107 lines
6.9 KiB
Plaintext
107 lines
6.9 KiB
Plaintext
---
|
||
title: LieTrotter (dev version)
|
||
description: API reference for qiskit.synthesis.LieTrotter in the dev version of qiskit
|
||
in_page_toc_min_heading_level: 1
|
||
python_api_type: class
|
||
python_api_name: qiskit.synthesis.LieTrotter
|
||
---
|
||
|
||
# LieTrotter
|
||
|
||
<Class id="qiskit.synthesis.LieTrotter" isDedicatedPage={true} github="https://github.com/Qiskit/qiskit/tree/main/qiskit/synthesis/evolution/lie_trotter.py#L25-L120" signature="qiskit.synthesis.LieTrotter(reps=1, insert_barriers=False, cx_structure='chain', atomic_evolution=None, wrap=False, preserve_order=True, *, atomic_evolution_sparse_observable=False)" modifiers="class">
|
||
Bases: [`SuzukiTrotter`](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 "(in Python v3.13)")) – The number of time steps.
|
||
* **insert\_barriers** ([*bool*](https://docs.python.org/3/library/functions.html#bool "(in Python v3.13)")) – Whether to insert barriers between the atomic evolutions.
|
||
* **cx\_structure** ([*str*](https://docs.python.org/3/library/stdtypes.html#str "(in Python v3.13)")) – 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\[\[*[*QuantumCircuit*](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*,* [*Pauli*](qiskit.quantum_info.Pauli "qiskit.quantum_info.Pauli") *|*[*SparsePauliOp*](qiskit.quantum_info.SparsePauliOp "qiskit.quantum_info.SparsePauliOp")*,* [*float*](https://docs.python.org/3/library/functions.html#float "(in Python v3.13)")*], None] | None*) – A function to apply the evolution of a single [`Pauli`](qiskit.quantum_info.Pauli "qiskit.quantum_info.Pauli"), or [`SparsePauliOp`](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.
|
||
* **wrap** ([*bool*](https://docs.python.org/3/library/functions.html#bool "(in Python v3.13)")) – 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 "(in Python v3.13)")) – 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.
|
||
* **atomic\_evolution\_sparse\_observable** ([*bool*](https://docs.python.org/3/library/functions.html#bool "(in Python v3.13)")) – If a custom `atomic_evolution` is passed, which does not yet support [`SparseObservable`](qiskit.quantum_info.SparseObservable "qiskit.quantum_info.SparseObservable")s as input, set this argument to `False` to automatically apply a conversion to [`SparsePauliOp`](qiskit.quantum_info.SparsePauliOp "qiskit.quantum_info.SparsePauliOp"). This argument is supported until Qiskit 2.2, at which point all atomic evolutions are required to support [`SparseObservable`](qiskit.quantum_info.SparseObservable "qiskit.quantum_info.SparseObservable")s as input.
|
||
|
||
## Attributes
|
||
|
||
### settings
|
||
|
||
<Attribute id="qiskit.synthesis.LieTrotter.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 "(in Python v3.13)") – If a custom atomic evolution is set, which cannot be serialized.
|
||
</Attribute>
|
||
|
||
## Methods
|
||
|
||
### expand
|
||
|
||
<Function id="qiskit.synthesis.LieTrotter.expand" github="https://github.com/Qiskit/qiskit/tree/main/qiskit/synthesis/evolution/suzuki_trotter.py#L119-L175" signature="expand(evolution)">
|
||
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*](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 "(in Python v3.13)")\[[tuple](https://docs.python.org/3/library/stdtypes.html#tuple "(in Python v3.13)")\[[str](https://docs.python.org/3/library/stdtypes.html#str "(in Python v3.13)"), [list](https://docs.python.org/3/library/stdtypes.html#list "(in Python v3.13)")\[[int](https://docs.python.org/3/library/functions.html#int "(in Python v3.13)")], ParameterValueType]]
|
||
</Function>
|
||
|
||
### synthesize
|
||
|
||
<Function id="qiskit.synthesis.LieTrotter.synthesize" github="https://github.com/Qiskit/qiskit/tree/main/qiskit/synthesis/evolution/product_formula.py#L125-L147" signature="synthesize(evolution)">
|
||
Synthesize a [`PauliEvolutionGate`](qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate").
|
||
|
||
**Parameters**
|
||
|
||
**evolution** ([*PauliEvolutionGate*](qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate")) – The evolution gate to synthesize.
|
||
|
||
**Returns**
|
||
|
||
A circuit implementing the evolution.
|
||
|
||
**Return type**
|
||
|
||
[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
|
||
</Function>
|
||
</Class>
|
||
|