Your First Encoding

Now we arrive at FockMap’s marquee feature: fermion-to-qubit encoding.

The problem: quantum computers have qubits (Pauli operators), but chemistry uses fermions (ladder operators). We need a mapping between the two worlds.

Jordan-Wigner: the classic

The simplest encoding is Jordan-Wigner (1928). It maps a ladder operator to Pauli strings by inserting a chain of Z operators on all preceding qubits:

open Encodings
open Encodings.JordanWigner
// Encode a†₂ (creation on mode 2) for a 4-qubit system:
let result = jordanWignerTerms Raise 2u 4u

// Print the Pauli terms:
for term in result.DistributeCoefficient.SummandTerms do
    printfn "%s %s" term.PhasePrefix term.Signature
//  0.5 ZZXI
// -0.5i ZZYI

The result is $a^\dagger_2 = \tfrac{1}{2}(ZZXI) - \tfrac{i}{2}(ZZYI)$.

Notice the two Z operators before the X/Y — they track the parity of electrons in modes 0 and 1. This is how the encoding preserves fermionic anti-symmetry on a qubit register.

Ordering convention. In a PauliRegister string, character position i is mode/qubit i, so mode 0 is the leftmost character ("ZZXI" = Z₀ Z₁ X₂ I₃). Occupation integers weight mode j by 2ʲ (mode 0 = least-significant bit), so the H₂ Hartree–Fock state |modes 0,1 occupied⟩ is the integer 3 = 0b0011. To read a string in that occupation basis, or to produce a Qiskit-style Pauli-label string, reverse it (mode 0 becomes rightmost / bit 2⁰). OpenQASM gates use explicit indexed operands, so map qubit i to q[i] rather than relying on a string-order convention.

The Z-chain problem

Watch what happens as you encode higher modes:

for j in 0u .. 7u do
    let terms = jordanWignerTerms Raise j 8u
    let weight =
        terms.DistributeCoefficient.SummandTerms.[0].Signature
        |> Seq.filter (fun c -> c <> 'I')
        |> Seq.length
    printfn "a†_%d → weight %d" j weight

// a†_0 → weight 1
// a†_1 → weight 2
// a†_2 → weight 3
// ...
// a†_7 → weight 8

The Pauli weight grows linearly — $O(n)$. For a 100-qubit molecule, the last operator touches all 100 qubits. This is expensive to measure on real hardware, and it’s why we need better encodings.

Next: Six Encodings, One Interface — comparing all available encodings