With the advent of physical qubits exhibiting strong noise bias, it becomes increasingly relevant to identify which quantum gates can be efficiently implemented on error-correcting s designed to address a single dominant error type. Here, we consider $[n,k,d]$-classical stabilizer s addressing bit-flip errors where $n$, $k$, and $d$ are the numbers of physical and logical qubits, and the code distance, respectively. We prove that operations essential for achieving a universal logical gate set necessarily require complex unitary circuits to be implemented. Specifically, these implementation circuits either consists of $h$ layers of $r$-transversal operations on $c$ code blocks such that $c^{h-1}{r}^h\ge d$ or of $h$ gates, each operating on at most $r$ physical qubits on the same code block, such that $hr\ge d$. Similar constraints apply not only to classical codes designed to correct phase-flip errors, but also to quantum stabilizer codes tailored to biased noise. This motivates a closer examination of alternative logical gate constructions using, e.g., magic state distillation and cultivation within the framework of biased-noise stabilizer codes.

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• Quantum error correction
• Quantum gates