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suzuki_trotter.py
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suzuki_trotter.py
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# This code is part of Qiskit.
#
# (C) Copyright IBM 2021.
#
# This code is licensed under the Apache License, Version 2.0. You may
# obtain a copy of this license in the LICENSE.txt file in the root directory
# of this source tree or at http://www.apache.org/licenses/LICENSE-2.0.
#
# Any modifications or derivative works of this code must retain this
# copyright notice, and modified files need to carry a notice indicating
# that they have been altered from the originals.
"""The Suzuki-Trotter product formula."""
from typing import Callable, Optional, Union
import numpy as np
from qiskit.circuit.quantumcircuit import QuantumCircuit
from qiskit.quantum_info.operators import SparsePauliOp, Pauli
from .product_formula import ProductFormula
class SuzukiTrotter(ProductFormula):
r"""The (higher order) Suzuki-Trotter product formula.
The Suzuki-Trotter formulas improve the error of the Lie-Trotter approximation.
For example, the second order decomposition is
.. math::
e^{A + B} \approx e^{B/2} e^{A} e^{B/2}.
Higher order decompositions are based on recursions, see Ref. [1] for more details.
In this implementation, the operators are provided as sum terms of a Pauli operator.
For example, in the second order Suzuki-Trotter decomposition we approximate
.. math::
e^{-it(XX + ZZ)} = e^{-it/2 ZZ}e^{-it XX}e^{-it/2 ZZ} + \mathcal{O}(t^3).
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>`_
"""
def __init__(
self,
order: int = 2,
reps: int = 1,
insert_barriers: bool = False,
cx_structure: str = "chain",
atomic_evolution: Optional[
Callable[[Union[Pauli, SparsePauliOp], float], QuantumCircuit]
] = None,
) -> None:
"""
Args:
order: The order of the product formula.
reps: The number of time steps.
insert_barriers: Whether to insert barriers between the atomic evolutions.
cx_structure: 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.
atomic_evolution: A function to construct the circuit for the evolution of single
Pauli string. Per default, a single Pauli evolution is decomposed in a CX chain
and a single qubit Z rotation.
Raises:
ValueError: If order is not even
"""
if order % 2 == 1:
raise ValueError(
"Suzuki product formulae are symmetric and therefore only defined "
"for even orders."
)
super().__init__(order, reps, insert_barriers, cx_structure, atomic_evolution)
def synthesize(self, evolution):
# get operators and time to evolve
operators = evolution.operator
time = evolution.time
if not isinstance(operators, list):
pauli_list = [(Pauli(op), np.real(coeff)) for op, coeff in operators.to_list()]
else:
pauli_list = [(op, 1) for op in operators]
ops_to_evolve = self._recurse(self.order, time / self.reps, pauli_list)
# construct the evolution circuit
single_rep = QuantumCircuit(operators[0].num_qubits)
first_barrier = False
for op, coeff in ops_to_evolve:
# add barriers
if first_barrier:
if self.insert_barriers:
single_rep.barrier()
else:
first_barrier = True
single_rep.compose(self.atomic_evolution(op, coeff), wrap=True, inplace=True)
evolution_circuit = QuantumCircuit(operators[0].num_qubits)
first_barrier = False
for _ in range(self.reps):
# add barriers
if first_barrier:
if self.insert_barriers:
single_rep.barrier()
else:
first_barrier = True
evolution_circuit.compose(single_rep, inplace=True)
return evolution_circuit
@staticmethod
def _recurse(order, time, pauli_list):
if order == 1:
return pauli_list
elif order == 2:
halves = [(op, coeff * time / 2) for op, coeff in pauli_list[:-1]]
full = [(pauli_list[-1][0], time * pauli_list[-1][1])]
return halves + full + list(reversed(halves))
else:
reduction = 1 / (4 - 4 ** (1 / (order - 1)))
outer = 2 * SuzukiTrotter._recurse(
order - 2, time=reduction * time, pauli_list=pauli_list
)
inner = SuzukiTrotter._recurse(
order - 2, time=(1 - 4 * reduction) * time, pauli_list=pauli_list
)
return outer + inner + outer