qml.gradients.param_shift_cv

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

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

Parameters

• tape (QNode or QuantumTape) – quantum circuit to differentiate

• dev (pennylane.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 gradient 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

The transformed circuit as described in qml.transform. Executing this circuit will provide the Jacobian in the form of a tensor, a tuple, or a nested tuple depending upon the nesting structure of measurements in the original circuit.

Return type

qnode (QNode) 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:

• Expectation values are restricted to observables that are first- and second-order in \(\hat{x}\) and \(\hat{p}\) only. This includes X, P, QuadOperator, PolyXP, and NumberOperator.

  For second-order observables, the device must support PolyXP.

• Variances are restricted to observables that are first-order in \(\hat{x}\) and \(\hat{p}\) only. This includes X, P, QuadOperator, and some parameter values of PolyXP.

  The device must support PolyXP.

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
      qml.FockState(np.array(2, requires_grad=False), wires=0)
      qml.Kerr(np.array(0.654, requires_grad=False), wires=1)
  
      # 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])

Usage Details

This gradient transform can be applied directly to QNode objects. However, for performance reasons, we recommend providing the gradient transform as the diff_method argument of the QNode decorator, and differentiating with your preferred machine learning framework.

>>> @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)
>>> qml.gradients.param_shift_cv(circuit, dev)(params)
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]
>>> ops = [qml.Squeezing(r0, phi0, wires=0), qml.Squeezing(r1, phi1, wires=0)]
>>> tape = qml.tape.QuantumTape(ops, [qml.expval(qml.NumberOperator(0))])
>>> gradient_tapes, fn = qml.gradients.param_shift_cv(tape, dev)
>>> gradient_tapes
[<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)
>>> fn(qml.execute(gradient_tapes, dev, None))
(-0.32487113372219933,
 -0.4054074025310772,
 -0.8704985300843778,
 0.4054074025310775)

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