Jordan–Wigner number mapping

Why the number operator is the cheap one. Under Jordan–Wigner a single creation or annihilation operator drags a string of $Z$s across every mode below it, which is the whole reason the encoding costs $O(m)$ per fermionic gate. The number operator does not, and the cancellation is worth seeing rather than asserting.

Bravyi and Kitaev give the Majorana operators at Eq. (24),

$c_{2k} = a_k + a^\dagger_k = \sigma^x[k]\prod_{j<k}\sigma^z[j], \qquad c_{2k+1} = \frac{a_k - a^\dagger_k}{i} = \sigma^y[k]\prod_{j<k}\sigma^z[j],$

and state separately, in §8, the identity $B_k = -i\,c_{2k}c_{2k+1} = 1 - 2a^\dagger_k a_k$. Multiplying the two expressions in Eq. (24), the $Z$ strings are identical and square to the identity, so they cancel and leave $c_{2k}c_{2k+1} = \sigma^x\sigma^y[k] = i\sigma^z[k]$. Substituting into the §8 identity gives $1 - 2n_k = \sigma^z[k]$, that is

$n_p \;\mapsto\; \frac{I - Z_p}{2}.$

One product, no string. The strings cancel because both Majorana operators for mode $k$ carry the same prefix. That is why occupation is locally measurable under an encoding whose defining feature is nonlocality, and it is why a Hamiltonian written only in number operators — the Coulomb-repulsion diagonal of a Hubbard model, say — maps to a diagonal Pauli sum with no string cost at all.

What this record deliberately does not cite. Jordan and Wigner's 1928 Über das Paulische Äquivalenzverbot is where the transformation comes from, and it is not the source of the statement above, because it was not read. Its full text is behind a publisher paywall; a free scan of the whole 1928 volume exists but is in German and reached only through imperfect OCR of a 924-page file. Naming it as the origin costs nothing and claims nothing. Citing it for a qubit-form identity nobody here has read would be the failure this catalog most wants to avoid, and the identity is available from a paper that is free, in English, and was read end to end.