Quantum Codes with Transversal $CCZ$ Gates and Sublinear $Z$-Stabilizers

Abstract

We construct asymmetric quantum CSS codes with transversal \(CCZ\) gates from algebraic expander codes \cite{KT26}. For every fixed \(m\ge 3\), our growing-alphabet codes have length \(N\), dimension \(Θ(N)\), and distances \[ d_X=Θ(N), \qquad d_Z=Θ(N^{1/m}). \] Moreover, the \(Z\)-stabilizer space has an explicit generating set of weight \(O(N^{1/m})\). We build on the algebraic puncturing framework of Golowich and Guruswami \cite{GG24}, which turns classical codes with the required Schur-product and distance conditions into CSS codes with transversal \(CCZ\). However, applying the framework directly to the algebraic expander codes runs into their small dual distance, and therefore produces only sublinear dimension. Our main technical step is a refined puncturing theorem in which the global dual-distance assumption is replaced by a condition only on the selected puncturing set. We also reduce the alphabet to a fixed prime field using a projective-multiplicity version of multiplication-friendly codes. The resulting fixed-prime-field CSS code triples, of length \(n\), still have transversal \(CCZ\) gates. Their dimension is \(Θ(n/(\log n)^4)\), with distances \[ d_X=Ω\!\left(\frac{n}{(\log n)^4}\right), \qquad d_Z=Ω\!\left(\frac{n^{1/m}}{(\log n)^{4/m}}\right), \] and the \(Z\)-stabilizer generating set remains sublinear.

The paper has been substantially revised. The main changes are: (i) the construction is now analyzed using the two separate CSS distances rather than only the minimum distance, revealing an asymmetric distance profile with one linear distance and one sublinear distance; (ii) the presentation has been reorganized to emphasize the asymmetric interpretation of the construction and its implications

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