qml.labs.resource_estimation.ResourceTrotterTHC

class ResourceTrotterTHC(compact_ham, num_steps, order, wires=None)

Bases: ResourceOperator

An operation representing the Suzuki-Trotter product approximation for the complex matrix exponential of tensor hypercontracted 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:

• compact_ham (CompactHamiltonian) – a tensor hypercontracted Hamiltonian to be approximately exponentiated

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

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

• wires (list[int] or optional) – the wires on which the operator acts

Resources:

The resources are defined according to the recursive formula presented above. Specifically, each operator in the single step expansion of the exponentiation is called a number of times given by the formula:

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

Furthermore, the first and last terms of the Hamiltonian appear in pairs due to the symmetric form of the recursive formula. Those counts are further simplified by grouping like terms as:

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

The resources for a single step expansion of tensor hypercontracted Hamiltonian are calculated based on arXiv:2407.04432

See also

TrotterProduct

Example

The resources can be computed as:

>>> import pennylane.labs.resource_estimation as plre
>>> num_steps, order = (1, 2)
>>> compact_ham = plre.CompactHamiltonian.thc(num_orbitals=4, tensor_rank=4)
>>> res = plre.estimate_resources(plre.ResourceTrotterTHC(compact_ham, num_steps, order))
>>> print(res)
--- Resources: ---
 Total qubits: 8
 Total gates : 8.520E+3
 Qubit breakdown:
  clean qubits: 0, dirty qubits: 0, algorithmic qubits: 8
 Gate breakdown:
  {'T': 7.888E+3, 'S': 216.0, 'Z': 144.0, 'Hadamard': 144.0, 'CNOT': 128.0}

Attributes

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

num_wires = 0

resource_keys = {'compact_ham', 'num_steps', 'order'}

resource_params

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

Returns:

A dictionary containing the resource parameters:

• compact_ham (~pennylane.labs.resource_estimation.CompactHamiltonian): a tensor hypercontracted Hamiltonian to be approximately exponentiated

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

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

Return type:

dict

Methods

adjoint_resource_decomp(*args, **kwargs)	Returns a list of actions that define the resources of the operator.
controlled_resource_decomp(...)	Returns a list representing the resources for a controlled version of the operator.
default_adjoint_resource_decomp(*args, **kwargs)	Returns a list representing the resources for the adjoint of the operator.
default_controlled_resource_decomp(...)	Returns the controlled resource decomposition.
default_pow_resource_decomp(pow_z, *args, ...)	Returns a list representing the resources for an operator raised to a power.
default_resource_decomp(compact_ham, ...)	Returns a list representing the resources of the operator.
dequeue(op_to_remove[, context])	Remove the given resource operator(s) from the Operator queue.
pow_resource_decomp(pow_z, *args, **kwargs)	Returns a list representing the resources for an operator raised to a power.
queue([context])	Append the operator to the Operator queue.
resource_decomp(*args, **kwargs)	Returns a list of actions that define the resources of the operator.
resource_rep(compact_ham, num_steps, order)	Returns a compressed representation containing only the parameters of the Operator that are needed to compute the resources.
resource_rep_from_op()	Returns a compressed representation directly from the operator
set_resources(new_func[, override_type])	Set a custom function to override the default resource decomposition.
tracking_name(*args, **kwargs)	Returns a name used to track the operator during resource estimation.
tracking_name_from_op()	Returns the tracking name built with the operator's parameters.

classmethod adjoint_resource_decomp(*args, **kwargs)

Returns a list of actions that define the resources of the operator.

classmethod controlled_resource_decomp(ctrl_num_ctrl_wires, ctrl_num_ctrl_values, *args, **kwargs)

Returns a list representing the resources for a controlled version of the operator.

Parameters:

• ctrl_num_ctrl_wires (int) – the number of qubits the operation is controlled on

• ctrl_num_ctrl_values (int) – the number of control qubits, that are controlled when in the \(|0\rangle\) state

classmethod default_adjoint_resource_decomp(*args, **kwargs)

Returns a list representing the resources for the adjoint of the operator.

classmethod default_controlled_resource_decomp(compact_ham, num_steps, order, ctrl_num_ctrl_wires, ctrl_num_ctrl_values, **kwargs)

Returns the controlled resource decomposition.

Parameters:

• compact_ham (CompactHamiltonian) – a tensor hypercontracted Hamiltonian to be approximately exponentiated

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

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

• ctrl_num_ctrl_wires (int) – the number of control wires for the controlled operations

• ctrl_num_ctrl_values (int) – the number of control values for the controlled operations

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 default_pow_resource_decomp(pow_z, *args, **kwargs)

Returns a list representing the resources for an operator raised to a power.

Parameters:

pow_z (int) – exponent that the operator is being raised to

classmethod default_resource_decomp(compact_ham, num_steps, order, **kwargs)

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:

• compact_ham (CompactHamiltonian) – a tensor hypercontracted Hamiltonian to be approximately exponentiated

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

• order (int) – order of the 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]

static dequeue(op_to_remove, context=<class 'pennylane.queuing.QueuingManager'>)

Remove the given resource operator(s) from the Operator queue.

classmethod pow_resource_decomp(pow_z, *args, **kwargs)

Returns a list representing the resources for an operator raised to a power.

Parameters:

pow_z (int) – exponent that the operator is being raised to

queue(context=<class 'pennylane.queuing.QueuingManager'>)

Append the operator to the Operator queue.

classmethod resource_decomp(*args, **kwargs)

Returns a list of actions that define the resources of the operator.

classmethod resource_rep(compact_ham, num_steps, order)

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

Parameters:

• compact_ham (CompactHamiltonian) – a tensor hypercontracted Hamiltonian to be approximately exponentiated

• num_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:

CompressedResourceOp

resource_rep_from_op()

Returns a compressed representation directly from the operator

classmethod set_resources(new_func, override_type='base')

Set a custom function to override the default resource decomposition.

This method allows users to replace any of the resource_decomp, adjoint_resource_decomp, ctrl_resource_decomp, or pow_resource_decomp methods globally for every instance of the class.

classmethod tracking_name(*args, **kwargs)

Returns a name used to track the operator during resource estimation.

tracking_name_from_op()

Returns the tracking name built with the operator’s parameters.

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