qml.estimator.templates.TrotterTHC¶
- class TrotterTHC(thc_ham, num_steps, order, wires=None)[source]
Bases:
ResourceOperatorAn operation representing the Suzuki-Trotter product approximation for the complex matrix exponential of a tensor hypercontracted (THC) 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 \(m^{\text{th}}\) 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:
thc_ham (
THCHamiltonian) – a tensor hypercontracted Hamiltonian to be approximately exponentiatednum_steps (int) – number of Trotter steps to perform
order (int) – order of the approximation, must be
1or an even numberwires (list[int] | None) – the wires on which the operator acts
- 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 get grouped. This reduces the counts for those terms to:
\[\begin{split}\begin{align} C_{O_{0}} &= n \cdot 5^{\frac{m}{2} - 1} + 1, \\ C_{O_{N}} &= n \cdot 5^{\frac{m}{2} - 1}. \end{align}\end{split}\]The resources for a single step expansion of tensor hypercontracted Hamiltonian are calculated based on arXiv:2407.04432.
See also
See also
Example
The resources for this operation are computed using:
>>> import pennylane.estimator as qre >>> num_steps, order = (1, 2) >>> thc_ham = qre.THCHamiltonian(num_orbitals=4, tensor_rank=4) >>> res = qre.estimate(qre.TrotterTHC(thc_ham, num_steps, order)) >>> print(res) --- Resources: --- Total wires: 8 algorithmic wires: 8 allocated wires: 0 zero state: 0 any state: 0 Total gates : 8.520E+3 'T': 7.888E+3, 'CNOT': 128, 'Z': 144, 'S': 216, 'Hadamard': 144
Attributes
Returns a dictionary containing the minimal information needed to compute the resources.
- resource_keys = {'num_steps', 'order', 'thc_ham'}¶
- resource_params¶
Returns a dictionary containing the minimal information needed to compute the resources.
- Returns:
- A dictionary containing the resource parameters:
thc_ham (
THCHamiltonian): a tensor hypercontracted Hamiltonian to be approximately exponentiatednum_steps (int): number of Trotter steps to perform
order (int): order of the approximation, must be 1 or even
- Return type:
dict
Methods
controlled_resource_decomp(num_ctrl_wires, ...)Returns the controlled resource decomposition.
resource_decomp(thc_ham, num_steps, order)Returns a list representing the resources of the operator.
resource_rep(thc_ham, num_steps, order)Returns a compressed representation containing only the parameters of the Operator that are needed to compute the resources.
- classmethod controlled_resource_decomp(num_ctrl_wires, num_zero_ctrl, target_resource_params=None)[source]¶
Returns the controlled resource decomposition.
- Parameters:
num_ctrl_wires (int) – the number of qubits the operation is controlled on
num_zero_ctrl (int) – the number of control qubits, that are controlled when in the \(|0\rangle\) state
target_resource_params (dict) – dictionary containing the size of the larger of the two registers being added together
- 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]
- Resources:
The original resources are controlled only on the Z rotation gates
- classmethod resource_decomp(thc_ham, num_steps, order)[source]¶
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:
thc_ham (
THCHamiltonian) – a tensor hypercontracted Hamiltonian to be approximately exponentiatednum_steps (int) – number of Trotter steps to perform
order (int) – order of the approximation, must be 1 or even
- 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 get grouped. This reduces the counts for those terms to:
\[\begin{split}\begin{align} C_{O_{0}} &= n \cdot 5^{\frac{m}{2} - 1} + 1, \\ C_{O_{N}} &= n \cdot 5^{\frac{m}{2} - 1}. \end{align}\end{split}\]The resources for a single step expansion of tensor hypercontracted Hamiltonian are calculated based on arXiv:2407.04432.
- 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(thc_ham, num_steps, order)[source]¶
Returns a compressed representation containing only the parameters of the Operator that are needed to compute the resources.
- Parameters:
thc_ham (
THCHamiltonian) – a tensor hypercontracted Hamiltonian to be approximately exponentiatednum_steps (int) – number of Trotter steps to perform
order (int) – order of the approximation, must be 1 or even
- Returns:
the operator in a compressed representation
- Return type: