Quantum error-correcting codes are essential for protecting quantum information against decoherence. This work investigates the construction of quantum CSS codes from classical self-orthogonal binary linear codes derived from extended quadratic residue codes. By exploiting the large automorphism groups of these codes, the minimum-weight codewords are partitioned according to the action of maximal subgroups of the automorphism group. Codes generated from these partitions are shown to be self-orthogonal and to retain large automorphism groups, allowing the construction of associated quantum CSS codes with good parameters. The resulting codes include new quantum codes with lengths ranging from 16 to 48 and minimum distances up to 11. Their automorphism groups and permutation-decoding properties are also analyzed, including the number of permutations required for complete decoding. The results provide new families of quantum CSS codes combining good error-correcting parameters, large automorphism groups, and efficient permutation decoding.
Quantum error-correcting codes are essential for protecting quantum information against decoherence. This work investigates the construction of quantum CSS codes from classical self-orthogonal binary linear codes derived from extended quadratic residue codes. By exploiting the large automorphism groups of these codes, the minimum-weight codewords are partitioned according to the action of maximal subgroups of the automorphism group. Codes generated from these partitions are shown to be self-orthogonal and to retain large automorphism groups, allowing the construction of associated quantum CSS codes with good parameters. The resulting codes include new quantum codes with lengths ranging from 16 to 48 and minimum distances up to 11. Their automorphism groups and permutation-decoding properties are also analyzed, including the number of permutations required for complete decoding. The results provide new families of quantum CSS codes combining good error-correcting parameters, large automorphism groups, and efficient permutation decoding.
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.
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.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define X and Z-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with X-, Y-, and Z-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the Y-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
Divisible codes are defined by the property that codeword weights share a common divisor greater than one. They are used to design signals for communications and sensing, and this paper explores how they can be used to protect quantum information as it is transformed by logical gates. Given a CSS code C, we derive conditions that are both necessary and sufficient for a transversal diagonal physical operator UZ to preserve C and induce UL. The group of Z-stabilizers in a CSS code C is determined by the dual of a classical [n,k1] binary code C1, and the group of X-stabilizers is determined by a classical [n,k2] binary code C2 that is contained in C1. The requirement that a diagonal physical operator UZ fixes a CSS code C leads to constraints on the congruence of weights in cosets of C2. These constraints are a perfect fit to divisible codes, and represent an opportunity to take advantage of the extensive literature on classical codes with two or three weights. We construct new families of CSS codes using cosets of the first order Reed Muller code defined by quadratic forms. We provide a simple alternative to the standard method of deriving the coset weight distributions (based on Dickson normal form) that may be of independent interest. Finally, we develop an approach to circumventing the Eastin-Knill Theorem which states that no QECC can implement a universal set of logical gates through transversal gates alone. The essential idea is to design stabilizer codes in layers, with N1 inner qubits and N2 outer qubits, and to assemble a universal set of fault tolerant gates on the inner qubits.
We introduce an adaptive simulated annealing algorithm to search for moderately-sized quantum error-correcting codes with high encoding rates and large distances. Our search targets two classes of stabilizer codes: (1) CSS codes, and (2) a subclass of CSS codes that we call ``self-dual with equivalent logicals'' (SWEL) codes, the latter of which which admit transversal implementations of logical Hadamard and phase gates that can be leveraged to construct fault-tolerant gate sets. The search is guided by an energy function that acts as a surrogate for the logical error rate in a code-capacity noise model, combining code distance with a count of minimum-weight logical operators to resolve the discrete plateaux that impede naïve distance optimization. For block lengths of up to 50 physical qubits, our search finds state-of-the-art CSS and SWEL codes whose distances frequently meet or exceed the variants of the quantum Gilbert-Varshamov bound. In addition to providing favorable seed codes for fault-tolerant architectures based on code concatenation, the codes found in this work are promising candidates for high-rate code demonstrations on near-term quantum computing hardware.