Multivariate multicycle (MM) code[1,2]

Description

Multi-block CSS code whose \(t\) commuting matrices represent elements \(F_1,\ldots,F_t\) of the group algebra of a finite Abelian group, written as polynomials in \(D\) variables. Each element defines a two-term complex \(S\xrightarrow{F_i}S\) over the group algebra \(S\), and the tensor product of the \(t\) complexes is a Koszul complex with \(t+1\) terms whose boundary maps are the check and metacheck matrices [1,2]. For \(t \geq 4\), the code has metachecks in both bases and admits complete single-shot decoding, while \(t=2\) and \(t=3\) recover Abelian 2BGA and tricycle codes [2].

Writing \(G \cong \mathbb{Z}_{\ell_1}\times\cdots\times\mathbb{Z}_{\ell_D}\), the group algebra is 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 \(F_1,\ldots,F_t \in S\), each represented by a sum of Kronecker products of circulant matrices [1,2]. The number of polynomials \(t\) (the number of “cycles”) and the number of variables \(D\) can be chosen independently. The number of variables is a matter of presentation. By the Chinese remainder theorem, coprime cyclic factors can be split or merged, so \(\mathbb{Z}_{15}\) and \(\mathbb{Z}_{5}\times\mathbb{Z}_{3}\) present the same code with \(D=1\) and \(D=2\), respectively.

The level-\(q\) space of the Koszul complex \(K_{\bullet}(F_1,\ldots,F_t;S)\) is \(S^{\binom{t}{q}}\), and its boundary maps \(\partial_q\) are available in closed form [2]. 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 \(P_Z=\partial_{q+1}\). The \(X\)-type metacheck matrix is \(M_X = \partial_{q-1}^T\) whenever \(t\geq 4\), and the \(Z\)-type metacheck matrix is \(M_Z=\partial_{q+2}\) whenever \(t\geq 3\). The multi-block-complex formulation [1] allows qudits to be placed at any level \(0<j<t\) of the same complex. It also admits a general Abelian group presentation whose relators define a periodicity lattice on \(\mathbb{Z}^D\), i.e., a torus [1].

A CSS code together with metacheck matrices \(M_X\) and \(M_Z\) satisfying \(M_X P_X = 0\) and \(M_Z P_Z = 0\) forms a five-term chain complex and is dubbed a metacheck CSS (mCSS) code [2]. An mCSS code is balanced when its outermost pair and its next-to-outermost pair of spaces each have equal dimensions [2]. Any five-term segment of a longer chain complex yields an mCSS code, and the middle five terms of the Koszul complex of an MM code with even \(t\) yield a balanced one.

Fixing \(t\) and \(D\) recovers several known code families, each with explicit MM polynomials [2]. Abelian 2BGA codes (\(t=2\)) include GB codes (\(D=1\)), bivariate bicycle codes (\(D=2\)), and higher-variable Abelian 2BGA codes. They also include cyclic hypergraph-product codes (\(D=2\) with polynomials in disjoint variables) and La-cross codes with periodic boundary conditions (\(D=2\), with \(G=\mathbb{Z}_n\times\mathbb{Z}_n\)). Haah cubic codes on a cubic lattice have \(t=2\) and \(D=3\). 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].

A \([[144,12,12]]\) MM code has the same parameters as, but is distinct from, the gross code [2,3]. 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]. See Ref. [2] for tables of MM codes with \(t \geq 4\), including a \([[648,60,(9,9)]]\) code whose check and metacheck matrices are written out explicitly.