Accessible Quantum Gates on Classical Stabilizer Codes

• URL: http://arxiv.org/abs/2507.05408v1
• Date: Mon, 07 Jul 2025 18:47:58 GMT
• Title: Accessible Quantum Gates on Classical Stabilizer Codes
• Authors: Victor Barizien, Hugo Jacinto, Nicolas Sangouard,
• Abstract summary: 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.<n>We prove that operations essential for achieving a universal logical gate set necessarily require complex unitary circuits to be implemented.<n>Similar constraints apply not only to classical codes designed to correct phase-flip errors, but also to quantum stabilizer codes tailored to biased noise.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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