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IBM Quantum Platform

LieTrotter

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

GitHub

Bases: 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:

eA+B≈eAeB.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−itXIe−itZZ+O(t2).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 [2]: N. Hatano and M. Suzuki, “Finding Exponential Product Formulas of Higher Orders” (2005). arXiv:math-ph/0506007

Parameters

  • reps (int) – The number of time steps.
  • insert_barriers (bool) – Whether to insert barriers between the atomic evolutions.
  • 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. This only takes effect when atomic_evolution is None.
  • atomic_evolution (Callable[[Pauli |SparsePauliOp, float], QuantumCircuit] | Callable[[QuantumCircuit, Pauli |SparsePauliOp, float], None] | None) – A function to apply the evolution of a single Pauli, or 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) – Whether to wrap the atomic evolutions into custom gate objects. This only takes effect when atomic_evolution is None.
  • preserve_order (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 – If a custom atomic evolution is set, which cannot be serialized.

Methods

expand

expand(evolution)

GitHub

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:

("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 PP over time tt, which is eitPe^{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) – The evolution gate to expand.

Returns

The Pauli network implementing the Trotter expansion.

Return type

list[tuple[str, list[int], ParameterValueType]]

synthesize

synthesize(evolution)

GitHub

Synthesize a PauliEvolutionGate.

Parameters

evolution (PauliEvolutionGate) – The evolution gate to synthesize.

Returns

A circuit implementing the evolution.

Return type

QuantumCircuit

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