qrunch.quantum.operators.mappers.jordan_wigner.mapper

Module containing the logic for implementing the Jordan-Wigner transformation.

Classes

JordanWignerMapper

Class for mapping operators in second quantization space into operators in Pauli space.
class JordanWignerMapper

Bases: object

Class for mapping operators in second quantization space into operators in Pauli space.

The Jordan-Wigner transformation is a method used to map fermionic creation and annihilation operators onto Pauli strings, enabling the representation of fermionic systems in a qubit-based quantum computing framework.

The transformation is defined as follows:

  • A fermionic creation operator \(\hat{a}^\dagger_j\) acting on the \(j\)-th mode

    is mapped to a Pauli string:

    \[\hat{a}^\dagger_j \rightarrow \frac{1}{2} \left( X_j - i Y_j \right) \prod_{k=1}^{j-1} Z_k\]

  • A fermionic annihilation operator \(\hat{a}_j\) acting on the \(j\)-th mode is mapped to a Pauli string:

    \[\hat{a}_j \rightarrow \frac{1}{2} \left( X_j + i Y_j \right) \prod_{k=1}^{j-1} Z_k\]

These transformations allow for the simulation of fermionic systems by qubit-based quantum computers, making the Jordan-Wigner transformation a crucial tool in quantum computational chemistry and physics.

map(operator: FermionHermitianSumProtocol | PairedHardcoreBosonHermitianSumProtocol, tolerance: float = 1e-16) → HermitianPauliSum
map(operator: Expression[FermionOperators] | Expression[PairedHardcoreBosonOperators] | int | float | complex, tolerance: float = 1e-16) → Expression[PauliOperators] | int | float | complex

Map an operator involving second quantization operators into one using Pauli strings.

An expression as input will return an expression, which allows for non-hermitian operators. A SecondQuantizationHamiltonian will map to a HermitianPauliSum directly.

Removes terms with coefficient less than the tolerance.

Parameters:
Returns:

The simplified Pauli string or collection of Pauli strings given by the Jordan-Wigner transformation of the expression.

Return type:

HermitianPauliSum | Expression[PauliOperators] | int | float | complex