---
title: LieTrotter (v1.4)
description: API reference for qiskit.synthesis.LieTrotter in qiskit v1.4
source: https://quantum.cloud.ibm.com/docs/en/api/qiskit/1.4/qiskit.synthesis.LieTrotter
---

# LieTrotter

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

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.4/qiskit/synthesis/evolution/lie_trotter.py#L25-L117)

Bases: [`SuzukiTrotter`](/docs/api/qiskit/1.4/qiskit.synthesis.SuzukiTrotter "qiskit.synthesis.evolution.suzuki_trotter.SuzukiTrotter")

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

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

**References**

\[1]: D. Berry, G. Ahokas, R. Cleve and B. Sanders, “Efficient quantum algorithms for simulating sparse Hamiltonians” (2006). [arXiv:quant-ph/0508139](https://arxiv.org/abs/quant-ph/0508139) \[2]: N. Hatano and M. Suzuki, “Finding Exponential Product Formulas of Higher Orders” (2005). [arXiv:math-ph/0506007](https://arxiv.org/pdf/math-ph/0506007.pdf)

**Parameters**

- **reps** ([*int*](https://docs.python.org/3/library/functions.html#int)) – The number of time steps.
- **insert\_barriers** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – Whether to insert barriers between the atomic evolutions.
- **cx\_structure** ([*str*](https://docs.python.org/3/library/stdtypes.html#str)) – 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. This only takes effect when `atomic_evolution is None`.
- **atomic\_evolution** (*Callable\[\[*[*Pauli*](/docs/api/qiskit/1.4/qiskit.quantum_info.Pauli "qiskit.quantum_info.Pauli")  *|*[*SparsePauliOp*](/docs/api/qiskit/1.4/qiskit.quantum_info.SparsePauliOp "qiskit.quantum_info.SparsePauliOp")*,* [*float*](https://docs.python.org/3/library/functions.html#float)*],* [*QuantumCircuit*](/docs/api/qiskit/1.4/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*] | Callable\[\[*[*QuantumCircuit*](/docs/api/qiskit/1.4/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*,* [*Pauli*](/docs/api/qiskit/1.4/qiskit.quantum_info.Pauli "qiskit.quantum_info.Pauli")  *|*[*SparsePauliOp*](/docs/api/qiskit/1.4/qiskit.quantum_info.SparsePauliOp "qiskit.quantum_info.SparsePauliOp")*,* [*float*](https://docs.python.org/3/library/functions.html#float)*], None] | None*) – A function to apply the evolution of a single [`Pauli`](/docs/api/qiskit/1.4/qiskit.quantum_info.Pauli "qiskit.quantum_info.Pauli"), or [`SparsePauliOp`](/docs/api/qiskit/1.4/qiskit.quantum_info.SparsePauliOp "qiskit.quantum_info.SparsePauliOp") of only commuting terms, to a circuit. The function takes in three arguments: the circuit to append the evolution to, the Pauli operator to evolve, and the evolution time. By default, a single Pauli evolution is decomposed into a chain of `CX` gates and a single `RZ` gate. Alternatively, the function can also take Pauli operator and evolution time as inputs and returns the circuit that will be appended to the overall circuit being built.
- **wrap** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – Whether to wrap the atomic evolutions into custom gate objects. This only takes effect when `atomic_evolution is None`.
- **preserve\_order** ([*bool*](https://docs.python.org/3/library/functions.html#bool)) – If `False`, allows reordering the terms of the operator to potentially yield a shallower evolution circuit. Not relevant when synthesizing operator with a single term.

## 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**](https://docs.python.org/3/library/exceptions.html#NotImplementedError) – If a custom atomic evolution is set, which cannot be serialized.

## Methods

### expand

`expand(evolution)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.4/qiskit/synthesis/evolution/suzuki_trotter.py#L131-L182)

Expand the Hamiltonian into a Suzuki-Trotter sequence of sparse gates.

For example, the Hamiltonian `H = IX + ZZ` for an evolution time `t` and 1 repetition for an order 2 formula would get decomposed into a list of 3-tuples containing `(pauli, indices, rz_rotation_angle)`, that is:

```text
("X", [0], t), ("ZZ", [0, 1], 2t), ("X", [0], t)
```

Note that the rotation angle contains a factor of 2, such that that evolution of a Pauli $P$ over time $t$, which is $e^{itP}$, is represented by `(P, indices, 2 * t)`.

For `N` repetitions, this sequence would be repeated `N` times and the coefficients divided by `N`.

**Parameters**

**evolution** ([*PauliEvolutionGate*](/docs/api/qiskit/1.4/qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate")) – The evolution gate to expand.

**Returns**

The Pauli network implementing the Trotter expansion.

**Return type**

[list](https://docs.python.org/3/library/stdtypes.html#list)\[[tuple](https://docs.python.org/3/library/stdtypes.html#tuple)\[[str](https://docs.python.org/3/library/stdtypes.html#str), [list](https://docs.python.org/3/library/stdtypes.html#list)\[[int](https://docs.python.org/3/library/functions.html#int)], ParameterValueType]]

### synthesize

`synthesize(evolution)`

[GitHub](https://github.com/Qiskit/qiskit/tree/stable/1.4/qiskit/synthesis/evolution/product_formula.py#L145-L167)

Synthesize a [`PauliEvolutionGate`](/docs/api/qiskit/1.4/qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate").

**Parameters**

**evolution** ([*PauliEvolutionGate*](/docs/api/qiskit/1.4/qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate")) – The evolution gate to synthesize.

**Returns**

A circuit implementing the evolution.

**Return type**

[QuantumCircuit](/docs/api/qiskit/1.4/qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")