Accessible Quantum Gates on Classical Stabilizer Codes

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 codes designed to address a single dominant error type. Here, we consider $[n,k,d]$-classical stabilizer codes 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 $c$ codeblocks such that $c^{h-1}r^h \geq d$ or of $h$ gates, each operating on at most $r$ physical qubits on the same codeblock, such that $hr\geq d$. 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 eg.~magic state distillation and cultivation within the framework of biased-noise stabilizer codes.