qml.estimator.templates.TrotterProduct

class TrotterProduct(first_order_expansion, num_steps, order, wires=None)

Bases: ResourceOperator

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

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 \(m^{\text{th}}\) order, \(n\)-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:

• first_order_expansion (list[ResourceOperator]) – A list of operators constituting the first order expansion of the Hamiltonian to be approximately exponentiated.

• num_steps (int) – number of Trotter steps to perform

• order (int) – order of the Suzuki-Trotter approximation; must be 1 or an even number

• wires (list[int] | None) – The wires on which the operator acts. If provided, these wire labels will be used instead of the wires provided by the ResourceOperators in the first_order_expansion.

Resources:

The resources are defined according to the recursive formula presented above. The number of times an operator \(e^{itO_{j}}\) is applied depends on the number of Trotter steps (n) and the order of the approximation (m) and is given by:

\[C_{O_j} = 2 * n \cdot 5^{\frac{m}{2} - 1}\]

Furthermore, because of the symmetric form of the recursive formula, the first and last terms are grouped. This reduces the counts for those terms to:

\[\begin{split}\begin{align} C_{O_{0}} &= n \cdot 5^{\frac{m}{2} - 1} + 1, \\ C_{O_{N}} &= n \cdot 5^{\frac{m}{2} - 1}. \end{align}\end{split}\]

See also

The corresponding PennyLane operation TrotterProduct

See also

TrotterCDF, TrotterTHC, TrotterVibrational, TrotterVibronic

Example

The resources for this operation are computed using:

>>> import pennylane.estimator as qre
>>> num_steps, order = (1, 2)
>>> first_order_expansion = [qre.RX(), qre.RY()] # H = X + Y
>>> gate_set = {"RX", "RY"}
>>> res = qre.estimate(qre.TrotterProduct(first_order_expansion, num_steps, order), gate_set=gate_set)
>>> print(res)
--- Resources: ---
 Total wires: 1
    algorithmic wires: 1
    allocated wires: 0
         zero state: 0
         any state: 0
 Total gates : 3
  'RX': 2,
  'RY': 1

Attributes

resource_keys
resource_params	Returns a dictionary containing the minimal information needed to compute the resources.

resource_keys = {'first_order_expansion', 'num_steps', 'num_wires', 'order'}

resource_params

Returns a dictionary containing the minimal information needed to compute the resources.

Returns:

A dictionary containing the resource parameters:

• first_order_expansion (list[CompressedResourceOp]): A list of operators, in the compressed representation, constituting the first order expansion of the Hamiltonian to be approximately exponentiated.

• num_steps (int): number of Trotter steps to perform

• order (int): order of the Suzuki-Trotter approximation, must be 1 or even

• num_wires (int): number of wires the operator acts on

Return type:

dict

Methods

resource_decomp(first_order_expansion, ...)	Returns a list representing the resources of the operator.
resource_rep(first_order_expansion, ...)	Returns a compressed representation containing only the parameters of the Operator that are needed to compute a resource estimation.

classmethod resource_decomp(first_order_expansion, num_steps, order, num_wires)

Returns a list representing the resources of the operator. Each object represents a quantum gate and the number of times it occurs in the decomposition.

Parameters:

• first_order_expansion (list[CompressedResourceOp]) – A list of operators, in the compressed representation, constituting the first order expansion of the Hamiltonian to be approximately exponentiated.

• num_steps (int) – number of Trotter steps to perform

• order (int) – order of the Suzuki-Trotter approximation, must be 1 or even

Returns:

A list of GateCount objects, where each object represents a specific quantum gate and the number of times it appears in the decomposition.

Return type:

list[GateCount]

classmethod resource_rep(first_order_expansion, num_steps, order, num_wires)

Returns a compressed representation containing only the parameters of the Operator that are needed to compute a resource estimation.

Parameters:

• first_order_expansion (list[CompressedResourceOp]) – A list of operators, in the compressed representation, constituting the first order expansion of the Hamiltonian to be approximately exponentiated.

• num_steps (int) – number of Trotter steps to perform

• order (int) – order of the Suzuki-Trotter approximation, must be 1 or even

• num_wires (int) – number of wires the operator acts on

Returns:

the operator in a compressed representation

Return type:

CompressedResourceOp

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