# LieTrotter#

class qiskit.synthesis.LieTrotter(reps=1, insert_barriers=False, cx_structure='chain', atomic_evolution=None)[source]#

Bases: ProductFormula

The Lie-Trotter product formula.

The Lie-Trotter formula approximates the exponential of two non-commuting operators with products of their exponentials up to a second order error:

$e^{A + B} \approx e^{A}e^{B}.$

In this implementation, the operators are provided as sum terms of a Pauli operator. For example, we approximate

$e^{-it(XX + ZZ)} = e^{-it XX}e^{-it ZZ} + \mathcal{O}(t^2).$

References

[1]: D. Berry, G. Ahokas, R. Cleve and B. Sanders, â€śEfficient quantum algorithms for simulating sparse Hamiltoniansâ€ť (2006). arXiv:quant-ph/0508139 [2]: N. Hatano and M. Suzuki, â€śFinding Exponential Product Formulas of Higher Ordersâ€ť (2005). arXiv:math-ph/0506007

Parameters:
• reps (int) â€“ The number of time steps.

• insert_barriers (bool) â€“ Whether to insert barriers between the atomic evolutions.

• cx_structure (str) â€“ How to arrange the CX gates for the Pauli evolutions, can be â€śchainâ€ť, where next neighbor connections are used, or â€śfountainâ€ť, where all qubits are connected to one.

• atomic_evolution (Callable[[Pauli | SparsePauliOp, float], QuantumCircuit] | None) â€“ A function to construct the circuit for the evolution of single Pauli string. Per default, a single Pauli evolution is decomposed in a CX chain and a single qubit Z rotation.

Attributes

settings#

Return the settings in a dictionary, which can be used to reconstruct the object.

Returns:

A dictionary containing the settings of this product formula.

Raises:

NotImplementedError â€“ If a custom atomic evolution is set, which cannot be serialized.

Methods

synthesize(evolution)[source]#

Synthesize an qiskit.circuit.library.PauliEvolutionGate.

Parameters:

evolution (PauliEvolutionGate) â€“ The evolution gate to synthesize.

Returns:

A circuit implementing the evolution.

Return type:

QuantumCircuit