BosonicLinearMapper#

class BosonicLinearMapper(max_occupation)[source]#

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))

নোট

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

প্যারামিটার:: 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.

প্যারামিটার:

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.

রিটার্নস:

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.

রিটার্ন টাইপ:

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

classmethod mode_based_mapping(second_q_op, register_length=None)#

Utility method to map a SparseLabelOp to a qubit operator using a pauli table.

প্যারামিটার:

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

রিটার্নস:

The qubit operator corresponding to the problem-Hamiltonian in the qubit space.

রেইজেস:

QiskitNatureError -- If number length of pauli table does not match the number of operator modes, or if the operator has unexpected label content

রিটার্ন টাইপ:

SparsePauliOp

classmethod pauli_table(register_length)#

Generates a Pauli-lookup table mapping from modes to pauli pairs.

The generated table is processed by QubitMapper.sparse_pauli_operators().

প্যারামিটার:: register_length (int) -- the register length for which to generate the table.
রিটার্নস:: A list of tuples in which the first and second Pauli operator the real and imaginary Pauli strings, respectively.
রিটার্ন টাইপ:: list[tuple[qiskit.quantum_info.operators.symplectic.pauli.Pauli, qiskit.quantum_info.operators.symplectic.pauli.Pauli]]

classmethod sparse_pauli_operators(register_length)#

Generates the cached SparsePauliOp terms.

This uses QubitMapper.pauli_table() to construct a list of operators used to translate the second-quantization symbols into qubit operators.

প্যারামিটার:: register_length (int) -- the register length for which to generate the operators.
রিটার্নস:: Two lists stored in a tuple, consisting of the creation and annihilation operators, applied on the individual modes.
রিটার্ন টাইপ:: tuple[list[qiskit.quantum_info.operators.symplectic.sparse_pauli_op.SparsePauliOp], list[qiskit.quantum_info.operators.symplectic.sparse_pauli_op.SparsePauliOp]]