qml.TrotterProduct¶

class TrotterProduct(hamiltonian, time, n=1, order=1, check_hermitian=True, id=None)[source]¶

Bases: pennylane.resource.error.error.ErrorOperation, pennylane.resource.resource.ResourcesOperation

An operation representing the Suzuki-Trotter product approximation for the complex matrix exponential of a given Hamiltonian.

The Suzuki-Trotter product formula provides a method to approximate the matrix exponential of Hamiltonian expressed as a linear combination of terms which in general do not commute. Consider the Hamiltonian \(H = \Sigma^{N}_{j=0} O_{j}\), the product formula is constructed using symmetrized products of the terms in the Hamiltonian. The symmetrized products of order \(m \in [1, 2, 4, ..., 2k]\) with \(k \in \mathbb{N}\) are given by:

\[\begin{split}\begin{align} S_{1}(t) &= \Pi_{j=0}^{N} \ e^{i t O_{j}} \\ S_{2}(t) &= \Pi_{j=0}^{N} \ e^{i \frac{t}{2} O_{j}} \cdot \Pi_{j=N}^{0} \ e^{i \frac{t}{2} O_{j}} \\ &\vdots \\ S_{m}(t) &= S_{m-2}(p_{m}t)^{2} \cdot S_{m-2}((1-4p_{m})t) \cdot S_{m-2}(p_{m}t)^{2}, \end{align}\end{split}\]

where the coefficient is \(p_{m} = 1 / (4 - \sqrt[m - 1]{4})\). The \(m\)-step Suzuki-Trotter approximation is then defined as:

\[e^{iHt} \approx \left [S_{m}(t / n) \right ]^{n}.\]

For more details see J. Math. Phys. 32, 400 (1991).

Parameters

hamiltonian (Union[Hamiltonian, Sum, SProd]) – The Hamiltonian written as a linear combination of operators with known matrix exponentials.
time (float) – The time of evolution, namely the parameter \(t\) in \(e^{iHt}\)
n (int) – An integer representing the number of Trotter steps to perform
order (int) – An integer (\(m\)) representing the order of the approximation (must be 1 or even)
check_hermitian (bool) – A flag to enable the validation check to ensure this is a valid unitary operator

Raises

TypeError – The hamiltonian is not of type Hamiltonian, or Sum.
ValueError – The hamiltonian must have atleast two terms.
ValueError – One or more of the terms in hamiltonian are not Hermitian.
ValueError – The order is not one or a positive even integer.

Example

coeffs = [0.25, 0.75]
ops = [qml.X(0), qml.Z(0)]
H = qml.dot(coeffs, ops)

dev = qml.device("default.qubit", wires=2)
@qml.qnode(dev)
def my_circ():
    # Prepare some state
    qml.Hadamard(0)

    # Evolve according to H
    qml.TrotterProduct(H, time=2.4, order=2)

    # Measure some quantity
    return qml.state()

>>> my_circ()
array([-0.13259524+0.59790098j,  0.        +0.j        , -0.13259524-0.77932754j,  0.        +0.j        ])

Warning

The Trotter-Suzuki decomposition depends on the order of the summed observables. Two mathematically identical Hamiltonian objects may undergo different time evolutions due to the order in which those observables are stored. The order of observables can be queried using the terms() method.

Warning

TrotterProduct does not automatically simplify the input Hamiltonian, allowing for a more fine-grained control over the decomposition but also risking an increased runtime and number of gates required. Simplification can be performed manually by applying simplify() to your Hamiltonian before using it in TrotterProduct.

Usage Details

An upper-bound for the error in approximating time-evolution using this operator can be computed by calling error(). It is computed using two different methods; the “one-norm-bound” scaling method and the “commutator-bound” scaling method. (see Childs et al. (2021))

>>> hamiltonian = qml.dot([1.0, 0.5, -0.25], [qml.X(0), qml.Y(0), qml.Z(0)])
>>> op = qml.TrotterProduct(hamiltonian, time=0.01, order=2)
>>> op.error(method="one-norm-bound")
SpectralNormError(8.039062500000003e-06)
>>> op.error(method="commutator-bound")
SpectralNormError(6.166666666666668e-06)

This operation is similar to the ApproxTimeEvolution. One can recover the behaviour of ApproxTimeEvolution by taking the adjoint:

>>> qml.adjoint(qml.TrotterProduct(hamiltonian, time, order=1, n=n))

We can also compute the gradient with respect to the coefficients of the Hamiltonian and the evolution time:

@qml.qnode(dev)
def my_circ(c1, c2, time):
    # Prepare H:
    H = qml.dot([c1, c2], [qml.X(0), qml.Z(0)])

    # Prepare some state
    qml.Hadamard(0)

    # Evolve according to H
    qml.TrotterProduct(H, time, order=2)

    # Measure some quantity
    return qml.expval(qml.Z(0) @ qml.Z(1))

>>> args = np.array([1.23, 4.5, 0.1])
>>> qml.grad(my_circ)(*tuple(args))
(tensor(0.00961064, requires_grad=True), tensor(-0.12338274, requires_grad=True), tensor(-5.43401259, requires_grad=True))

Attributes

`arithmetic_depth`	Arithmetic depth of the operator.
`basis`	The basis of an operation, or for controlled gates, of the target operation.
`batch_size`	Batch size of the operator if it is used with broadcasted parameters.
`control_wires`	Control wires of the operator.
`grad_method`	Gradient computation method.
`grad_recipe`	Gradient recipe for the parameter-shift method.
`has_adjoint`
`has_decomposition`
`has_diagonalizing_gates`
`has_generator`
`has_matrix`
`has_sparse_matrix`
`hash`	Integer hash that uniquely represents the operator.
`hyperparameters`	Dictionary of non-trainable variables that this operation depends on.
`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`	Number of dimensions per trainable parameter of the operator.
`num_params`	Number of trainable parameters that the operator depends on.
`num_wires`	Number of wires the operator acts on.
`parameter_frequencies`	Returns the frequencies for each operator parameter with respect to an expectation value of the form \(\langle \psi \| U(\mathbf{p})^\dagger \hat{O} U(\mathbf{p})\|\psi\rangle\).
`parameters`	Trainable parameters that the operator depends on.
`pauli_rep`	A `PauliSentence` representation of the Operator, or `None` if it doesn't have one.
`wires`	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 and CNOT have basis = "X", whereas ControlledPhaseShift and RZ have basis = "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 on ndim_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 is None.

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 length 0.

Returns: The control wires of the operation.
Return type: Wires

grad_method¶

Gradient computation method.

'A': analytic differentiation using the parameter-shift method.
'F': finite difference numerical differentiation.
None: the operation may not be differentiated.

Default is 'F', or None if the Operation has zero parameters.

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 = False¶

has_decomposition = True¶

has_diagonalizing_gates = False¶

has_generator = False¶

has_matrix = False¶

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¶

Number of dimensions per trainable parameter of the operator.

By default, this property returns the numbers of dimensions of the parameters used for the operator creation. If the parameter sizes for an operator subclass are fixed, this property can be overwritten to return the fixed value.

Returns: Number of dimensions for each trainable parameter.
Return type: tuple

num_params¶

Number of trainable parameters that the operator depends on.

By default, this property returns as many parameters as were used for the operator creation. If the number of parameters for an operator subclass is fixed, this property can be overwritten to return the fixed value.

Returns: number of parameters
Return type: int

num_wires = -1¶: Number of wires the operator acts on.

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, or None if it doesn’t have one.

wires¶

Wires that the operator acts on.

Returns: wires
Return type: Wires

Methods

`adjoint`()	Create an operation that is the adjoint of this one.
`compute_decomposition`(args, *kwargs)	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`(params, *hyperparams)	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).
`decomposition`()	Representation of the operator as a product of other operators.
`diagonalizing_gates`()	Sequence of gates that diagonalize the operator in the computational basis.
`eigvals`()	Eigenvalues of the operator in the computational basis.
`error`([method, fast])	Compute an upper-bound on the spectral norm error for approximating the time-evolution of the base Hamiltonian using the Suzuki-Trotter product formula.
`generator`()	Generator of an operator that is in single-parameter-form.
`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.
`resources`()	The resource requirements for a given instance of the Suzuki-Trotter product.
`simplify`()	Reduce the depth of nested operators to the minimum.
`single_qubit_rot_angles`()	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()¶

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(*args, **kwargs)[source]¶

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

decomposition().

Parameters

*params (list) – trainable parameters of the operator, as stored in the parameters attribute
wires (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

diagonalizing_gates().

Parameters

params (list) – trainable parameters of the operator, as stored in the parameters attribute
wires (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.

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(*params, **hyperparams)¶

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

Operator.matrix() and qml.matrix()

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

matrix representation

Return type

tensor_like

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

sparse_matrix()

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()¶

Representation of the operator as a product of other operators.

\[O = O_1 O_2 \dots O_n\]

A DecompositionUndefinedError is raised if no representation by decomposition is defined.

See also

compute_decomposition().

Returns: decomposition of the operator
Return type: list[Operator]

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

compute_diagonalizing_gates().

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

compute_eigvals()

Returns: eigenvalues
Return type: tensor_like

error(method='commutator-bound', fast=True)[source]¶

Compute an upper-bound on the spectral norm error for approximating the time-evolution of the base Hamiltonian using the Suzuki-Trotter product formula.

The error in the Suzuki-Trotter product formula is defined as

\[|| \ e^{iHt} - \left [S_{m}(t / n) \right ]^{n} \ ||,\]

Where the norm \(||\cdot||\) is the spectral norm. This function supports two methods from literature for upper-bounding the error, the “one-norm” error bound and the “commutator” error bound.

Example:

The “one-norm” error bound can be computed by passing the kwarg method="one-norm-bound", and is defined according to Section 2.3 (lemma 6, equation 22 and 23) of Childs et al. (2021).

>>> hamiltonian = qml.dot([1.0, 0.5, -0.25], [qml.X(0), qml.Y(0), qml.Z(0)])
>>> op = qml.TrotterProduct(hamiltonian, time=0.01, order=2)
>>> op.error(method="one-norm-bound")
SpectralNormError(8.039062500000003e-06)

The “commutator” error bound can be computed by passing the kwarg method="commutator-bound", and is defined according to Appendix C (equation 189) Childs et al. (2021).

>>> hamiltonian = qml.dot([1.0, 0.5, -0.25], [qml.X(0), qml.Y(0), qml.Z(0)])
>>> op = qml.TrotterProduct(hamiltonian, time=0.01, order=2)
>>> op.error(method="commutator-bound")
SpectralNormError(6.166666666666668e-06)

Parameters

method (str, optional) – Options include “one-norm-bound” and “commutator-bound” and specify the method with which the error is computed. Defaults to “commutator-bound”.
fast (bool, optional) – Uses more approximations to speed up computation. Defaults to True.

Raises

ValueError – The method is not supported.

Returns

The spectral norm error.

Return type

SpectralNormError

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 and SparseHamiltonian respectively).

The default value to return is None, indicating that the operation has no defined generator.

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)[source]¶

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: Operator

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

compute_matrix()

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'>)[source]¶: Append the operator to the Operator queue.

resources()[source]¶

The resource requirements for a given instance of the Suzuki-Trotter product.

Returns: The resources for an instance of TrotterProduct.
Return type: Resources

simplify()¶

Reduce the depth of nested operators to the minimum.

Returns: simplified operator
Return type: Operator

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

compute_sparse_matrix()

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]]

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