Accessible Quantum Gates on Classical Stabilizer Codes

TL;DR

This paper investigates the implementation complexity of universal quantum gates on classical stabilizer codes tailored for biased noise, revealing that such implementations require complex circuits or many gates, thus motivating alternative approaches.

Contribution

It proves that implementing universal gates on biased-noise stabilizer codes necessitates complex circuits or numerous gates, highlighting the need for alternative logical gate constructions.

Findings

Complex unitary circuits are required for universal gate implementation.

Constraints relate to code parameters like distance and number of qubits.

Alternative methods like magic state distillation are motivated.

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.