Gauged sheaf QLTC [1]

Description

Qubit code whose dimension has a prime divisor that grows with \(n\), obtained by gauging three sheaf-code QLTC blocks with a cup-product trilinear form [1; Lemma 11]. The family has constant rate, linear distance, and constant operator soundness [1; Thm. 4].

The blocks come from the \(c^3\)-QLTC construction of Ref. [2], with the choices of Ref. [3] that make the trilinear form nonzero [1; App. B.2]. Gauging keeps the \(Z\)-type checks of the blocks [1; Sec. 4.1]. Each \(X\)-type generator of a block is multiplied by a phase set by the trilinear form of its cochain with the other two blocks’ cochains [4][1; Sec. 4.1].

The third block, with cochains \(C^j\) and coboundary \(\delta\), is replaced by \(E^j=C^j\oplus C^{j-1}\) with \(\delta_E(y,x)=(\delta y,\,y-\rho y+\delta x)\), where \(\rho\) is a deck transformation of prime order \(p\) [1; App. B.3]. Cohomology classes of \(E\) then project to classes fixed by \(\rho\) [1; App. B.3]. The trilinear form is evaluated on \(b-\rho b\) for the second block’s cochain \(b\), and the third block’s generators are assigned eigenvalues \((-1)^{\mathrm{Tr}\,\lambda(w)}\) for a nonzero linear functional \(\lambda\) [1; App. B.3]. Acting with \(\rho\) on the first two blocks permutes the triples of cohomology classes labeling a codeword basis without fixed points, so \(p\) divides the code dimension [1; Lemma 11].