Quantum Hamming bound (${\sum }_{j=0}^t{3}^j\left(\genfrac{}{}{0ex}{}{n}{j}\right)\le 2^{n-k}$)

For unknown-location Pauli operators of weight at most $t$, there are $N(t)={\sum }_{j=0}^t{3}^j\left(\genfrac{}{}{0ex}{}{n}{j}\right)$ physically distinct errors modulo phase, including the identity. A nondegenerate quantum code assigns them mutually orthogonal error-image subspaces, each of dimension $2^k$, within the $2^n$-dimensional physical Hilbert space. Hence $N(t)2^k\le 2^n$. The hypothesis of nondegeneracy is essential to this dimension count.

A nondegenerate quantum code is perfect at correction radius $t$ if its distinct correctable Pauli error images exhaust the physical Hilbert space. Each image has dimension $2^k$, and their orthogonal direct sum is the whole $2^n$-dimensional space precisely when the quantum Hamming bound is an equality. This perfectness concerns unknown-location errors of weight at most $t$; it does not imply correction of every higher-weight error.