10 Quantum Hamming Bound
10.1 Overview
This chapter proves the quantum Hamming bound for non-degenerate qubit stabilizer codes. Working in the Pauli-string formalism, it shows that if a quantum code \(C \le \mathcal{H}_n\) satisfies the Knill–Laflamme conditions for \(t\)-error correction and is non-degenerate, then
10.2 Pauli strings
The four Pauli basis elements \(\{ I,X,Y,Z\} \), represented as an inductive type.
A Pauli string of length \(n\) is a function \(p : \mathrm{Fin}\, n \to \mathrm{PauliBasis}\).
The support \(\mathrm{supp}(p) \subseteq \mathrm{Fin}\, n\) consists of coordinates where \(p(i) \neq I\); the weight is \(\mathrm{wt}(p) = |\mathrm{supp}(p)|\).
The set of all Pauli strings of weight at most \(t\):
10.3 \(n\)-qubit Hilbert space
The \(n\)-qubit Hilbert space \(\mathcal{H}_n = \ell ^2\! \bigl(\{ 0,1\} ^n,\mathbb {C}\bigr)\), implemented as .
\(\dim _{\mathbb {C}}(\mathcal{H}_n) = 2^n\).
For \(p \in \mathrm{PauliString}\, n\), the associated Pauli operator \(\hat{p} : \mathcal{H}_n \to \mathcal{H}_n\).
10.4 Knill–Laflamme conditions
A subspace \(C \le \mathcal{H}_n\) satisfies the Knill–Laflamme condition for \(t\)-error correction if for all Pauli strings \(E,F\) with \(\mathrm{wt}(E),\mathrm{wt}(F) \le t\) there exists \(\lambda _{EF}\in \mathbb {C}\) such that \(P_C\, E^\dagger F\, P_C = \lambda _{EF}\, P_C\), where \(P_C\) is the orthogonal projection onto \(C\).
A code is non-degenerate if it satisfies the Knill–Laflamme condition and additionally \(P_C E^\dagger F P_C = 0\) whenever \(E \neq F\).
10.5 Error sphere and sphere-packing
The error sphere \(\mathrm{ES}(C,t)\) is the subspace \(\bigvee _{\mathrm{wt}(p)\le t} \hat{p}(C)\), i.e. the supremum of the Pauli images of \(C\) over all \(t\)-errors.
If \(C\) is non-degenerate and \(E \neq F\) both have weight \(\le t\), then \(\hat{E}(C) \perp \hat{F}(C)\).
If \(C\) is non-degenerate,
10.6 Quantum Hamming bound
If \(C \le \mathcal{H}_n\) is non-degenerate, then