The Jordan–Wigner Transform

Second quantization gave us a Hamiltonian written in creation and annihilation operators. To run it on a quantum computer we need those fermionic operators expressed as Pauli operators on qubits. The Jordan–Wigner (JW) transform is the original and most direct fermion-to-qubit mapping.

The occupation-to-qubit dictionary

JW makes the simplest possible identification: spin-orbital $p$ is represented by qubit $p$ , and the occupation number becomes the qubit's computational-basis label,

|n_0\, n_1\, \cdots n_{n-1}\rangle_{\text{fermion}} \;\longleftrightarrow\; |n_0\rangle \otimes |n_1\rangle \otimes \cdots \otimes |n_{n-1}\rangle_{\text{qubit}}.

An empty orbital is $|0\rangle$ , an occupied orbital is $|1\rangle$ . The number operator becomes $\hat{n}_p = a_p^\dagger a_p \mapsto \tfrac{1}{2}(I - Z_p)$ , which reads off the qubit: it returns $0$ for $|0\rangle$ and $1$ for $|1\rangle$ .

Encoding the fermionic phase

The hard part is the antisymmetry sign $(-1)^{\sigma_p}$ , where $\sigma_p$ counts occupied orbitals before $p$ . A bare qubit flip on site $p$ cannot know about the occupation of qubits $0,\dots,p-1$ . JW supplies that knowledge with a parity string of $Z$ operators:

a_p = \left(\prod_{q<p} Z_q\right) \frac{X_p + i Y_p}{2}, \qquad a_p^\dagger = \left(\prod_{q<p} Z_q\right) \frac{X_p - i Y_p}{2}.

The local operator $\tfrac{1}{2}(X_p \mp i Y_p)$ raises or lowers qubit $p$ , while the leading $Z_0 Z_1 \cdots Z_{p-1}$ multiplies in exactly the $\pm 1$ phase demanded by the anticommutation relations. You can verify that with these definitions $\{a_p, a_q^\dagger\} = \delta_{pq}$ holds.

A worked single term

Consider the one-body hopping term $a_1^\dagger a_0 + a_0^\dagger a_1$ on two qubits. Substituting the JW definitions and simplifying (the $Z_0$ strings on the two adjacent sites combine) gives the clean two-qubit Pauli form

a_1^\dagger a_0 + a_0^\dagger a_1 = \tfrac{1}{2}\left(X_0 X_1 + Y_0 Y_1\right).

This is the standard "XY" hopping interaction. Every term of a molecular Hamiltonian reduces, after JW, to a weighted sum of Pauli strings like this one — which is precisely the input VQE needs.

Preparing occupation states is easy

Because occupation maps directly to qubit values, preparing a specific occupation string costs only $X$ gates: flip qubit $p$ to $|1\rangle$ for each occupied orbital $p$ . The parity-string signs do not change which basis state results when you build it from the vacuum in increasing-index order; they only matter when operators act within the Hamiltonian. This is exactly how the Hartree–Fock reference state (lesson 6) is prepared.

Try it

On 4 spin-orbitals (qubits 0–3), prepare the occupation state with orbitals 0 and 2 filled, i.e. apply $a_2^\dagger a_0^\dagger$ to the vacuum. Under Jordan–Wigner this is just the computational basis state with qubits 0 and 2 set to $|1\rangle$ . The grader checks the full statevector.

Run your code to see the quantum state.

After running, the statevector panel should show a single non-zero amplitude at the basis state 1010-pattern (qubits 0 and 2 high), confirming the JW occupation encoding.