Quantum gradient transforms are strategies for computing the gradient of a quantum circuit that work by transforming the quantum circuit into one or more gradient circuits. These gradient circuits, once executed and post-processed, return the gradient of the original circuit.

Examples of quantum gradient transforms include finite-differences and parameter-shift rules.

This module provides a selection of device-independent, differentiable quantum gradient transforms. As such, these quantum gradient transforms can be used to compute the gradients of quantum circuits on both simulators and hardware.

These quantum gradient transforms can be used in two ways:

• Transforming quantum circuits directly

• Registering a quantum gradient strategy for use when performing autodifferentiation with a QNode.

## Overview¶

 finite_diff(tape[, argnum, h, approx_order, …]) Transform a QNode to compute the finite-difference gradient of all gate parameters with respect to its inputs. param_shift(tape[, argnum, shift, …]) Transform a QNode to compute the parameter-shift gradient of all gate parameters with respect to its inputs. param_shift_cv(tape, dev[, argnum, shift, …]) Transform a continuous-variable QNode to compute the parameter-shift gradient of all gate parameters with respect to its inputs.

 gradient_transform(*args, **kwargs) Decorator for defining quantum gradient transforms.

### Utility functions¶

 finite_diff_coeffs(n, approx_order, strategy) Generate the finite difference shift values and corresponding term coefficients for a given derivative order, approximation accuracy, and strategy. generate_shifted_tapes(tape, idx, shifts[, …]) Generate a list of tapes where the corresponding trainable parameter index has been shifted by the values given. compute_vjp(dy, jac) Convenience function to compute the vector-Jacobian product for a given vector of gradient outputs and a Jacobian. batch_vjp(tapes, dys, gradient_fn[, …]) Generate the gradient tapes and processing function required to compute the vector-Jacobian products of a batch of tapes. vjp(tape, dy, gradient_fn[, gradient_kwargs]) Generate the gradient tapes and processing function required to compute the vector-Jacobian products of a tape.

All PennyLane QNodes are automatically differentiable, and can be included seamlessly within an autodiff pipeline. When creating a QNode, the strategy for determining the optimal differentiation strategy is automated, and takes into account the circuit, device, autodiff framework, and metadata (such as whether a finite number of shots are used).

dev = qml.device("default.qubit", wires=2, shots=1000)

@qml.qnode(dev, interface="tf")
def circuit(weights):
...


In particular:

• When using a simulator device with exact measurement statistics, backpropagation is preferred due to performance and memory improvements.

• When using a hardware device, or a simulator with a finite number of shots, a quantum gradient transform—such as the parameter-shift rule—is preferred.

If you would like to specify a particular quantum gradient transform to use when differentiating your quantum circuit, this can be passed when creating the QNode:

@qml.qnode(dev, gradient_fn=qml.gradients.param_shift)
def circuit(weights):
...


When using your preferred autodiff framework to compute the gradient of your hybrid quantum-classical cost function, the specified gradient transform for each QNode will be used.

Note

A single cost function may include multiple QNodes, each with their own quantum gradient transform registered.

## Transforming QNodes¶

Alternatively, quantum gradient transforms can be applied manually to QNodes.

dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev)
def circuit(weights):
qml.RX(weights[0], wires=0)
qml.RY(weights[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RX(weights[2], wires=1)
return qml.probs(wires=1)

>>> weights = np.array([0.1, 0.2, 0.3], requires_grad=True)
>>> circuit(weights)
tensor([[-0.04673668, -0.09442394, -0.14409127],


Comparing this to autodifferentiation:

>>> qml.grad(circuit)(weights)
array([[-0.04673668, -0.09442394, -0.14409127],
[ 0.04673668,  0.09442394,  0.14409127]])


Quantum gradient transforms can also be applied as decorators to QNodes, if only gradient information is needed. Evaluating the QNode will then automatically return the gradient:

dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev)
def decorated_circuit(weights):
qml.RX(weights[0], wires=0)
qml.RY(weights[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RX(weights[2], wires=1)
return qml.probs(wires=1)

>>> decorated_circuit(weights)
tensor([[-0.04673668, -0.09442394, -0.14409127],


Note

If your circuit contains any operations not supported by the gradient transform, the transform will attempt to automatically decompose the circuit into only operations that support gradients.

Note

If you wish to only return the purely quantum component of the gradient—that is, the gradient of the output with respect to gate arguments, not QNode arguments—pass hybrid=False when applying the transform:

>>> qml.gradients.param_shift(circuit, hybrid=False)(weights)


dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev)
def circuit(weights):
qml.RX(weights[0], wires=0)
qml.RY(weights[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RX(weights[2], wires=1)
return qml.expval(qml.PauliZ(1))

>>> weights = np.array([0.1, 0.2, 0.3], requires_grad=True)
>>> circuit(weights)
array([[-0.09347337, -0.18884787, -0.28818254]])
array([[[-0.9316158 ,  0.01894799,  0.0289147 ],
[ 0.01894799, -0.9316158 ,  0.05841749],
[ 0.0289147 ,  0.05841749, -0.9316158 ]]])


## Transforming tapes¶

Gradient transforms can be applied to low-level QuantumTape objects, a datastructure representing variational quantum algorithms:

weights = np.array([0.1, 0.2, 0.3], requires_grad=True)

with qml.tape.JacobianTape() as tape:
qml.RX(weights[0], wires=0)
qml.RY(weights[1], wires=1)
qml.CNOT(wires=[0, 1])
qml.RX(weights[2], wires=1)
qml.expval(qml.PauliZ(1))


Unlike when transforming a QNode, transforming a tape directly will perform no implicit quantum device evaluation. Instead, it returns the processed tapes, and a post-processing function, which together define the gradient:

>>> gradient_tapes, fn = qml.gradients.param_shift(tape)
[<JacobianTape: wires=[0, 1], params=3>,
<JacobianTape: wires=[0, 1], params=3>,
<JacobianTape: wires=[0, 1], params=3>,
<JacobianTape: wires=[0, 1], params=3>,
<JacobianTape: wires=[0, 1], params=3>,
<JacobianTape: wires=[0, 1], params=3>]


This can be useful if the underlying circuits representing the gradient computation need to be analyzed.

The output tapes can then be evaluated and post-processed to retrieve the gradient:

>>> dev = qml.device("default.qubit", wires=2)
[[-0.09347337 -0.18884787 -0.28818254]]


Note that the post-processing function fn returned by the gradient transform is applied to the flat list of results returned from executing the gradient tapes.

Using the gradient_transform decorator, custom gradient transforms can be created:

@gradient_transform
...


Once created, a custom gradient transform can be applied directly to QNodes, or registered as the quantum gradient transform to use during autodifferentiation.

For more details, please see the gradient_transform documentation.