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1. Classical Coding Problem from Transversal T Gates

   https://doi.org/10.1109/ISIT44484.2020.9174408  

   Rengaswamy, Narayanan; Calderbank, Robert; Newman, Michael; Pfister, Henry D. (June 2020, 2020 IEEE International Symposium on Information Theory (ISIT))

   null (Ed.)

   Universal quantum computation requires the implementation of a logical non-Clifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical T and T † gates. For example, this could enable magic state distillation with non-CSS codes and, thus, provide better parameters than CSS-based protocols. However, among non-degenerate stabilizer codes that support transversal T, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal T via physical transversal T. Using our algebraic approach, we reveal new purely-classical coding problems that are intimately related to the realization of logical operations via transversal T. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum Reed-Muller codes. This result is generalized to finer angle Z-rotations in https://arxiv.org/abs/1910.09333. 

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2. Designing the Quantum Channels Induced by Diagonal Gates

   https://doi.org/10.22331/q-2022-09-08-802  

   Hu, Jingzhen; Liang, Qingzhong; Calderbank, Robert (September 2022, Quantum)

   The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of Z -stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal Z -rotation through π / 2 l , that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on l . The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes. 

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3. An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance

   https://doi.org/10.1007/s11128-024-04427-5  

   Camps-Moreno, Eduardo; López, Hiram H; Matthews, Gretchen L; Ruano, Diego; San-José, Rodrigo; Soprunov, Ivan (June 2024, Quantum Information Processing)

   Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes$$(C_1, C_2)$$ $(C_1,C_2)$such that$$C_1$$ $C_1$contains$$C_2$$ $C_2$,$$C_2$$ $C_2$is even, and the shortening of the dual of$$C_1$$ $C_1$with respect to the support of each codeword of$$C_2$$ $C_2$is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes$$(C_1, C_2)$$ $(C_1,C_2)$is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. 

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4. Quantum stabilizer codes, lattices, and CFTs

   https://doi.org/10.1007/JHEP03(2021)160  

   Dymarsky, Anatoly; Shapere, Alfred (March 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E 8 theory, which is based on the root lattice of E 8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples. 

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5. CSS codes that are oblivious to coherent noise

   https://doi.org/10.1109/ISIT45174.2021.9518206  

   Hu, J.: Liang; Rengaswamy, N.; Calderbank, R (July 2021, IEEE International Symposium on Information Theory)

   Physical platforms such as trapped ions suffer from coherent noise that does not follow a simple stochastic model. We view coherent errors as rotations about a particular axis, and observe that since they can accumulate coherently over time, they can be more damaging. It is natural to consider coherent noise acting transversally giving rise to an effective error, which is a Z-rotation on each qubit by some angle. Rather than addressing coherent noise through active error correction, we instead investigate passive mitigation through decoherence free subspaces. In the language of stabilizer codes, we require the noise to preserve the code space, and to act trivially (as the logical identity operator) on the protected information. Thus, we develop necessary and sufficient conditions for all transversal Z-rotations to preserve the code space of a stabilizer code. 

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