Step: 15  Loss: 0.057335998862981796
Step: 20  Loss: 0.047165680676698685
Step: 25  Loss: 0.039545394480228424
Step: 30  Loss: 0.03321404010057449
Step: 35  Loss: 0.02776365913450718
Step: 40  Loss: 0.023554328829050064
Step: 45  Loss: 0.02116139605641365
Step: 50  Loss: 0.020479006692767143
Step: 55  Loss: 0.020495891571044922
Step: 60  Loss: 0.02018814906477928
Step: 65  Loss: 0.019282132387161255
Step: 70  Loss: 0.018022999167442322
Step: 75  Loss: 0.016644006595015526
Step: 80  Loss: 0.01525440439581871
Step: 85  Loss: 0.013919375836849213
Step: 90  Loss: 0.012653525918722153
Step: 95  Loss: 0.011443558149039745

Jitting the optimization loop

In the above example, we JIT compiled our cost function loss_fn (and JAXopt automatically JIT compiled the loss_and_grad function behind the scenes). However, we can also JIT compile the entire optimization loop; this means that the for-loop around optimization is not happening in Python, but is compiled and executed natively. This avoids (potentially costly) data transfer between Python and our JIT compiled cost function with each update step.

@jax.jit
def optimization_jit(params, data, targets, print_training=False):
    opt = jaxopt.GradientDescent(loss_and_grad, stepsize=0.3, value_and_grad=True)
    opt_state = opt.init_state(params)

    def update(i, args):
        params, opt_state = opt.update(*args, i)
        return (params, opt_state, *args[2:])

    args = (params, opt_state, data, targets, print_training)
    (params, opt_state, _, _, _) = jax.lax.fori_loop(0, 100, update, args)

    return params

Note that – similar to above – we use jax.lax.fori_loop and jax.lax.cond, rather than a standard Python for loop and if statement, to allow the control flow to be JIT compatible.

params = {"weights": weights, "bias": bias}
optimization_jit(params, data, targets, print_training=True)

Step: 0  Loss: 0.29232630133628845
Step: 5  Loss: 0.08928854763507843
Step: 10  Loss: 0.0715944766998291
Step: 15  Loss: 0.057335998862981796
Step: 20  Loss: 0.047165680676698685
Step: 25  Loss: 0.039545394480228424
Step: 30  Loss: 0.03321404010057449
Step: 35  Loss: 0.02776365913450718
Step: 40  Loss: 0.023554328829050064
Step: 45  Loss: 0.02116139605641365
Step: 50  Loss: 0.020479006692767143
Step: 55  Loss: 0.020495891571044922
Step: 60  Loss: 0.02018814906477928
Step: 65  Loss: 0.019282132387161255
Step: 70  Loss: 0.018022999167442322
Step: 75  Loss: 0.016644006595015526
Step: 80  Loss: 0.01525440439581871
Step: 85  Loss: 0.013919375836849213
Step: 90  Loss: 0.012653525918722153
Step: 95  Loss: 0.011443558149039745

{'bias': Array(-0.9073953, dtype=float32, weak_type=True), 'weights': Array([[ 1.5734292 ,  1.426786  ,  0.50623536],
       [ 0.1700654 ,  0.83469135,  1.9625024 ],
       [ 1.4379818 ,  1.1278569 ,  2.234396  ],
       [-0.25908187,  0.53192574,  1.3994696 ],
       [ 1.2112932 ,  1.6135522 ,  3.1225498 ]], dtype=float32)}

Appendix: Timing the two approaches

We can time the two approaches (JIT compiling just the cost function, vs JIT compiling the entire optimization loop) to explore the differences in performance:

from timeit import repeat

def optimization(params, data, targets):
    opt = jaxopt.GradientDescent(loss_and_grad, stepsize=0.3, value_and_grad=True)
    opt_state = opt.init_state(params)

    for i in range(100):
        params, opt_state = opt.update(params, opt_state, data, targets, False, i)

    return params

reps = 5
num = 2

times = repeat("optimization(params, data, targets)", globals=globals(), number=num, repeat=reps)
result = min(times) / num

print(f"Jitting just the cost (best of {reps}): {result} sec per loop")

times = repeat("optimization_jit(params, data, targets)", globals=globals(), number=num, repeat=reps)
result = min(times) / num

print(f"Jitting the entire optimization (best of {reps}): {result} sec per loop")

Jitting just the cost (best of 5): 0.6097046845000023 sec per loop
Jitting the entire optimization (best of 5): 0.005847669999994309 sec per loop

In this example, JIT compiling the entire optimization loop is significantly more performant.

About the authors

Josh Izaac

Josh Izaac

Josh is a theoretical physicist, software tinkerer, and occasional baker. At Xanadu, he contributes to the development and growth of Xanadu’s open-source quantum software products.

Maria Schuld

Maria Schuld

Dedicated to making quantum machine learning a reality one day.

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