# LieTrotter

*class *`qiskit.synthesis.LieTrotter(reps=1, insert_barriers=False, cx_structure='chain', atomic_evolution=None)`

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, “Efficient quantum algorithms for simulating sparse Hamiltonians” (2006). arXiv:quant-ph/0508139(opens in a new tab) [2]: N. Hatano and M. Suzuki, “Finding Exponential Product Formulas of Higher Orders” (2005). arXiv:math-ph/0506007(opens in a new tab)

**Parameters**

**reps**(*int*(opens in a new tab)) – The number of time steps.**insert_barriers**(*bool*(opens in a new tab)) – Whether to insert barriers between the atomic evolutions.**cx_structure**(*str*(opens in a new tab)) – How to arrange the CX gates for the Pauli evolutions, can be`"chain"`

, where next neighbor connections are used, or`"fountain"`

, where all qubits are connected to one.**atomic_evolution**(*Callable*(opens in a new tab)*[[**Pauli**|**SparsePauliOp**,**float*(opens in a new tab)*],**QuantumCircuit**] | None*) – 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**(opens in a new tab) – If a custom atomic evolution is set, which cannot be serialized.

## Methods

### synthesize

`synthesize(evolution)`

Synthesize an `qiskit.circuit.library.PauliEvolutionGate`

.

**Parameters**

**evolution** (*PauliEvolutionGate*) – The evolution gate to synthesize.

**Returns**

A circuit implementing the evolution.

**Return type**