majorana_jordan_wigner

majorana_jordan_wigner(op, num_qubits)

Map a MajoranaOperator to a SparseObservable under the Jordan-Wigner transformation. [1]

Majorana mode \(m\) acts on the fermionic mode \(\lfloor m/2 \rfloor\), which is mapped to the qubit of the same index. This follows Qiskit’s little-endian qubit ordering.

Parameters:

  • op – the Majorana operator to map.

  • num_qubits – the number of qubits for the resulting qubit operator. Note that this is counted in fermionic modes, so it must be strictly greater than the largest Majorana index in op divided by two (any additional qubits are padded with the identity).

Returns:

The mapped qubit operator. The result is not guaranteed to be fully simplified; call simplify() to combine any remaining duplicate terms. Duplicates are merged as the result is assembled, to bound the memory required, so the exact number of terms returned may vary with the number of threads used. This does not affect the operator that the result represents.

Raises:

ValueError – if num_qubits is too small to hold the operator’s support, i.e. if it is not larger than the largest fermionic mode acted upon by op.

Definition

With the MajoranaOperator convention that even indices carry \(\gamma_j = a^\dagger_j + a_j\) and odd ones \(\gamma'_j = i(a^\dagger_j - a_j)\), a single Majorana operator maps onto a single Pauli string,

\[\gamma_j \rightarrow \bigotimes_{k\lt j} \sigma^Z_k \otimes \sigma^X_j ~~\text{and}~~ \gamma'_j \rightarrow \bigotimes_{k\lt j} \sigma^Z_k \otimes \sigma^Y_j \, .\]

This is also what makes it cheaper than converting to a FermionOperator first and calling fermion_jordan_wigner(): each fermionic action maps onto a two-term sum, so that route inflates a single Pauli string into up to \(4^L\) terms for a term built from \(L\) Majorana operators, before merging them back down. The saving therefore grows with the length of the terms; for single-operator terms there is no blowup to avoid and the two routes cost about the same.

Usage

>>> from qiskit_fermions.mappers.library import majorana_jordan_wigner
>>> from qiskit_fermions.operators import MajoranaOperator, gamma
>>> mop = MajoranaOperator.from_dict(
...     {
...         (gamma(0, False), gamma(1, False)): 0.5,
...         (gamma(0, True), gamma(1, True)): 0.5,
...     }
... )
>>> majorana_jordan_wigner(mop, 2).simplify()
<SparseObservable with 2 terms on 2 qubits: (0+0.5j)(Y_1 X_0) + (0-0.5j)(X_1 Y_0)>