Stabilizer Codes

3.1 Checks as a definition of a subspace

3.1.1 Counting what commuting constraints leave free

A Pauli string is a tensor product of I,X,Y,Z, together with a phase. Two Pauli strings commute or anticommute. At a position where their nonidentity factors are different, exchanging them contributes a minus sign. Count such positions: an even number gives commutation, an odd number gives anticommutation.

Choose independent commuting Hermitian Pauli strings g₁,…,gᵣ whose generated group does not contain −I. Define the code by gⱼ|ψ⟩=|ψ⟩ for every j. The generated group S is its stabilizer. Anticommuting checks cannot share a nonzero eigenvector: if X and Z both fixed |ψ⟩, then XZ|ψ⟩ would equal |ψ⟩, while ZX=−XZ would force it to equal −|ψ⟩.

The projector onto the code is \(P=\prod_{j=1}^r(I+g_j)/2\). Each factor selects a +1 eigenspace, and commutation makes their product a projector. Expanding the product yields \(P=2^{-r}\sum_{g\in S}g\). Every nonidentity Pauli string has trace zero; independence and exclusion of −I leave one identity term. Thus \(\tr P=2^{n-r}\): the code contains k=n−r logical qubits.

This trace argument explains why independence matters. Adding a redundant generator does not halve the dimension again. For the repetition code, Z₁Z₃ is the product of the two existing checks; it supplies no new independent constraint. Keeping track of algebraic dependencies is essential when we later count checks on a surface.

3.2 Reading an error through its commutators

3.2.1 Syndromes without state-vector simulation

If |ψ⟩ is in the code and E is a Pauli error, then gⱼE|ψ⟩ equals either +E|ψ⟩ or −E|ψ⟩. The sign depends only on whether E commutes or anticommutes with gⱼ. It does not depend on the logical state. Our syndrome bit sⱼ is zero in the commuting case and one in the anticommuting case.

This makes a large calculation surprisingly small. Instead of applying E to every possible encoded superposition, compare E with the generators. For the repetition code, X₂ anticommutes with both Z₁Z₂ and Z₂Z₃, giving syndrome 11 immediately. Y₂ has the same syndrome, because its X component produces the same parity changes. The code cannot distinguish the accompanying phase error, which is exactly its weakness.

For automated calculations, ignore an overall Pauli phase and represent a string by binary vectors (x|z): I maps to (0,0), X to (1,0), Z to (0,1), and Y to (1,1). The commutation sign of two strings is

with dot products modulo two. This binary representation records commutation, not the full multiplication phase. If a calculation needs the sign of a product as an operator, it must track that phase separately. The simplification is useful precisely because syndrome calculations depend only on commutation.

3.3 A code that handles every single-qubit Pauli

3.3.1 The five-qubit construction

Consider five qubits with generators

Check a pair, such as g₁ and g₂. They have two positions with different nonidentity factors, hence commute. The same count works for the other pairs; the four generators are independent. Their common +1 space therefore encodes one logical qubit.

Now compute single-qubit syndromes in generator order. X₁ anticommutes only with g₄, giving 0001. Z₁ anticommutes with g₁ and g₃, giving 1010. Y₁ anticommutes wherever exactly one of X₁ and Z₁ does, giving 1011. Repeating this for all five positions gives fifteen distinct nonzero four-bit syndromes. They exhaust the fifteen available nonzero strings.

Each single-qubit Pauli therefore has a distinct syndrome, separate from identity. The products of distinct members of {I and all single-qubit Paulis} have nonzero syndrome and compress to zero on the code. Their diagonal products are identity. The Knill–Laflamme test consequently proves correction of that error span, including arbitrary noise on one unknown physical position.

The code is conventionally written [[5,1,3]]: five physical qubits, one logical qubit, and distance three. The next chapters explain the distance entry and how logical operations hide among strings with zero syndrome. The stabilizer formalism gives a systematic language for these constructions; here the syndrome calculation supplies a directly checkable certificate for the single-error claim.