I introduce a method to generate families of CSS codes with interesting code parameters. The object of study is Coxeter groups, both finite and infinite (reducible or not), and a geometrically motivated partial order of Coxeter group elements named after Bruhat. The Bruhat order is known to provide a link to algebraic topology -- it doubles as a face poset capturing the inclusion relations of the $p$-dimensional cells of a regular CW~complex and that is what makes it interesting for QEC code design. Assisted by the Bruhat face poset interval structure unique to Coxeter groups I show that the corresponding chain complexes can be turned into multitudes of CSS codes. Depending on the approach, I obtain CSS codes (and their families) with controlled stabilizer weights, for example $[6006, 924, \{{\leq14},{\leq7}\}]$ (stabilizer weights~14 and 9) and $[22880,3432,\{{\leq8},{\leq16}\}]$ (weights 16 and 10), and CSS codes with highly irregular stabilizer weight distributions such as $[571,199,\{5,5\}]$. For the latter, I develop a weight-reduction method to deal with rare heavy stabilizers. Finally, I show how to extract four-term (length three) chain complexes that can be interpreted as CSS codes with a metacheck.
I introduce a method to generate families of CSS codes with interesting code parameters. The object of study is Coxeter groups, both finite and infinite (reducible or not), and a geometrically motivated partial order of Coxeter group elements named after Bruhat. The Bruhat order is known to provide a link to algebraic topology -- it doubles as a face poset capturing the inclusion relations of the p-dimensional cells of a regular CWcomplex and that is what makes it interesting for QEC code design. Assisted by the Bruhat face poset interval structure unique to Coxeter groups I show that the corresponding chain complexes can be turned into multitudes of CSS codes. Depending on the approach, I obtain CSS codes (and their families) with controlled stabilizer weights, for example [6006,924,{≤14,≤7}] (stabilizer weights14 and 9) and [22880,3432,{≤8,≤16}] (weights 16 and 10), and CSS codes with highly irregular stabilizer weight distributions such as [571,199,{5,5}]. For the latter, I develop a weight-reduction method to deal with rare heavy stabilizers. Finally, I show how to extract four-term (length three) chain complexes that can be interpreted as CSS codes with a metacheck.
It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
An old question in sphere-packing turns out to yield an unexpected construction in quantum error correction. The face-centered cubic lattice densest sphere packing in three dimensions, proved by Hales generates a CSS code with encoding rate 2/3. We show that this construction extends to every Dn root lattice (n >= 3), yielding codes with parameters [n(n-1)L^n/2, (n-2)(n+1)L^n/2 + 2, 3] and encoding rate converging to 1 - 4/K as L -> infinity where K = 2n(n-1) is the coordination number. CSS validity, the rank formula, and the rate are proved analytically for all n >= 3. The first two members are computationally verified: the D3 (FCC) code and the D4 code [1536, 1282, 3] at L=4 confirmed at L=6 as [7776, 6482, 3]. For D4 at L=4, all 4096 triangles are logical operators, establishing d=3 exactly. As K grows along the densest-packing hierarchy the protected fraction approaches unity, making this, to our knowledge, the first proven infinite family of topological CSS codes whose rate converges to 1 at fixed distance.
In classical coding theory, the lift of a code is a practical and natural method for generatingfamilies of LDPC codes from a single chosen input code. This strategy has proven to bevery successful in producing LDPC codes of various blocklengths, reaching good parametersand approaching the capacity of certain channels in the asymptotic regime. It is also nowapplied in the 5G New Radio [RK18].It is widely believed that a quantum computer will integrate a quantum error correctingscheme, and the last three decades have seen the emergence of a variety of code-basedquantum error correction known as CSS codes. It is natural to ask whether we can performlifts of quantum codes, with the hope of having the same benefits as for classical codes.However, the method to lift linear codes cannot be directly applied to quantum CSS codes,as the constituent linear codes usually lose their orthogonality. This dissertation presentsthree new results on lifts of quantum CSS codes.Firstly, we propose a general notion of lift for quantum CSS codes. It is based on theexistence of a canonical complex associated to any CSS code, that we call the Tanner cone-complex, and over which we generate covering spaces. We show that our notion generalizesclassical code lifting and generate important CSS code families. Our main application ofthe lift is to devise new types of quantum CSS codes. So far, most seriously consideredconstructions rely on topology or some kind of product operations. Here, we intend toexplore beyond these paradigmatic ideas and for that we construct new non-topologicaland non-product CSS codes. We then apply the prescription to generate their lifts andshow that this procedure can improve their relative parameters, reaching parameter regionsnot explored prior to this work.Secondly, we introduce new families of quantum Tanner codes, a class of quantum codeswhich first appeared in the work of Leverrier and Zémor [LZ22]. We build these codesby generalizing the lift of classical Tanner codes, based on graph coverings, to the case ofCSS codes, based on coverings of 2D geometrical complexes. We present several explicitfamilies, and identify instances of moderate length quantum codes which are degenerate,have low check weight, and for which the distance is greater than the square root of thecode length.Lastly, we present a topological approach to lifting a quantum CSS code. Our originaldefinition of CSS code lifting was inspired by the work of Freedman and Hastings [FH21],which associates CSS codes with handlebodies. This thesis is a good opportunity to showhow the handlebody realization of a code can also be used to perform code lifting. Weprovide a detailed discussion of why in many cases this is essentially equivalent to ourprevious approach.
We generalize the unified framework in [Phys. Rev. A 113, 022438 (2026)] to accommodate arbitrary stabilizer codes (referred as non-CSS for emphasis). The generalization has immediate consequences in areas including logical measurement, quantum weight reduction and Euclidean embedding. For example, currently CSS (pure X,Z-type) logical measurements are well-understood based on the (height-1) cone, while non-CSS (e.g., Y-type) logical measurements are ad hoc (e.g., rely on local Clifford transform). Similarly, quantum weight reduction and optimal Euclidean embeddings only exist for CSS input codes. Here, we show that our generalized framework addresses this issue, and thus many constructions, including qLDPC surgery, Layer Codes and quantum weight reduction generalize to non-CSS codes in a relatively straightforward fashion. The previously mentioned examples are derived in detail for clarity.
The [[3L3,2L3+2,3]] CSS code on the Face Centered Cubic (FCC) lattice reaches a 67.7% encoding rate with weight-12 stabilizers. Its distance is 3. We classify every weight-3 logical operator of the code. The X-type ones are the 4L3 triangular faces of the lattice. The Z-type ones are the 4L3 claws, each formed by the three edges of a tetrahedron meeting at one of its vertices. The two families are exact duals under the exchange of vertices and octahedral voids, and each spans 2L3−1 independent logical classes. We prove this count analytically from the homology of the 3-torus, not only numerically: the cycle space has dimension 25L3+1 , the boundary space has dimension 25L3−2 , and subtracting the stabilizer rank leaves 2L3−1 for every even L. So k=(2L3−1)+3 , and the excess rate is carried entirely by weight-3 operators. We then show that this sector cannot be protected. The three-body parity ZeaZebZec at a face anticommutes with four octahedral stabilizers, so it is not measurable, and no measurement of any kind can distinguish ∣ψ⟩ from L^T∣ψ⟩ for arbitrary encoded ∣ψ⟩ . Removing the sector requires adjoining all 2L3−1 independent directions, which returns k=3 . Verified exactly over GF(2) at L=4,6,8,10 .
Calderbank-Shor-Steane (CSS) codes are a versatile quantum error-correcting family built out of commuting X- and Z-type checks. We introduce CSS-like codes on G-valued qudits for any finite group G that reduce to qubit CSS codes for G=Z2 yet generalize the Kitaev quantum double model for general groups. The X-checks of our group-CSS codes correspond to left and/or right multiplication by group elements, while Z-checks project onto solutions to group word equations. We describe quantum-double models on oriented two-dimensional CW complexes (which need not cellulate a manifold) and prove that, when G is non-Abelian and simple, every G-covariant group-CSS code with suitably upper-bounded Z-check weight and lower-bounded Z-distance reduces to a CW quantum double. We describe the codespace and logical operators of CW quantum doubles via the same intuition used to obtain logical structure of surface codes. We obtain distance bounds for codes on non-Abelian simple groups from the graph underlying the CW complex, and construct intrinsically non-Abelian code families with asymptotically optimal rate and distances. Adding "ghost vertices" to the CW complex generalizes quantum double models with defects and rough boundary conditions whose logical structure can be understood without reference to non-Abelian anyons or defects. Several non-invertible symmetry-protected topological states, both with ordinary and higher-form symmetries, are the unique codewords of simply-connected CW quantum doubles with a single ghost vertex.