qml.jordan_wigner

jordan_wigner(fermi_operator, **kwargs)

Convert a fermionic operator to a qubit operator using the Jordan-Wigner mapping.

The fermionic creation and annihilation operators are mapped to the Pauli operators as

\[a^{\dagger}_0 = \left (\frac{X_0 - iY_0}{2} \right ), \:\: \text{...,} \:\: a^{\dagger}_n = Z_0 \otimes Z_1 \otimes ... \otimes Z_{n-1} \otimes \left (\frac{X_n - iY_n}{2} \right ),\]

and

\[a_0 = \left (\frac{X_0 + iY_0}{2} \right ), \:\: \text{...,} \:\: a_n = Z_0 \otimes Z_1 \otimes ... \otimes Z_{n-1} \otimes \left (\frac{X_n + iY_n}{2} \right ),\]

where \(X\), \(Y\), and \(Z\) are the Pauli operators.

Parameters

• fermi_operator (FermiWord, FermiSentence) – the fermionic operator

• ps (bool) – whether to return the result as a PauliSentence instead of an Operator. Defaults to False.

• wire_map (dict) – a dictionary defining how to map the oribitals of the Fermi operator to qubit wires. If None, the integers used to order the orbitals will be used as wire labels. Defaults to None.

Returns

a linear combination of qubit operators

Return type

Union[PauliSentence, Operator]

Example

>>> w = FermiWord({(0, 0) : '+', (1, 1) : '-'})
>>> jordan_wigner(w)
(-0.25j*(PauliY(wires=[0]) @ PauliX(wires=[1]))) + ((0.25+0j)*(PauliY(wires=[0]) @ PauliY(wires=[1]))) +
((0.25+0j)*(PauliX(wires=[0]) @ PauliX(wires=[1]))) + (0.25j*(PauliX(wires=[0]) @ PauliY(wires=[1])))

>>> jordan_wigner(w, ps=True)
-0.25j * Y(0) @ X(1)
+ (0.25+0j) * Y(0) @ Y(1)
+ (0.25+0j) * X(0) @ X(1)
+ 0.25j * X(0) @ Y(1)

>>> jordan_wigner(w, ps=True, wire_map={0: 2, 1: 3})
-0.25j * Y(2) @ X(3)
+ (0.25+0j) * Y(2) @ Y(3)
+ (0.25+0j) * X(2) @ X(3)
+ 0.25j * X(2) @ Y(3)

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