Canonical Form and Finite Blocklength Bounds for Stabilizer Codes

First, a canonical form for stabilizer parity check matrices of arbitrary size and rank is derived. Next, it is shown that the closely related canonical form of the Clifford group can be computed in time $O(n^3)$ for $n$ qubits, which improves upon the previously known time $O(n^6)$. Finally, the related problem of finite blocklength bounds for stabilizer codes and Pauli noise is studied. A finite blocklength refinement of the hashing bound is derived, and it is shown that no argument that uses guessing the error as a substitute for guessing the coset can lead to a significantly better achievability bound.

💡 Research Summary

The paper makes two major contributions to the theory of quantum stabilizer codes. First, it introduces a canonical form for stabilizer parity‑check matrices of arbitrary size and rank. The construction is based on a variant of Gaussian elimination that operates on both rows and columns while preserving the symplectic inner product between rows at each step. To formalize the allowed elementary operations, the authors define a family of lower‑triangular matrix groups and show that these groups are generated by suitable Gaussian‑move matrices. By carefully tracking the symplectic constraints, they prove that every stabilizer parity‑check matrix can be transformed uniquely into a block‑diagonal canonical form. This form directly yields an explicit encoding circuit for the corresponding stabilizer code and also enables the generation of uniformly random parity‑check matrices of prescribed dimensions using the optimal number of random bits.

The second major result concerns the Clifford group. Existing algorithms compute a canonical Clifford element in two stages: an $O(n^3)$ “disentangling” stage followed by an $O(n^6)$ post‑processing stage. The authors observe that the disentangling stage already produces the desired canonical form when the Gaussian elimination is carried out with the symplectic constraints in mind. By providing a tighter analysis of this stage, they eliminate the need for the expensive second stage, thereby reducing the total runtime to $O(n^3)$. This improvement dramatically expands the practical range of qubit numbers for which the canonical Clifford form can be computed, facilitating tasks such as random Clifford sampling, circuit synthesis, and benchmarking.

The third part of the paper addresses finite‑blocklength performance bounds for stabilizer codes under Pauli noise. While the asymptotic hashing bound gives the achievable rate for large blocklengths, practical quantum communication systems operate with moderate $n$, and a refined analysis is needed. The authors define two families of quantities: $R_{\text{coset}}(p_{UV},\epsilon)$, the maximal rate achievable with a target error probability $\epsilon$ when decoding by coset identification, and $\epsilon_{\text{coset}}(p_{UV},r)$, the minimal error probability achievable at a fixed rate $r$. They then introduce a relaxed problem, $R_{\text{errguess}}$ and $\epsilon_{\text{errguess}}$, where the decoder guesses the full error vector rather than the stabilizer coset. By constructing explicit achievability bounds for independent qubit erasure and depolarizing channels, they show that these bounds can be evaluated efficiently (polynomial time) and that they constitute a finite‑blocklength refinement of the hashing bound.

A key theoretical insight is that any argument that replaces coset decoding by direct error guessing cannot substantially improve the achievability bound. Formally, they prove $\epsilon_{\text{coset}}(p_{UV},r)\le \epsilon_{\text{errguess}}(p_{UV},r)$ and $R_{\text{errguess}}(p_{UV},\epsilon)\le R_{\text{coset}}(p_{UV},\epsilon)$. This result justifies the widespread use of the error‑guessing relaxation in prior works (e.g., the finite‑blocklength bound of