Multivariate multicycle (MM) code[1,2]

Description

A multi-block CSS code whose \(t\) commuting matrices are built from an Abelian group algebra, parameterized by \(t\) polynomials in \(D\) variables via a Koszul complex. The construction unifies many known qLDPC families—including bivariate bicycle, Abelian 2BGA, generalized bicycle, tricycle, and toric codes—and, for \(t \geq 4\), yields codes with metachecks in both bases that support complete single-shot decoding.

Writing \(G \cong \mathbb{Z}_{\ell_1}\times\cdots\times\mathbb{Z}_{\ell_D}\), the group algebra is isomorphic to the quotient polynomial ring \(S=\mathbb{F}_q[x_1,\ldots,x_D]/\langle x_1^{\ell_1}-1,\ldots,x_D^{\ell_D}-1\rangle\), so that a code is specified by \(t\) polynomials \(F_1,\ldots,F_t \in S\) in \(D\) variables. The number of polynomials \(t\) (the number of “cycles”) and the number of variables \(D\) can be chosen independently. The number of variables is moreover a matter of presentation: by the Chinese remainder theorem, coprime cyclic factors can be split or merged, so, e.g., \(\mathbb{Z}_{15}\) and \(\mathbb{Z}_{5}\times\mathbb{Z}_{3}\) present the same code with \(D=1\) and \(D=2\), respectively.

The underlying chain complex is the Koszul complex \(K_{\bullet}(F_1,\ldots,F_t;S)\), equivalently the \(t\)-fold tensor product of the two-term complexes \(S \xrightarrow{F_i} S\), or the multi-block MBC complex of the commuting matrices representing the \(F_i\) as Kronecker products of circulants [1,2]. Its level-\(q\) space is \(S^{\binom{t}{q}}\), and its boundary maps \(\partial_q\) are available in closed form. In the Koszul-complex formulation [2], qubits are placed at the middle level \(q=\lfloor t/2\rfloor\), yielding \(P_X=\partial_q^T\) and \(P_Z=\partial_{q+1}\), an \(X\)-type metacheck matrix \(M_X = \partial_{q-1}^T\) whenever \(t\geq 4\), and a \(Z\)-type metacheck matrix \(M_Z=\partial_{q+2}\) whenever \(t\geq 3\). The multi-block-complex formulation [1] allows qudits to be placed at any level \(0<j<t\) and admits a general Abelian group presentation whose relators define a periodicity lattice on \(\mathbb{Z}^D\), i.e., a torus.

A CSS code equipped with metacheck matrices \(M_X\) and \(M_Z\) satisfying \(M_X P_X = 0\) and \(M_Z P_Z = 0\) can be organized into a five-term chain complex dubbed a metacheck CSS (mCSS) code, which is balanced when its outermost (next-to-outermost) pair of spaces have equal dimensions [2]. Any five-term segment of a longer chain complex yields an mCSS code, and the middle five terms of an MM code’s Koszul complex with even \(t\) yield a balanced mCSS code. MM codes with \(t \geq 4\) possess metachecks in both bases as well as high confinement, permitting complete single-shot decoding [2].

Subfamilies

Fixing \(t\) and \(D\) recovers several known code families, each admitting an explicit realization in terms of MM polynomials [2]. The case \(t=2\) yields Abelian 2BGA codes, which include GB codes (\(D=1\)), bivariate bicycle codes (\(D=2\)), multivariate bicycle codes, cyclic HGP codes and La-cross codes (\(D=2\) with polynomials in disjoint variables), and Haah cubic codes (\(D=3\)). The case \(t=3\) yields tricycle codes. Weight-two elements \(a_i=1+x_i\) with \(t=D\) yield \(D\)-dimensional toric codes on regular or twisted tori, with arbitrary twists realized via general group presentations [1].

Examples

New \(t \geq 4\) codes obtained via numerical search include \([[96,12,8]]\), \([[144,12,12]]\) (sharing the parameters of, but distinct from, the \([[144,12,12]]\) bivariate bicycle gross code [3]), \([[216,12,14]]\), \([[288,12,16]]\), \([[324,12,20]]\), \([[432,12,27]]\), \([[486,24,12]]\), \([[630,70,9]]\), and \([[648,18,23]]\) codes, as well as a \([[648,60,(9,9)]]\) code whose check and metacheck matrices are written out explicitly [2]. A family of MM codes locally equivalent to 4D toric codes, with weight-six stabilizer generators and \(k=6\), is constructed in Ref. [1]. Codes with \(t=5\) through \(t=9\) provide the first explicit instances of collapsed 5D through 9D higher-dimensional codes [2].