Second Quantization

Writing the electronic Hamiltonian directly in terms of electron positions forces us to drag the antisymmetry of the wavefunction around by hand: swapping any two electrons must flip the sign of $\Psi$ . Second quantization builds that antisymmetry into the operators themselves, giving a notation in which the Hamiltonian becomes a compact sum over orbitals — and, crucially, a form ready to be mapped onto qubits.

Occupation-number states

Fix a finite basis of $n$ spin-orbitals $\{\phi_0, \phi_1, \dots, \phi_{n-1}\}$ (each spatial orbital splits into a spin-up and a spin-down spin-orbital). Instead of tracking which electron sits in which orbital, we track only how many electrons occupy each orbital. Because electrons are fermions, each spin-orbital holds at most one (the Pauli exclusion principle), so an occupation number is either 0 or 1. A many-electron state is a string of occupation numbers:

|n_0\, n_1\, \cdots\, n_{n-1}\rangle, \qquad n_p \in \{0, 1\}.

This is already strikingly qubit-like: $n$ spin-orbitals give $2^n$ occupation strings, exactly the dimension of an $n$ -qubit register. That correspondence is what we exploit in the next lessons.

Creation and annihilation operators

For each spin-orbital $p$ define a creation operator $a_p^\dagger$ that adds an electron and an annihilation operator $a_p$ that removes one:

a_p^\dagger |\cdots 0_p \cdots\rangle = (-1)^{\sigma_p}\,|\cdots 1_p \cdots\rangle, \qquad a_p |\cdots 1_p \cdots\rangle = (-1)^{\sigma_p}\,|\cdots 0_p \cdots\rangle,

and they annihilate the state ( $\to 0$ ) if the orbital is already occupied (for $a_p^\dagger$ ) or already empty (for $a_p$ ). The sign $(-1)^{\sigma_p}$ , with $\sigma_p = \sum_{q<p} n_q$ counting the electrons in earlier orbitals, is the fermionic phase that enforces antisymmetry.

The anticommutation relations

The defining algebra of fermionic operators is the set of canonical anticommutation relations. Using $\{A, B\} = AB + BA$ :

\{a_p, a_q^\dagger\} = \delta_{pq}, \qquad \{a_p, a_q\} = 0, \qquad \{a_p^\dagger, a_q^\dagger\} = 0.

The last two relations encode antisymmetry: setting $p = q$ gives $a_p^\dagger a_p^\dagger = 0$ , which is the Pauli exclusion principle (you cannot create two electrons in the same spin-orbital). The operator $\hat{n}_p = a_p^\dagger a_p$ is the number operator; it returns the occupation $n_p \in \{0,1\}$ of orbital $p$ .

The Hamiltonian in second quantization

Projecting the electronic Hamiltonian onto the chosen orbital basis turns the differential operator into a sum of one-body and two-body terms with numerical coefficients:

\hat{H} = \sum_{p,q} h_{pq}\, a_p^\dagger a_q + \frac{1}{2} \sum_{p,q,r,s} h_{pqrs}\, a_p^\dagger a_q^\dagger a_r a_s.

The coefficients are molecular integrals computed once on a classical computer from the chosen basis set and nuclear geometry:

h_{pq} = \int \phi_p^*(\mathbf{x})\left(-\tfrac{1}{2}\nabla^2 - \sum_A \tfrac{Z_A}{|\mathbf{r}-\mathbf{R}_A|}\right)\phi_q(\mathbf{x})\, d\mathbf{x},

h_{pqrs} = \int \frac{\phi_p^*(\mathbf{x}_1)\phi_q^*(\mathbf{x}_2)\phi_r(\mathbf{x}_2)\phi_s(\mathbf{x}_1)}{|\mathbf{r}_1 - \mathbf{r}_2|}\, d\mathbf{x}_1\, d\mathbf{x}_2.

The one-body integrals $h_{pq}$ capture kinetic energy and electron–nucleus attraction; the two-body integrals $h_{pqrs}$ capture electron–electron repulsion. Tools such as PySCF or Psi4 produce them; on a quantum computer we treat them as known constants.

Why this is the right starting point

Three features make second quantization the launch pad for quantum simulation:

Fixed operator count. The Hamiltonian has at most $O(n^4)$ terms regardless of how many electrons are present, because antisymmetry is handled by the operators, not by enumerating configurations.
Qubit-shaped states. Occupation strings map one-to-one onto computational basis states of an $n$ -qubit register.
Algebraic, not analytic. Everything reduces to the anticommutation relations and a table of integrals — no wavefunction calculus survives into the quantum circuit.

In the next lesson we make the qubit correspondence precise with the Jordan–Wigner transform, which turns $a_p^\dagger$ and $a_p$ into concrete Pauli operators your simulator can run.