Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 342 / 2

A recovery operation must reverse every coherent superposition of the errors ${\tilde{E}}_a$ without learning or disturbing the encoded state. Thus the corrupted subspaces associated with distinguishable syndromes must be orthogonal, while errors with the same syndrome must have identical action on the logical information. For any code states $|\psi ⟩,|\varphi ⟩$, this meanswith coefficients independent of the encoded states. In projector form these are exactly the Knill--Laflamme conditionsDiagonalizing the positive matrix $c_{ab}$ chooses error combinations with mutually orthogonal syndrome spaces, which can be measured and reversed without revealing logical amplitudes.

The centralizer of a stabilizer group isThe nontrivial logical Pauli operators are represented byor more invariantly by the quotient $C(\mathcal{S})/\mathcal{S}$ after phases are removed.

For $P=E_a^{†}{E}_b$, there are three cases. If $P\in \mathcal{S}$, it acts as a scalar on the code. If $P\notin C(\mathcal{S})$, it anticommutes with a stabilizer and ${\Pi }_{\mathcal{L}}P{\Pi }_{\mathcal{L}}=0$. If $P\in C(\mathcal{S})\setminus \mathcal{S}$, it acts as a nontrivial logical operator and is not proportional to the identity. Therefore

If every $\mathrm{wt}(E_a)<d/2$, thenBy the definition of the distance of a stabilizer code, no Pauli operator of weight below $d$ is a nontrivial logical operator. Part (b) therefore proves the local correctability of a stabilizer code for this error set.

The converse fails because pairwise products, rather than individual weights, control correctability. For example, let $P$ be a high-weight Pauli outside $C(\mathcal{S})$ and take the error set $\{I,P\}$. If $P$ anticommutes with a stabilizer, then $\Pi P\Pi =0$, while $P^{†}P=I$; the KL conditions hold even when $\mathrm{wt}(P)\ge d/2$. A still simpler singleton set containing any known unitary Pauli error is always reversible regardless of its weight.

For Pauli errors, “suitably local” means that every pairwise product has support too small to carry a logical string:A sufficient geometric statement is $|\mathrm{supp}{E}_a\cup \mathrm{supp}{E}_b|<d$; in particular, each error having weight below $d/2$ suffices. Such a product is either a stabilizer or creates an excitation detected by at least one stabilizer. It is never a nontrivial logical operator, so the ground-space projector of the surface code obeys the Knill--Laflamme condition for the entire set.

For sufficiently weak local fields the bulk gap remains open, so quasi-adiabatic continuation supplies a quasi-local unitary $U$ mapping the unperturbed ground space to the span of the $2^k$ lowest eigenstates:If $S_j$, $j=1,\dots ,n-k$, are independent original stabilizer generators, defineThen ${\tilde{S}}_j|{\phi }_{\alpha }⟩=|{\phi }_{\alpha }⟩$, and independence is preserved by conjugation. The operators ${\tilde{S}}_j$ are quasilocal, with exponentially decaying tails, but a generic perturbation makes them non-Pauli; this is the dressed stabilizer under a weak local perturbation.

Exact error correction is transported with the code: the exactly correctable dressed errors are $U{E}_a{U}^{†}$. A bare Pauli error is generally not one of these dressed operators. Its expansion in the dressed algebra has exponentially small long-range components that can act within the logical space, soThus a set of bare Pauli errors satisfies the KL conditions only approximately, with deviations suppressed exponentially by the code distance relative to the dressing length, except at specially tuned points.