qdk_chemistry.algorithms.hamiltonian_unitary_builder.time_evolution.trotter module

QDK/Chemistry implementation of the Trotter decomposition Builder.

References

Childs, A. M., et al. “Theory of Trotter Error with Commutator Scaling.” Physical Review X 11.1 (2021): 011020.

Strang, G. “On the construction and comparison of difference schemes.” SIAM Journal on Numerical Analysis 5.3 (1968): 506-517.

Suzuki, M. “General theory of higher-order decomposition of exponential operators and symplectic integrators.” Physics Letters A 165.5-6 (1992): 387-395.

class qdk_chemistry.algorithms.hamiltonian_unitary_builder.time_evolution.trotter.Trotter(order=1, *, time=0.0, target_accuracy=0.0, num_divisions=0, error_bound='commutator', weight_threshold=1e-12, power=1, power_strategy='repeat')

Bases: TimeEvolutionBuilder

Trotter decomposition builder.

Parameters:

• order (int)

• time (float)

• target_accuracy (float)

• num_divisions (int)

• error_bound (str)

• weight_threshold (float)

• power (int)

• power_strategy (str)

__init__(order=1, *, time=0.0, target_accuracy=0.0, num_divisions=0, error_bound='commutator', weight_threshold=1e-12, power=1, power_strategy='repeat')

Initialize Trotter builder with specified Trotter decomposition settings.

The Trotter decomposition approximates the time evolution operator \(e^{-iHt}\) when the Hamiltonian \(H\) can be expressed as a sum of terms \(H = \sum_j \alpha_j P_j\) where \(P_j\) are Pauli strings and \(\alpha_j\) are scalar coefficients. Rather than exponentiating the full Hamiltonian at once, the Trotter method constructs an approximation by exponentiating each term separately and combining them in a product formula. For example, the first-order Trotter formula approximates the time evolution operator as

\(e^{-iHt} \approx S_1^N(t) = \left[\prod_j e^{-i\alpha_j P_j t/N}\right]^N\), where \(N\) is the number of divisions.

The number of divisions N can be determined automatically from target_accuracy, fixed explicitly via num_divisions, or both (in which case the larger value is used).

The error associated with the Trotter decomposition, \(S_k^N(t)\), can be expressted in terms of the spectral norm of the difference between the exact and approximate time evolution operators:

\(\lVert e^{-iHt} - S_k^N(t) \rVert \leq \epsilon\)

However, the cost of computing this norm is equivalent to computing the exact exponential itself. For this reason, we provide two approximate error-bound strategies to determine the number of divisions required to achieve a target accuracy at a particular Trotter order (used only when target_accuracy is set):

• "commutator" (default, tighter): uses the commutator-based bound from Childs et al. (2021). \(N = \lceil \frac{t^{2}}{2\epsilon} \sum_{j<k}\lVert[\alpha_jP_j,\alpha_kP_k]\rVert \rceil\)

• "naive": uses the triangle-inequality bound. \(N = \lceil (\sum_j|\alpha_j|)^{2}t^{2}/\epsilon \rceil\)

When the input QubitOperator carries a populated term_partition, the builder consumes it directly for schedule-level grouping. When no partition is present, each Pauli term is exponentiated as its own group.

Parameters:

• order (int) – Trotter decomposition order (1, 2, or any positive even integer). Defaults to 1.

• time (float) – The evolution time. Defaults to 0.0.

• target_accuracy (float) – Target accuracy for auto step computation. Use 0.0 (default) to disable.

• num_divisions (int) – Divisions per Trotter step. Max of this and auto value is used. Defaults to 0.

• error_bound (str) – Error bound strategy: "commutator" (default) or "naive".

• weight_threshold (float) – Threshold for filtering small coefficients. Defaults to 1e-12.

• power (int) – The power to raise the unitary to. Defaults to 1.

• power_strategy (str) – Strategy for U^power: "rescale" or "repeat" (default).

name()

Return the name of the unitary builder.

Return type:

str

type_name()

Return unitary_builder as the algorithm type name.

Return type:

str

class qdk_chemistry.algorithms.hamiltonian_unitary_builder.time_evolution.trotter.TrotterSettings

Bases: TimeEvolutionSettings

Settings for Trotter decomposition builder.

__init__()

Initialize TrotterSettings with default values.

order

The order of the Trotter decomposition (currently only first order is supported).

target_accuracy

Target accuracy for automatic step computation (0.0 means disabled).

num_divisions

Explicit number of divisions within a Trotter step (0 means automatic).

error_bound

Strategy for computing the Trotter error bound (“commutator” or “naive”).

weight_threshold

The absolute threshold for filtering small coefficients.