param_shift_cv(tape, dev, argnum=None, shifts=None, gradient_recipes=None, fallback_fn=<function finite_diff>, f0=None, force_order2=False)[source]

Transform a continuous-variable QNode to compute the parameter-shift gradient of all gate parameters with respect to its inputs.

Parameters
• tape (QuantumTape) – quantum tape to differentiate

• dev (Device) – device the parameter-shift method is to be computed on

• argnum (int or list[int] or None) – Trainable parameter indices to differentiate with respect to. If not provided, the derivative with respect to all trainable indices are returned.

• shifts (list[tuple[int or float]]) – List containing tuples of shift values. If provided, one tuple of shifts should be given per trainable parameter and the tuple should match the number of frequencies for that parameter. If unspecified, equidistant shifts are assumed.

• gradient_recipes (tuple(list[list[float]] or None)) –

List of gradient recipes for the parameter-shift method. One gradient recipe must be provided per trainable parameter.

This is a tuple with one nested list per parameter. For parameter $$\phi_k$$, the nested list contains elements of the form $$[c_i, a_i, s_i]$$ where $$i$$ is the index of the term, resulting in a gradient recipe of

$\frac{\partial}{\partial\phi_k}f = \sum_{i} c_i f(a_i \phi_k + s_i).$

If None, the default gradient recipe containing the two terms $$[c_0, a_0, s_0]=[1/2, 1, \pi/2]$$ and $$[c_1, a_1, s_1]=[-1/2, 1, -\pi/2]$$ is assumed for every parameter.

• fallback_fn (None or Callable) – a fallback grdient function to use for any parameters that do not support the parameter-shift rule.

• f0 (tensor_like[float] or None) – Output of the evaluated input tape. If provided, and the gradient recipe contains an unshifted term, this value is used, saving a quantum evaluation.

• force_order2 (bool) – if True, use the order-2 method even if not necessary

Returns

• If the input is a QNode, a tensor representing the output Jacobian matrix of size (number_outputs, number_gate_parameters) is returned.

• If the input is a tape, a tuple containing a list of generated tapes, in addition to a post-processing function to be applied to the evaluated tapes.

Return type

tensor_like or tuple[list[QuantumTape], function]

This transform supports analytic gradients of Gaussian CV operations using the parameter-shift rule. This gradient method returns exact gradients, and can be computed directly on quantum hardware.

Analytic gradients of photonic circuits that satisfy the following constraints with regards to measurements are supported:

Warning

Fock state probabilities (tapes that return probs() or expectation values of FockStateProjector) are not supported.

In addition, the operations must fulfill the following requirements:

• Only Gaussian operations are differentiable.

• Non-differentiable Fock states and Fock operations may precede all differentiable Gaussian, operations. For example, the following is permissible:

@qml.qnode(dev)
def circuit(weights):
# Non-differentiable Fock operations

# differentiable Gaussian operations
qml.Displacement(weights[0], weights[1], wires=0)
qml.Beamsplitter(weights[2], weights[3], wires=[0, 1])

return qml.expval(qml.NumberOperator(0))

• If a Fock operation succeeds a Gaussian operation, the Fock operation must not contribute to any measurements. For example, the following is allowed:

@qml.qnode(dev)
def circuit(weights):
qml.Displacement(weights[0], weights[1], wires=0)
qml.Beamsplitter(weights[2], weights[3], wires=[0, 1])
qml.Kerr(np.array(0.654, requires_grad=False), wires=1)  # there is no measurement on wire 1
return qml.expval(qml.NumberOperator(0))


If any of the above constraints are not followed, the tape cannot be differentiated via the CV parameter-shift rule. Please use numerical differentiation instead.

Example

This transform can be registered directly as the quantum gradient transform to use during autodifferentiation:

>>> dev = qml.device("default.gaussian", wires=2)
>>> @qml.qnode(dev, diff_method="parameter-shift")
... def circuit(params):
...     qml.Squeezing(params[0], params[1], wires=[0])
...     qml.Squeezing(params[2], params[3], wires=[0])
...     return qml.expval(qml.NumberOperator(0))
>>> params = np.array([0.1, 0.2, 0.3, 0.4], requires_grad=True)
>>> qml.jacobian(circuit)(params)
array([ 0.87516064,  0.01273285,  0.88334834, -0.01273285])


This gradient transform can be applied directly to QNode objects:

>>> @qml.qnode(dev)
... def circuit(params):
...     qml.Squeezing(params[0], params[1], wires=[0])
...     qml.Squeezing(params[2], params[3], wires=[0])
...     return qml.expval(qml.NumberOperator(0))
>>> params = np.array([0.1, 0.2, 0.3, 0.4], requires_grad=True)
tensor([[ 0.87516064,  0.01273285,  0.88334834, -0.01273285]], requires_grad=True)


This quantum gradient transform can also be applied to low-level QuantumTape objects. This will result in no implicit quantum device evaluation. Instead, the processed tapes, and post-processing function, which together define the gradient are directly returned:

>>> r0, phi0, r1, phi1 = [0.4, -0.3, -0.7, 0.2]
>>> with qml.tape.QuantumTape() as tape:
...     qml.Squeezing(r0, phi0, wires=[0])
...     qml.Squeezing(r1, phi1, wires=[0])
...     qml.expval(qml.NumberOperator(0))  # second-order
[<QuantumTape: wires=[0], params=4>,
<QuantumTape: wires=[0], params=4>,
<QuantumTape: wires=[0], params=4>,
<QuantumTape: wires=[0], params=4>]


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.gaussian", wires=2)