# qml.Displacement¶

class Displacement(a, phi, wires)[source]

Phase space displacement.

$D(a,\phi) = D(\alpha) = \exp(\alpha \ad -\alpha^* \a) = \exp\left(-i\sqrt{\frac{2}{\hbar}}(\re(\alpha) \hat{p} -\im(\alpha) \hat{x})\right).$

where $$\alpha = ae^{i\phi}$$ has magnitude $$a\geq 0$$ and phase $$\phi$$. The result of applying a displacement to the vacuum is a coherent state $$D(\alpha)\ket{0} = \ket{\alpha}$$.

Details:

• Number of wires: 1

• Number of parameters: 2

• Gradient recipe: $$\frac{d}{da}f(D(a,\phi)) = \frac{1}{2s} \left[f(D(a+s, \phi)) - f(D(a-s, \phi))\right]$$, where $$s$$ is an arbitrary real number ($$0.1$$ by default) and $$f$$ is an expectation value depending on $$D(a,\phi)$$.

• Heisenberg representation:

$\begin{split}M = \begin{bmatrix} 1 & 0 & 0 \\ 2a\cos\phi & 1 & 0 \\ 2a\sin\phi & 0 & 1\end{bmatrix}\end{split}$
Parameters
• a (float) – displacement magnitude $$a=|\alpha|$$

• phi (float) – phase angle $$\phi$$

• wires (Sequence[int] or int) – the wire the operation acts on

 a base_name Get base name of the operator. eigvals Eigenvalues of an instantiated operator. generator Generator of the operation. grad_method grad_recipe Gradient recipe for the parameter-shift method. id String for the ID of the operator. inverse Boolean determining if the inverse of the operation was requested. matrix Matrix representation of an instantiated operator in the computational basis. multiplier name Get and set the name of the operator. num_params num_wires par_domain parameters Current parameter values. shift string_for_inverse supports_heisenberg supports_parameter_shift wires Wires of this operator.
a = 1
base_name

Get base name of the operator.

eigvals
generator

Generator of the operation.

A length-2 list [generator, scaling_factor], where

• generator is an existing PennyLane operation class or $$2\times 2$$ Hermitian array that acts as the generator of the current operation

• scaling_factor represents a scaling factor applied to the generator operation

For example, if $$U(\theta)=e^{i0.7\theta \sigma_x}$$, then $$\sigma_x$$, with scaling factor $$s$$, is the generator of operator $$U(\theta)$$:

generator = [PauliX, 0.7]


Default is [None, 1], indicating the operation has no generator.

grad_method = 'A'
grad_recipe = ([[5.0, 1, 0.1], [-5.0, 1, -0.1]], None)

Gradient recipe for the parameter-shift method.

This is a tuple with one nested list per operation 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.

Type

tuple(Union(list[list[float]], None)) or None

id

String for the ID of the operator.

inverse

Boolean determining if the inverse of the operation was requested.

matrix
multiplier = 5.0
name

Get and set the name of the operator.

num_params = 2
num_wires = 1
par_domain = 'R'
parameters

Current parameter values.

shift = 0.1
string_for_inverse = '.inv'
supports_heisenberg = True
supports_parameter_shift = True
wires

Wires of this operator.

Returns

wires

Return type

Wires

 adjoint([do_queue]) Create an operation that is the adjoint of this one. decomposition(*params, wires) Returns a template decomposing the operation into other quantum operations. Returns a tape containing the decomposed operations, rather than a list. get_parameter_shift(idx[, shift]) Multiplier and shift for the given parameter, based on its gradient recipe. heisenberg_expand(U, wires) Expand the given local Heisenberg-picture array into a full-system one. Partial derivative of the Heisenberg picture transform matrix. heisenberg_tr(wires[, inverse]) Heisenberg picture representation of the linear transformation carried out by the gate at current parameter values. Inverts the operation, such that the inverse will be used for the computations by the specific device. Append the operator to the Operator queue.
adjoint(do_queue=False)[source]

Create an operation that is the adjoint of this one.

Adjointed operations are the conjugated and transposed version of the original operation. Adjointed ops are equivalent to the inverted operation for unitary gates.

Parameters

do_queue – Whether to add the adjointed gate to the context queue.

Returns

The adjointed operation.

static decomposition(*params, wires)

Returns a template decomposing the operation into other quantum operations.

expand()

Returns a tape containing the decomposed operations, rather than a list.

Returns

Returns a quantum tape that contains the operations decomposition, or if not implemented, simply the operation itself.

Return type

JacobianTape

get_parameter_shift(idx, shift=1.5707963267948966)

Multiplier and shift for the given parameter, based on its gradient recipe.

Parameters

idx (int) – parameter index

Returns

list of multiplier, coefficient, shift for each term in the gradient recipe

Return type

list[[float, float, float]]

heisenberg_expand(U, wires)

Expand the given local Heisenberg-picture array into a full-system one.

Parameters
• U (array[float]) – array to expand (expected to be of the dimension 1+2*self.num_wires)

• wires (Wires) – wires on the device the array U should be expanded to apply to

Raises

ValueError – if the size of the input matrix is invalid or num_wires is incorrect

Returns

expanded array, dimension 1+2*num_wires

Return type

array[float]

heisenberg_pd(idx)

Partial derivative of the Heisenberg picture transform matrix.

Computed using grad_recipe.

Parameters

idx (int) – index of the parameter with respect to which the partial derivative is computed.

Returns

partial derivative

Return type

array[float]

heisenberg_tr(wires, inverse=False)

Heisenberg picture representation of the linear transformation carried out by the gate at current parameter values.

Given a unitary quantum gate $$U$$, we may consider its linear transformation in the Heisenberg picture, $$U^\dagger(\cdot) U$$.

If the gate is Gaussian, this linear transformation preserves the polynomial order of any observables that are polynomials in $$\mathbf{r} = (\I, \x_0, \p_0, \x_1, \p_1, \ldots)$$. This also means it maps $$\text{span}(\mathbf{r})$$ into itself:

$U^\dagger \mathbf{r}_i U = \sum_j \tilde{U}_{ij} \mathbf{r}_j$

For Gaussian CV gates, this method returns the transformation matrix for the current parameter values of the Operation. The method is not defined for non-Gaussian (and non-CV) gates.

Parameters
• wires (Wires) – wires on the device that the observable gets applied to

• inverse (bool) – if True, return the inverse transformation instead

Raises

RuntimeError – if the specified operation is not Gaussian or is missing the _heisenberg_rep method

Returns

$$\tilde{U}$$, the Heisenberg picture representation of the linear transformation

Return type

array[float]

inv()

Inverts the operation, such that the inverse will be used for the computations by the specific device.

This method concatenates a string to the name of the operation, to indicate that the inverse will be used for computations.

Any subsequent call of this method will toggle between the original operation and the inverse of the operation.

Returns

operation to be inverted

Return type

Operator

queue()

Append the operator to the Operator queue.