BosonicLinearMapper

class BosonicLinearMapper(max_occupation)

Bases: BosonicMapper

The Linear boson-to-qubit mapping.

This mapper generates a linear encoding of the Bosonic operator \(b_k^\dagger, b_k\) to qubit operators (linear combinations of Pauli strings). In this linear encoding each bosonic mode is represented via \(n_k^{max} + 1\) qubits, where \(n_k^{max}\) is the max occupation of the mode (meaning the number of states used in the expansion of the mode, or equivalently the state at which the maximum excitation can take place). The mode \(|k\rangle\) is then mapped to the occupation number vector \(|0_{n_k^{max}}, 0_{n_k^{max} - 1},..., 0_{n_k + 1}, 1_{n_k}, 0_{n_k - 1},..., 0_{0_k}\rangle\)

It implements the formula in Section II.C of Reference [1]:

\[b_k^\dagger = \sum_{n_k =0}^{n_k^{max}-1}(\sqrt{n_k +1}\sigma_{n_k}^{+}\sigma_{n_k + 1}^{-})\]

from \(n_k = 0\) to \(n_k^{max} + 1\) where \(n_k^{max}\) is the maximum occupation (defined by the user). In the following implementation, we explicit the operators \(\sigma^+\) and \(\sigma^-\) with the Pauli matrices:

\[ \begin{align}\begin{aligned}\sigma_{n_k}^+ := S_j^+ = 0.5 * (X_j + \textit{i}Y_j)\\\sigma_{n_k}^- := S_j^- = 0.5 * (X_j - \textit{i}Y_j)\end{aligned}\end{align} \]

The length of the qubit register is:

BosonicOp.num_modes * (BosonicLinearMapper.max_occupation + 1)

To use this mapper one can for example:

from qiskit_nature.second_q.mappers import BosonicLinearMapper
from qiskit_nature.second_q.operators import BosonicOp

mapper = BosonicLinearMapper(max_occupation=1)
qubit_op = mapper.map(BosonicOp({'+_0 -_0': 1}, num_modes=1))

Note

Since this mapper truncates the maximum occupation of a bosonic state as represented in the qubit register, the commutation relation after the mapping differ from the standard ones. Please refer to Section 4, equation 22 of Reference [2] for more details

References

[1] A. Miessen et al., Quantum algorithms for quantum dynamics: A performance study on the spin-boson model, Phys. Rev. Research 3, 043212. https://link.aps.org/doi/10.1103/PhysRevResearch.3.043212

[2] R. Somma et al., Quantum Simulations of Physics Problems, Arxiv https://doi.org/10.48550/arXiv.quant-ph/0304063

Parameters:

max_occupation (int) – defines the excitation space of the k-th bosonic state. Together with the number of modes required to represent the bosonic operator, it defines the minimum length of the qubit register. The minimum value is 1.

Attributes

max_occupation

The maximum occupation of any bosonic state.

Methods

map(second_q_ops, *, register_length=None)

Maps a second quantized operator or a list, dict of second quantized operators based on the current mapper.

Parameters:

• second_q_ops (BosonicOp | List[BosonicOp | None] | Dict[str, BosonicOp]) – A second quantized operator, or list thereof.

• register_length (int | None) – when provided, this will be used to overwrite the register_length attribute of the SparseLabelOp being mapped. This is possible because the register_length is considered a lower bound in a SparseLabelOp.

Returns:

A qubit operator in the form of a SparsePauliOp, or list (resp. dict) thereof if a list (resp. dict) of second quantized operators was supplied.

Return type:

SparsePauliOp | List[SparsePauliOp | None] | Dict[str, SparsePauliOp]