Second Quantization
The operator formalism that expresses many-fermion Hamiltonians in terms of creation and annihilation operators, the starting point for qubit encodings.
From wavefunctions to operators
First quantization tracks the coordinates of each particle explicitly and enforces antisymmetry by hand through Slater determinants. Second quantization instead describes states by orbital occupation numbers and builds them with creation operators a_p^dagger and annihilation operators a_p that add or remove an electron in orbital p. Antisymmetry is baked into the operators' algebra.
The fermionic anticommutation relations
Fermionic operators obey anticommutation relations: {a_p, a_q^dagger} = delta_pq and {a_p, a_q} = 0. The anticommutators, rather than commutators, enforce the Pauli exclusion principle: applying a_p^dagger twice gives zero, so no orbital holds two identical fermions.
The molecular Hamiltonian
In this language the electronic Hamiltonian is H = sum over p,q of h_pq a_p^dagger a_q + (1/2) sum over p,q,r,s of h_pqrs a_p^dagger a_q^dagger a_r a_s. The one-body coefficients h_pq encode kinetic energy and nuclear attraction; the two-body h_pqrs encode electron-electron repulsion. Both are integrals computed classically from the chosen orbitals.
- a_p^dagger creates an electron in orbital p; a_p removes one.
- Number operator n_p = a_p^dagger a_p counts occupation (0 or 1).
- The vacuum |0> has no electrons; determinants are products of a^dagger on |0>.
- One-body terms hop electrons; two-body terms scatter pairs.
Why it precedes qubit encoding
Quantum computers store qubits, not fermions. To simulate H you must map the fermionic operators onto qubit operators (Pauli strings) that reproduce the anticommutation relations. That mapping, Jordan-Wigner, Bravyi-Kitaev, or parity, is the bridge from second quantization to a runnable circuit. Second quantization is therefore the canonical intermediate representation: chemistry integrals go in, a fermionic Hamiltonian comes out, and an encoding turns it into qubit gates.