MajoranaMapping

The MajoranaMapping class defines a fermion-to-qubit encoding. It exposes the bilinear \(i\,\gamma_j\,\gamma_k\) as the unified primitive available across every encoding, and (for Majorana-atomic encodings) individual Majorana operators \(\gamma_k\) as an additional capability. It follows QDK/Chemistry’s data-container conventions for immutable fermion-to-qubit encoding data.

Overview

Fermion-to-qubit mappings transform fermionic creation and annihilation operators into qubit (Pauli) operators. The MajoranaMapping class encapsulates such an encoding as data, making the QubitMapper algorithm encoding-agnostic: the mapper receives the encoding as data rather than selecting it internally.

Bilinears as the unified primitive

Every fermion-to-qubit encoding can express the bilinear product \(i\,\gamma_j\,\gamma_k\) as a Pauli string. This makes the bilinear the most general building block that all encodings share.

Why bilinears? Physical (parity-conserving) fermionic operators can always be written as products of bilinears, so any Hamiltonian can be mapped through bilinears alone — even if the encoding does not assign a Pauli image to individual Majorana operators \(\gamma_k\).

For the formal foundations, see Bravyi and Kitaev (2002), which develops the Majorana-operator perspective on fermion-to-qubit encodings.

Individual Majorana operators \(\gamma_k\) have a Pauli image only in Majorana-atomic encodings (e.g. Jordan-Wigner, Bravyi-Kitaev, Parity). In bilinear-only encodings — where the number of qubits exceeds the number of fermionic modes — single Majoranas have no representation in the physical subspace; only the bilinears are observable.

The MajoranaMapping supports both forms:

bilinear() — the unified primitive available on every encoding.
majorana() — the additional capability provided by Majorana-atomic encodings; gated by is_majorana_atomic.

Majorana-atomic mappings are constructed from a Pauli-string table (the constructor or factory methods). Bilinear-only mappings are constructed via from_bilinears().

Convention

num_modes: The number of fermionic modes (spin-orbitals) in the system.

Pauli strings use little-endian qubit ordering, consistent with the rest of QDK/Chemistry’s PauliOperator layer. majorana() and bilinear() return Pauli strings in the encoding’s native (pre-taper) qubit basis.

Built-in encodings

Factory methods construct standard encodings for a given number of modes. Each returns a MajoranaMapping with the appropriate Pauli-string table and a descriptive name.

Jordan-Wigner

from qdk_chemistry.data import MajoranaMapping

mapping = MajoranaMapping.jordan_wigner(num_modes=12)

# Bilinear (works on every encoding):
coeff, pauli_str = mapping.bilinear(0, 1)

# Single Majorana (Majorana-atomic encodings only):
if mapping.is_majorana_atomic:
    gamma_0 = mapping.majorana(0)

Encodes each fermionic mode in a single qubit. See Jordan-Wigner Jordan-Wigner1928 for a description of the encoding.

Bravyi-Kitaev

mapping = MajoranaMapping.bravyi_kitaev(num_modes=12)

Uses a binary-tree structure to reduce average Pauli-string weight. See Bravyi-Kitaev Seeley2012 for a description of the encoding.

Parity

mapping = MajoranaMapping.parity(num_modes=12)

Encodes cumulative electron-number parities. See Parity Seeley2012 for a description of the encoding.

Custom encodings

A custom Majorana-atomic encoding can be defined by providing a sparse Pauli-word table directly:

from qdk_chemistry.data import MajoranaMapping

# Provide one sparse Pauli word per Majorana operator.
# Entries are (qubit_index, operator_code), with X=1, Y=2, Z=3.
mapping = MajoranaMapping.from_table([...], name="my-custom-encoding")

Serialization

MajoranaMapping supports the same serialization formats as other QDK/Chemistry data classes:

from qdk_chemistry.data import MajoranaMapping

mapping = MajoranaMapping.jordan_wigner(num_modes=12)

# JSON round-trip
json_str = mapping.to_json()
restored = MajoranaMapping.from_json(json_str)

# HDF5 round-trip
mapping.to_hdf5_file("mapping.h5")
restored = MajoranaMapping.from_hdf5_file("mapping.h5")