qml.SpecialUnitary¶
- class SpecialUnitary(theta, wires, id=None)[source]¶
Bases:
pennylane.operation.Operation
Gate from the group \(SU(N)\) with \(N=2^n\) for \(n\) qubits.
We use the following parametrization of a special unitary operator:
\[\begin{split}U(\bm{\theta}) &= \exp\{A(\bm{\theta})\}\\ A(\bm{\theta}) &= \sum_{m=1}^d i \theta_m P_m\\ P_m &\in \{I, X, Y, Z\}^{\otimes n} \setminus \{I^{\otimes n}\}\end{split}\]This means, \(U(\bm{\theta})\) is the exponential of the operator \(A(\bm{\theta})\), which in turn is a linear combination of Pauli words with coefficients \(i\bm{\theta}\) and satisfies \(A(\bm{\theta})^\dagger=-A(\bm{\theta})\) (it is skew-Hermitian). Note that this gate takes an exponential number \(d=4^n-1\) of parameters. See below for more theoretical background and details regarding differentiability.
Details:
Number of wires: Any
Number of parameters: 1
Number of dimensions per parameter: (1,)
Gradient recipe:
\[\begin{split}\frac{\partial}{\partial\theta_\ell} f(U(\bm{\theta})) &= \sum_{m=1}^d 2i\omega_{\ell m} \frac{\mathrm{d}}{\mathrm{d} t} f\left(\exp\left\{-i\frac{t}{2}G_m\right\} U(\bm{\theta})\right)\\ &= \sum_{m=1}^d i\omega_{\ell m} \left[ f\left(\exp\left\{-i\frac{\pi}{4}G_m\right\} U(\bm{\theta})\right) -f\left(\exp\left\{i\frac{\pi}{4}G_m\right\} U(\bm{\theta})\right) \right]\end{split}\]where \(f\) is an expectation value depending on \(U(\bm{\theta})\) and the derivative of the Pauli rotation gates \(\exp\left\{-i\frac{t}{2}G_m\right\}\) follows from the two-term parameter-shift rule (also see:
PauliRot
). For details on the gradient recipe, also consider the theoretical background section below.- Parameters
theta (tensor_like) – Pauli coordinates of the exponent \(A(\bm{\theta})\). See details below for the order of the Pauli words.
wires (Sequence[int] or int) – The wire(s) the operation acts on
id (str or None) – String representing the operation (optional)
- Raises
ValueError – If the shape of the input does not match the Lie algebra dimension \(d=4^n-1\) for \(n\) wires.
Note
This operation should only be used to parametrize special unitaries that can not easily be represented by other operations, as it incurs computational cost that scale exponentially with the wires it acts on.
The parameter
theta
refers to all Pauli words (except for the identity) in lexicographical order, which looks like the following for one and two qubits:>>> qml.ops.qubit.special_unitary.pauli_basis_strings(1) # 4**1-1 = 3 Pauli words ['X', 'Y', 'Z'] >>> qml.ops.qubit.special_unitary.pauli_basis_strings(2) # 4**2-1 = 15 Pauli words ['IX', 'IY', 'IZ', 'XI', 'XX', 'XY', 'XZ', 'YI', 'YX', 'YY', 'YZ', 'ZI', 'ZX', 'ZY', 'ZZ']
See also
For more details on using this operator in applications, see the SU(N) gate demo.
Warning
This operation only is differentiable when using the JAX, Torch or TensorFlow interfaces, even when using hardware-compatible differentiation techniques like the parameter-shift rule.
Warning
This operation supports broadcasting and hardware-compatible differentiation techniques like the parameter-shift rule. However, the two features can not be used simultaneously.
Examples
Simple examples of this operation are single-qubit Pauli rotation gates, which can be created by setting all but one parameter \(\theta_m\) to zero:
>>> x = 0.412 >>> theta = x * np.array([1, 0, 0]) # The first entry belongs to the Pauli word "X" >>> su = qml.SpecialUnitary(theta, wires=0) >>> prot = qml.PauliRot(-2 * x, "X", wires=0) # PauliRot introduces a prefactor -0.5 >>> rx = qml.RX(-2 * x, 0) # RX introduces a prefactor -0.5 >>> qml.math.allclose(su.matrix(), prot.matrix()) True >>> qml.math.allclose(su.matrix(), rx.matrix()) True
Note that for specific operations like the
RX
rotation gate above, it is strongly recommended to use the specialized implementationqml.RX
rather thanPauliRot
orSpecialUnitary
. However,SpecialUnitary
gates go beyond such rotations: Multiple Pauli words can be activated simultaneously, giving access to more complex operations. For two qubits, this could look like this:>>> wires = [0, 1] # Activating the Pauli words ["IY", "IZ", "XX", "XY", "YY", "YZ", "ZY", "ZZ"] >>> theta = 0.3 * np.array([0, 1, 2, 0, -1, 1, 0, 0, 0, 1, 1, 1, 0, 0, -1]) >>> len(theta) == 4 ** len(wires) - 1 # theta contains one parameter per Pauli word True >>> su = qml.SpecialUnitary(theta, wires=wires) >>> su.matrix() array([[ 0.56397118+0.52139241j, 0.30652227+0.02438052j, 0.13555302+0.22630716j, 0.0689876 -0.49110826j], [-0.15454843+0.00998377j, 0.88294943+0.01496327j, -0.25396275-0.10785888j, -0.26041566+0.22857073j], [-0.2876174 -0.2443733j , 0.25423439+0.05896445j, 0.71621665+0.50686226j, 0.1380692 +0.02252197j], [-0.34495668-0.35307844j, 0.10817019-0.21404059j, -0.29040522+0.00830631j, 0.15015337-0.76933485j]])
The
SpecialUnitary
operation also can be differentiated with hardware-compatible differentiation techniques if the JAX, Torch or TensorFlow interface is used. See the theoretical background section below for details.Theoretical background
We consider special unitaries parametrized in the following way:
\[\begin{split}U(\bm{\theta}) &= \exp\{A(\bm{\theta})\}\\ A(\bm{\theta}) &= \sum_{m=1}^d i \theta_m G_m\\ G_m &\in \mathcal{P^{(n)}}=\{I, X, Y, Z\}^{\otimes n} \setminus \{I^{\otimes n}\}\end{split}\]where \(n\) is the number of qubits and \(\theta_m\) are \(d=4^n-1\) real-valued parameters. This parametrization allows us to express any special unitary for the given number of qubits.
Note that this differs from a sequence of all possible Pauli rotation gates because for non-commuting Pauli words \(G_1, G_2\) we have \(\exp\{i\theta_1G_1\}\exp\{i\theta_2G_2\}\neq \exp\{i(\theta_1G_1+\theta_2G_2)\}\).
Differentiation
The partial derivatives of \(U(\bm{\theta})\) above can be expressed as
\[\begin{split}\frac{\partial}{\partial \theta_\ell} U(\bm{\theta}) &= U(\bm{\theta}) \frac{\mathrm{d}}{\mathrm{d}t}\exp\left(t\Omega_\ell(\bm{\theta})\right)\large|_{t=0}\\ &=U(\bm{\theta})\Omega_\ell(\bm{\theta})\end{split}\]where \(\Omega_\ell(\bm{\theta})\) is the effective generator belonging to the partial derivative \(\partial_\ell U(\bm{\theta})\) at the parameters \(\bm{\theta}\). It can be computed via
\[\Omega_\ell(\bm{\theta}) = U(\bm{\theta})^\dagger \left(\frac{\partial}{\partial \theta_\ell}\mathfrak{Re}[U(\bm{\theta})] +i\frac{\partial}{\partial \theta_\ell}\mathfrak{Im}[U(\bm{\theta})]\right)\]where we may compute the derivatives of the real and imaginary parts of \(U(\bm{\theta})\) using auto-differentiation.
Each effective generator \(\Omega_\ell(\bm{\theta})\) that reproduces a partial derivative can be decomposed in the Pauli basis of \(\mathfrak{su}(N)\) via
\[\Omega_\ell(\bm{\theta}) = \sum_{m=1}^d \omega_{\ell m}(\bm{\theta}) G_m.\]As the Pauli words are orthonormal with respect to the trace, or Frobenius, inner product (rescaled by \(2^{-n}\)), we can compute the coefficients using this inner product (\(G_m\) is Hermitian, so we skip the adjoint \({}^\dagger\)):
\[\omega_{\ell m}(\bm{\theta}) = \frac{1}{2^n}\operatorname{tr} \left\{G_m \Omega_\ell(\bm{\theta}) \right\}\]The coefficients satisfy \(\omega_{\ell m}^\ast=-\omega_{\ell m}\) because \(\Omega_\ell(\bm{\theta})\) is skew-Hermitian. Therefore they are purely imaginary.
Now we turn to the derivative of an expectation value-based function which uses a circuit with a
SpecialUnitary
operation. Absorbing the remaining circuit into the quantum state \(\rho\) and the observable \(H\), this can be written as\[\begin{split}f(U(\bm{\theta})) &= \operatorname{Tr}\left\{H U(\bm{\theta})\rho U^\dagger(\bm{\theta})\right\}\\ \partial_\ell f(U(\bm{\theta})) &= \operatorname{Tr}\left\{H U(\bm{\theta}) [\Omega_\ell(\bm{\theta}),\rho] U^\dagger(\bm{\theta})\right\}\end{split}\]Inserting the decomposition for the effective generator from above, we may rewrite this as a combination of derivatives of Pauli rotation gates:
\[\begin{split}\partial_\ell f(U(\bm{\theta})) &= \operatorname{Tr}\left\{H U(\bm{\theta}) \left[\sum_{m=1}^d \omega_{\ell m}(\bm{\theta}) G_m,\rho\right] U^\dagger(\bm{\theta})\right\}\\ &= \sum_{m=1}^d 2i\omega_{\ell m}(\bm{\theta}) \frac{\mathrm{d}}{\mathrm{d}t}f\left(R_{G_m}(t)U(\bm{\theta})\right)\bigg|_{t=0}.\end{split}\]Here we abbreviated a Pauli rotation gate as \(R_{G_m}(t) = \exp\{-i\frac{t}{2} G_m\}\). As all partial derivatives are combinations of these Pauli rotation derivatives, we may write the gradient of \(f\) as
\[\nabla f(U(\bm{\theta})) = \overline{\omega}(\bm{\theta}) \cdot \bm{\mathrm{d}f}\]where we defined the matrix \(\overline{\omega_{\ell m}}=2i\omega_{\ell m}\) and the vector of Pauli rotation derivatives \(\mathrm{d}f_m=\frac{\mathrm{d}}{\mathrm{d}t} f\left(R_{G_m}(t)U(\bm{\theta})\right)\bigg|_{t=0}\). These derivatives can be computed with the standard parameter-shift rule because Pauli words satisfy the condition \(G_m^2=1\) (see e.g. Mitarai et al. (2018)). Provided that we can implement the
SpecialUnitary
gate itself, this allows us to compute \(\bm{\mathrm{d}f}\) in a hardware-compatible manner using \(2d\) quantum circuits.Note that for
SpecialUnitary
we frequently handle objects that have one or multiple dimensions of exponentially large size \(d=4^n-1\), and that the number of quantum circuits for the differentiation scales accordingly. This strongly affects the number of qubits to which we can apply aSpecialUnitary
gate in practice.Attributes
Arithmetic depth of the operator.
The basis of an operation, or for controlled gates, of the target operation.
Batch size of the operator if it is used with broadcasted parameters.
Control wires of the operator.
Gradient computation method.
Gradient recipe for the parameter-shift method.
Integer hash that uniquely represents the operator.
Dictionary of non-trainable variables that this operation depends on.
Custom string to label a specific operator instance.
This property determines if an operator is hermitian.
String for the name of the operator.
Number of dimensions per trainable parameter that the operator depends on.
Number of trainable parameters that the operator depends on.
Number of wires that the operator acts on.
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle\).
Trainable parameters that the operator depends on.
A
PauliSentence
representation of the Operator, orNone
if it doesn't have one.Wires that the operator acts on.
- arithmetic_depth¶
Arithmetic depth of the operator.
- basis¶
The basis of an operation, or for controlled gates, of the target operation. If not
None
, should take a value of"X"
,"Y"
, or"Z"
.For example,
X
andCNOT
havebasis = "X"
, whereasControlledPhaseShift
andRZ
havebasis = "Z"
.- Type
str or None
- batch_size¶
Batch size of the operator if it is used with broadcasted parameters.
The
batch_size
is determined based onndim_params
and the provided parameters for the operator. If (some of) the latter have an additional dimension, and this dimension has the same size for all parameters, its size is the batch size of the operator. If no parameter has an additional dimension, the batch size isNone
.- Returns
Size of the parameter broadcasting dimension if present, else
None
.- Return type
int or None
- control_wires¶
Control wires of the operator.
For operations that are not controlled, this is an empty
Wires
object of length0
.- Returns
The control wires of the operation.
- Return type
- grad_method = None¶
Gradient computation method.
- grad_recipe = 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
- has_adjoint = True¶
- has_decomposition = True¶
- has_diagonalizing_gates = False¶
- has_generator = False¶
- has_matrix = True¶
- has_sparse_matrix = False¶
- hash¶
Integer hash that uniquely represents the operator.
- Type
int
- hyperparameters¶
Dictionary of non-trainable variables that this operation depends on.
- Type
dict
- id¶
Custom string to label a specific operator instance.
- is_hermitian¶
This property determines if an operator is hermitian.
- name¶
String for the name of the operator.
- ndim_params = (1,)¶
Number of dimensions per trainable parameter that the operator depends on.
- Type
tuple[int]
- num_params = 1¶
Number of trainable parameters that the operator depends on.
- Type
int
- num_wires = -1¶
Number of wires that the operator acts on.
- Type
int
- parameter_frequencies¶
Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi | U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})|\psi\rangle\).
These frequencies encode the behaviour of the operator \(U(\mathbf{p})\) on the value of the expectation value as the parameters are modified. For more details, please see the
pennylane.fourier
module.- Returns
Tuple of frequencies for each parameter. Note that only non-negative frequency values are returned.
- Return type
list[tuple[int or float]]
Example
>>> op = qml.CRot(0.4, 0.1, 0.3, wires=[0, 1]) >>> op.parameter_frequencies [(0.5, 1), (0.5, 1), (0.5, 1)]
For operators that define a generator, the parameter frequencies are directly related to the eigenvalues of the generator:
>>> op = qml.ControlledPhaseShift(0.1, wires=[0, 1]) >>> op.parameter_frequencies [(1,)] >>> gen = qml.generator(op, format="observable") >>> gen_eigvals = qml.eigvals(gen) >>> qml.gradients.eigvals_to_frequencies(tuple(gen_eigvals)) (1.0,)
For more details on this relationship, see
eigvals_to_frequencies()
.
- parameters¶
Trainable parameters that the operator depends on.
- pauli_rep¶
A
PauliSentence
representation of the Operator, orNone
if it doesn’t have one.
Methods
adjoint
()Create an operation that is the adjoint of this one.
compute_decomposition
(*params[, wires])Representation of the operator as a product of other operators (static method).
compute_diagonalizing_gates
(*params, wires, ...)Sequence of gates that diagonalize the operator in the computational basis (static method).
compute_eigvals
(*params, **hyperparams)Eigenvalues of the operator in the computational basis (static method).
compute_matrix
(theta, num_wires)Representation of the operator as a canonical matrix in the computational basis (static method).
compute_sparse_matrix
(*params, **hyperparams)Representation of the operator as a sparse matrix in the computational basis (static method).
Representation of the operator as a product of other operators.
Sequence of gates that diagonalize the operator in the computational basis.
eigvals
()Eigenvalues of the operator in the computational basis.
Generator of an operator that is in single-parameter-form.
get_one_parameter_coeffs
(interface)Compute the Pauli basis coefficients of the generators of one-parameter groups that reproduce the partial derivatives of a special unitary gate.
get_one_parameter_generators
([interface])Compute the generators of one-parameter groups that reproduce the partial derivatives of a special unitary gate.
label
([decimals, base_label, cache])A customizable string representation of the operator.
map_wires
(wire_map)Returns a copy of the current operator with its wires changed according to the given wire map.
matrix
([wire_order])Representation of the operator as a matrix in the computational basis.
pow
(z)A list of new operators equal to this one raised to the given power.
queue
([context])Append the operator to the Operator queue.
simplify
()Reduce the depth of nested operators to the minimum.
The parameters required to implement a single-qubit gate as an equivalent
Rot
gate, up to a global phase.sparse_matrix
([wire_order])Representation of the operator as a sparse matrix in the computational basis.
terms
()Representation of the operator as a linear combination of other operators.
- adjoint()[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.
- Returns
The adjointed operation.
- static compute_decomposition(*params, wires=None, **hyperparameters)¶
Representation of the operator as a product of other operators (static method).
\[O = O_1 O_2 \dots O_n.\]Note
Operations making up the decomposition should be queued within the
compute_decomposition
method.See also
- Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
**hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- Returns
decomposition of the operator
- Return type
list[Operator]
- static compute_diagonalizing_gates(*params, wires, **hyperparams)¶
Sequence of gates that diagonalize the operator in the computational basis (static method).
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
See also
- Parameters
params (list) – trainable parameters of the operator, as stored in the
parameters
attributewires (Iterable[Any], Wires) – wires that the operator acts on
hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- Returns
list of diagonalizing gates
- Return type
list[Operator]
- static compute_eigvals(*params, **hyperparams)¶
Eigenvalues of the operator in the computational basis (static method).
If
diagonalizing_gates
are specified and implement a unitary \(U^{\dagger}\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
See also
- Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- Returns
eigenvalues
- Return type
tensor_like
- static compute_matrix(theta, num_wires)[source]¶
Representation of the operator as a canonical matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
See also
- Parameters
theta (tensor_like) – Pauli coordinates of the exponent \(A(\theta)\).
num_wires (int) – The number of wires the matrix acts on.
- Returns
- canonical matrix of the special unitary corresponding to
theta
. It has the shape
(2**num_wires, 2**num_wires)
.
- canonical matrix of the special unitary corresponding to
- Return type
tensor_like
Compute the matrix of an element in SU(N), given by the Pauli basis coordinated of the associated Lie algebra element. The \(4^n-1\) Pauli basis elements of the Lie algebra \(\mathfrak{su}(N)\) for \(n\) qubits are \(P_m\in\{I, X, Y, Z\}^{\otimes n}\setminus\{I^{\otimes n}\}\), and the special unitary matrix is computed as
\[U(\theta) = \exp(i\sum_{m=1}^d \theta_m P_m)\]See the main class documentation above for the ordering of Pauli words.
Note
Note that this method internally handles a complex-valued tensor of size
(4**num_wires, 2**num_wires, 2**num_wires)
, which requires at least4 ** (2 * num_wires - 13)
GB of memory (at default precision).Example
>>> theta = np.array([0.5, 0.1, -0.3]) >>> qml.SpecialUnitary.compute_matrix(theta, num_wires=1) array([[ 0.83004499-0.28280371j, 0.0942679 +0.47133952j], [-0.0942679 +0.47133952j, 0.83004499+0.28280371j]])
- static compute_sparse_matrix(*params, **hyperparams)¶
Representation of the operator as a sparse matrix in the computational basis (static method).
The canonical matrix is the textbook matrix representation that does not consider wires. Implicitly, this assumes that the wires of the operator correspond to the global wire order.
See also
- Parameters
*params (list) – trainable parameters of the operator, as stored in the
parameters
attribute**hyperparams (dict) – non-trainable hyperparameters of the operator, as stored in the
hyperparameters
attribute
- Returns
sparse matrix representation
- Return type
scipy.sparse._csr.csr_matrix
- decomposition()[source]¶
Representation of the operator as a product of other operators.
\[O = O_1 O_2 \dots O_n\]This
Operation
is decomposed into the correspondingQubitUnitary
.- Returns
decomposition of the operator
- Return type
list[Operator]
Example:
>>> theta = np.array([0.5, 0.1, -0.3]) >>> qml.SpecialUnitary(theta, wires=[0]).decomposition() [QubitUnitary(array([[ 0.83004499-0.28280371j, 0.0942679 +0.47133952j], [-0.0942679 +0.47133952j, 0.83004499+0.28280371j]]), wires=[0])]
- diagonalizing_gates()¶
Sequence of gates that diagonalize the operator in the computational basis.
Given the eigendecomposition \(O = U \Sigma U^{\dagger}\) where \(\Sigma\) is a diagonal matrix containing the eigenvalues, the sequence of diagonalizing gates implements the unitary \(U^{\dagger}\).
The diagonalizing gates rotate the state into the eigenbasis of the operator.
A
DiagGatesUndefinedError
is raised if no representation by decomposition is defined.See also
- Returns
a list of operators
- Return type
list[Operator] or None
- eigvals()¶
Eigenvalues of the operator in the computational basis.
If
diagonalizing_gates
are specified and implement a unitary \(U^{\dagger}\), the operator can be reconstructed as\[O = U \Sigma U^{\dagger},\]where \(\Sigma\) is the diagonal matrix containing the eigenvalues.
Otherwise, no particular order for the eigenvalues is guaranteed.
Note
When eigenvalues are not explicitly defined, they are computed automatically from the matrix representation. Currently, this computation is not differentiable.
A
EigvalsUndefinedError
is raised if the eigenvalues have not been defined and cannot be inferred from the matrix representation.See also
- Returns
eigenvalues
- Return type
tensor_like
- generator()¶
Generator of an operator that is in single-parameter-form.
For example, for operator
\[U(\phi) = e^{i\phi (0.5 Y + Z\otimes X)}\]we get the generator
>>> U.generator() 0.5 * Y(0) + Z(0) @ X(1)
The generator may also be provided in the form of a dense or sparse Hamiltonian (using
Hamiltonian
andSparseHamiltonian
respectively).The default value to return is
None
, indicating that the operation has no defined generator.
- get_one_parameter_coeffs(interface)[source]¶
Compute the Pauli basis coefficients of the generators of one-parameter groups that reproduce the partial derivatives of a special unitary gate.
- Parameters
interface (str) – The auto-differentiation framework to be used for the computation.
- Returns
The Pauli basis coefficients of the effective generators that reproduce the partial derivatives of the special unitary gate defined by
theta
. There are \(d=4^n-1\) generators for \(n\) qubits and \(d\) Pauli coefficients per generator, so that the output shape is(4**num_wires-1, 4**num_wires-1)
.- Return type
tensor_like
Given a generator \(\Omega\) of a one-parameter group that reproduces a partial derivative of a special unitary gate, it can be decomposed in the Pauli basis of \(\mathfrak{su}(N)\) via
\[\Omega = \sum_{m=1}^d \omega_m P_m\]where \(d=4^n-1\) is the size of the basis for \(n\) qubits and \(P_m\) are the Pauli words making up the basis. As the Pauli words are orthonormal with respect to the trace or Frobenius inner product (rescaled by \(2^n\)), we can compute the coefficients using this inner product (\(P_m\) is Hermitian, so we skip the adjoint \({}^\dagger\)):
\[\omega_m = \frac{1}{2^n}\operatorname{tr}\left[P_m \Omega \right]\]The coefficients satisfy \(\omega_m^\ast=-\omega_m\) because \(\Omega\) is skew-Hermitian. Therefore they are purely imaginary.
Warning
An auto-differentiation framework is required by this function. The matrix exponential is not differentiable in Autograd. Therefore this function only supports JAX, Torch and Tensorflow.
See also
- get_one_parameter_generators(interface=None)[source]¶
Compute the generators of one-parameter groups that reproduce the partial derivatives of a special unitary gate.
- Parameters
interface (str) – The auto-differentiation framework to be used for the computation. Has to be one of
["jax", "tensorflow", "tf", "torch"]
.- Raises
NotImplementedError – If the chosen interface is
"autograd"
. Autograd does not support differentiation oflinalg.expm
.ValueError – If the chosen interface is not supported.
- Returns
The generators for one-parameter groups that reproduce the partial derivatives of the special unitary gate. There are \(d=4^n-1\) generators for \(n\) qubits, so that the output shape is
(4**num_wires-1, 2**num_wires, 2**num_wires)
.- Return type
tensor_like
Consider a special unitary gate parametrized in the following way:
\[\begin{split}U(\theta) &= \exp\{A(\theta)\}\\ A(\theta) &= \sum_{m=1}^d i \theta_m P_m\\ P_m &\in \{I, X, Y, Z\}^{\otimes n} \setminus \{I^{\otimes n}\}\end{split}\]Then the partial derivatives of the gate can be shown to be given by
\[\frac{\partial}{\partial \theta_\ell} U(\theta) = U(\theta) \frac{\mathrm{d}}{\mathrm{d}t}\exp\left(t\Omega_\ell(\theta)\right)\large|_{t=0}\]where \(\Omega_\ell(\theta)\) is the one-parameter generator belonging to the partial derivative \(\partial_\ell U(\theta)\) at the parameters \(\theta\). It can be computed via
\[\Omega_\ell(\theta) = U(\theta)^\dagger \left(\frac{\partial}{\partial \theta_\ell}\mathfrak{Re}[U(\theta)] +i\frac{\partial}{\partial \theta_\ell}\mathfrak{Im}[U(\theta)]\right)\]where we may compute the derivatives \(\frac{\partial}{\partial \theta_\ell} U(\theta)\) using auto-differentiation.
Warning
An auto-differentiation framework is required for this function. The matrix exponential is not differentiable in Autograd. Therefore this function only supports JAX, Torch and TensorFlow.
- label(decimals=None, base_label=None, cache=None)¶
A customizable string representation of the operator.
- Parameters
decimals=None (int) – If
None
, no parameters are included. Else, specifies how to round the parameters.base_label=None (str) – overwrite the non-parameter component of the label
cache=None (dict) – dictionary that carries information between label calls in the same drawing
- Returns
label to use in drawings
- Return type
str
Example:
>>> op = qml.RX(1.23456, wires=0) >>> op.label() "RX" >>> op.label(base_label="my_label") "my_label" >>> op = qml.RX(1.23456, wires=0, id="test_data") >>> op.label() "RX("test_data")" >>> op.label(decimals=2) "RX\n(1.23,"test_data")" >>> op.label(base_label="my_label") "my_label("test_data")" >>> op.label(decimals=2, base_label="my_label") "my_label\n(1.23,"test_data")"
If the operation has a matrix-valued parameter and a cache dictionary is provided, unique matrices will be cached in the
'matrices'
key list. The label will contain the index of the matrix in the'matrices'
list.>>> op2 = qml.QubitUnitary(np.eye(2), wires=0) >>> cache = {'matrices': []} >>> op2.label(cache=cache) 'U(M0)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True)] >>> op3 = qml.QubitUnitary(np.eye(4), wires=(0,1)) >>> op3.label(cache=cache) 'U(M1)' >>> cache['matrices'] [tensor([[1., 0.], [0., 1.]], requires_grad=True), tensor([[1., 0., 0., 0.], [0., 1., 0., 0.], [0., 0., 1., 0.], [0., 0., 0., 1.]], requires_grad=True)]
- map_wires(wire_map)¶
Returns a copy of the current operator with its wires changed according to the given wire map.
- Parameters
wire_map (dict) – dictionary containing the old wires as keys and the new wires as values
- Returns
new operator
- Return type
- matrix(wire_order=None)¶
Representation of the operator as a matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.If the matrix depends on trainable parameters, the result will be cast in the same autodifferentiation framework as the parameters.
A
MatrixUndefinedError
is raised if the matrix representation has not been defined.See also
- Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
- Returns
matrix representation
- Return type
tensor_like
- pow(z)¶
A list of new operators equal to this one raised to the given power.
- Parameters
z (float) – exponent for the operator
- Returns
list[
Operator
]
- queue(context=<class 'pennylane.queuing.QueuingManager'>)¶
Append the operator to the Operator queue.
- simplify()¶
Reduce the depth of nested operators to the minimum.
- Returns
simplified operator
- Return type
- single_qubit_rot_angles()¶
The parameters required to implement a single-qubit gate as an equivalent
Rot
gate, up to a global phase.- Returns
A list of values \([\phi, \theta, \omega]\) such that \(RZ(\omega) RY(\theta) RZ(\phi)\) is equivalent to the original operation.
- Return type
tuple[float, float, float]
- sparse_matrix(wire_order=None)¶
Representation of the operator as a sparse matrix in the computational basis.
If
wire_order
is provided, the numerical representation considers the position of the operator’s wires in the global wire order. Otherwise, the wire order defaults to the operator’s wires.A
SparseMatrixUndefinedError
is raised if the sparse matrix representation has not been defined.See also
- Parameters
wire_order (Iterable) – global wire order, must contain all wire labels from the operator’s wires
- Returns
sparse matrix representation
- Return type
scipy.sparse._csr.csr_matrix
- terms()¶
Representation of the operator as a linear combination of other operators.
\[O = \sum_i c_i O_i\]A
TermsUndefinedError
is raised if no representation by terms is defined.- Returns
list of coefficients \(c_i\) and list of operations \(O_i\)
- Return type
tuple[list[tensor_like or float], list[Operation]]