80 lines
3.3 KiB
Plaintext
80 lines
3.3 KiB
Plaintext
---
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title: LieTrotter
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description: API reference for qiskit.synthesis.LieTrotter
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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/0.23/qiskit/synthesis/evolution/lie_trotter.py" signature="LieTrotter(reps=1, insert_barriers=False, cx_structure='chain', atomic_evolution=None)" modifiers="class">
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Bases: [`qiskit.synthesis.evolution.product_formula.ProductFormula`](qiskit.synthesis.ProductFormula "qiskit.synthesis.evolution.product_formula.ProductFormula")
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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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$$
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e^{-it(XX + ZZ)} = e^{-it XX}e^{-it ZZ} + \mathcal{O}(t^2).
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$$
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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`) – The number of time steps.
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* **insert\_barriers** (`bool`) – Whether to insert barriers between the atomic evolutions.
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* **cx\_structure** (`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.
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* **atomic\_evolution** (`Optional`\[`Callable`\[\[`Union`\[[`Pauli`](qiskit.quantum_info.Pauli "qiskit.quantum_info.operators.symplectic.pauli.Pauli"), [`SparsePauliOp`](qiskit.quantum_info.SparsePauliOp "qiskit.quantum_info.operators.symplectic.sparse_pauli_op.SparsePauliOp")], `float`], [`QuantumCircuit`](qiskit.circuit.QuantumCircuit "qiskit.circuit.quantumcircuit.QuantumCircuit")]]) – A function to construct the circuit for the evolution of single Pauli string. Per default, a single Pauli evolution is decomopsed in a CX chain and a single qubit Z rotation.
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## Methods
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### synthesize
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<Function id="qiskit.synthesis.LieTrotter.synthesize" signature="LieTrotter.synthesize(evolution)">
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Synthesize an `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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## 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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**Return type**
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`Dict`\[`str`, `Any`]
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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** – If a custom atomic evolution is set, which cannot be serialized.
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</Attribute>
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</Class>
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