Toric varieties and their associated toric codes, as well as determination of their parameters with intersection theory, are presented in the two dimensional case. Linear Secret Sharing Schemes with strong multiplication are constructed from toric varieties and codes by the J. L. Massey construction. Asymmetric Quantum Codes are obtained from toric codes by the A.R. Calderbank, P.W. Shor and A.M. Steane construction of stabilizer codes from linear codes containing their dual codes. Decoding of a class of toric codes is presented.
Abstract : We consider quantum error-correcting codes over alphabets of arbitrary size. We generalize quantum stabilizer codes and develop Calderbank-Shor-Steane construction over quantum systems of arbitrary dimensions using the group structure of alphabets. We also develop a methodology systematically conjoining error-correcting codes into a new class of error-correcting codes. With the help of these methods we give several ways to construct quantum maximum distance separable (MDS) codes and present many families of quantum MDS codes.
We study the algebraic compatibility between quantum homomorphic encryption and stabilizer quantum error correction in an honest-but-curious model, restricting the encryption to the transversal block-Pauli mask
(XaZb)⊗n controlled by two secret key bits
a and
b. We prove a necessary and sufficient criterion for an
[[n,1,d]] stabilizer code to remain compatible with this mask, namely that the transversal
X and
Z operators commute with every stabilizer generator, and we specialize it to a classical-code criterion for Calderbank--Shor--Steane codes. Combining code-space preservation with the logical action of the mask, every stabilizer code falls, with respect to each of the two transversal mask operators, into exactly one of three classes: incompatible, compatible but trivially acting, or compatible with a nontrivial logical Pauli action; the two operators need not fall into the same class. We verify this classification by exact symplectic computation for rotated and planar surface codes, toric codes, triangular and toric color codes, the
[[15,1,3]] code and several small codes, and find that the class is fixed by the lattice size and the boundary presentation rather than by the topological phase: the surface-code family realizes all three classes, whereas color codes realize only the two compatible ones. For encrypted computation we show that the Shor code does not implement the logical
T gate by a transversal
T⊗9, whereas suitable triorthogonal codes do so up to a Clifford correction. We formulate logical-Pauli masking as a code-independent algebraic template. Syndrome extraction and recovery are key-independent whenever the mask preserves the code space, so that the client only needs the block size and the class of the code to decrypt.
Abstract In the domain of quantum error correction, a critical task involves identifying logical operations on logical qubits for various quantum codes. However, owing to the inherent complexity of many quantum codes, devising an efficient method to implement the desired logical operations utilizing the structure of these codes presents a significant challenge. In previous studies, several methods were used to realize specific logical operations for certain quantum codes; however, they usually do not work for other quantum codes. In this study, we propose a systematic method to find solutions to any Clifford logical operation for Calderbank–Shor–Steane codes, which represent a broad category of many quantum codes. In contrast to previous geometrical methods, our method is based on an algebraic approach. The solved logical operation can be further decomposed into a sequence of single-qubit and two-qubit gates that can be directly realized using quantum hardware. By applying our method, we can find many geometrically difficult logical operations for two different types of quantum code, that is, toric and hyperbolic surface codes. We believe that the present study will find wide applications in the design of logical operations for many quantum low-density parity-check codes, which can play an important role in future fault-tolerant quantum computing.