106 lines
6.7 KiB
Plaintext
106 lines
6.7 KiB
Plaintext
---
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title: LieTrotter (v1.4)
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description: API reference for qiskit.synthesis.LieTrotter in qiskit v1.4
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in_page_toc_min_heading_level: 1
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python_api_type: class
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python_api_name: qiskit.synthesis.LieTrotter
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---
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# LieTrotter
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<Class id="qiskit.synthesis.LieTrotter" isDedicatedPage={true} github="https://github.com/Qiskit/qiskit/tree/stable/1.4/qiskit/synthesis/evolution/lie_trotter.py#L25-L117" signature="qiskit.synthesis.LieTrotter(reps=1, insert_barriers=False, cx_structure='chain', atomic_evolution=None, wrap=False, preserve_order=True)" modifiers="class">
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Bases: [`SuzukiTrotter`](qiskit.synthesis.SuzukiTrotter "qiskit.synthesis.evolution.suzuki_trotter.SuzukiTrotter")
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The Lie-Trotter product formula.
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The Lie-Trotter formula approximates the exponential of two non-commuting operators with products of their exponentials up to a second order error:
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$$
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e^{A + B} \approx e^{A}e^{B}.
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$$
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In this implementation, the operators are provided as sum terms of a Pauli operator. For example, we approximate
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$$
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e^{-it(XI + ZZ)} = e^{-it XI}e^{-it ZZ} + \mathcal{O}(t^2).
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$$
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**References**
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\[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)
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**Parameters**
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* **reps** ([*int*](https://docs.python.org/3/library/functions.html#int "(in Python v3.13)")) – The number of time steps.
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* **insert\_barriers** ([*bool*](https://docs.python.org/3/library/functions.html#bool "(in Python v3.13)")) – Whether to insert barriers between the atomic evolutions.
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* **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`.
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* **atomic\_evolution** (*Callable\[\[*[*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)")*],* [*QuantumCircuit*](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")*] | 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. 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.
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* **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`.
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* **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.
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## Attributes
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### settings
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<Attribute id="qiskit.synthesis.LieTrotter.settings">
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Return the settings in a dictionary, which can be used to reconstruct the object.
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**Returns**
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A dictionary containing the settings of this product formula.
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**Raises**
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[**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.
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</Attribute>
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## Methods
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### expand
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<Function id="qiskit.synthesis.LieTrotter.expand" github="https://github.com/Qiskit/qiskit/tree/stable/1.4/qiskit/synthesis/evolution/suzuki_trotter.py#L131-L182" signature="expand(evolution)">
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Expand the Hamiltonian into a Suzuki-Trotter sequence of sparse gates.
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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:
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```text
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("X", [0], t), ("ZZ", [0, 1], 2t), ("X", [0], t)
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```
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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)`.
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For `N` repetitions, this sequence would be repeated `N` times and the coefficients divided by `N`.
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**Parameters**
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**evolution** ([*PauliEvolutionGate*](qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate")) – The evolution gate to expand.
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**Returns**
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The Pauli network implementing the Trotter expansion.
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**Return type**
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[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]]
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</Function>
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### synthesize
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<Function id="qiskit.synthesis.LieTrotter.synthesize" github="https://github.com/Qiskit/qiskit/tree/stable/1.4/qiskit/synthesis/evolution/product_formula.py#L145-L167" signature="synthesize(evolution)">
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Synthesize a [`PauliEvolutionGate`](qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate").
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**Parameters**
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**evolution** ([*PauliEvolutionGate*](qiskit.circuit.library.PauliEvolutionGate "qiskit.circuit.library.PauliEvolutionGate")) – The evolution gate to synthesize.
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**Returns**
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A circuit implementing the evolution.
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**Return type**
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[QuantumCircuit](qiskit.circuit.QuantumCircuit "qiskit.circuit.QuantumCircuit")
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</Function>
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</Class>
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