Jordan-Wigner Fermion-to-Qubit Encoding

Molecular Register

Each sphere is one spin-orbital, Jordan-Wigner-mapped onto one qubit. Bright = occupied (n=1), dark = empty (n=0). Click a sphere to toggle it directly.

Spin-orbitals (qubits) N 6

Single Excitation a†_pa_q

Create at p (empty) Annihilate at q (occ.)

|n₊₋₁…n₀⟩ = |000000⟩

Electrons Σn_k

Parity ⟩Z⊗N⟨

JW string length

Last sign factor

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How it works

Fermionic creation/annihilation operators must anti-commute (Pauli exclusion), but qubit operators on different qubits naturally commute. The Jordan-Wigner transform fixes this by attaching a "parity string" of Z gates to every fermionic operator:

a†_p = ( ⊗_{k<p} Z_k ) ⊗ σ⁺_p ,   σ⁺ = (X - iY)/2
a_p  = ( ⊗_{k<p} Z_k ) ⊗ σ⁻_p ,   σ⁻ = (X + iY)/2
Parity:  P = Z_0 Z_1 … Z_{N-1} = (-1)^(Σ n_k)

Applying a single excitation a†_pa_q (move one electron from occupied orbital q into empty orbital p) needs the strings of both operators; the Z factors on qubits strictly between p and q survive and the ones coincident with p, q collapse into the ladder operators. The net result on a computational-basis occupation state is:

a†_p a_q |...n_q...n_p...⟩ = 0                      if n_q=0 or n_p=1
                       = (-1)^(Σ_{min<k<max} n_k) |...n_q→0...n_p→1...⟩   otherwise

That alternating sign — one minus sign per occupied orbital sitting between p and q — is exactly the fermionic exchange phase, and it is why every real quantum-chemistry circuit (VQE/UCCSD ansatze, quantum phase estimation on a molecular Hamiltonian) needs a ladder of CNOTs or a literal Z-string threaded between the two qubits it excites between, not just a two-qubit gate on p and q alone. This simulator computes that sign exactly from the current occupation and animates the string.

N slider — size of the spin-orbital / qubit register (a minimal active space, e.g. N=4 for H₂ in a minimal STO-3G basis: 2 bonding + 2 antibonding spin-orbitals).
Hartree-Fock ref. — fills the lowest N/2 orbitals, the standard starting reference determinant.
p / q selectors + Apply excitation — runs a†_pa_q on the current state; the highlighted arc is the Jordan-Wigner Z-string, and the readouts show its length and the resulting sign.
Click a sphere — toggle any orbital's occupation directly to explore parity and string length for any bitstring.

Occupied spin-orbital (n=1) Empty spin-orbital (n=0) Active Jordan-Wigner Z-string

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