Research Projects

This project centers on the characterization and construction of quantum CSS-T codes, contributing to advancements in quantum information theory and addressing key challenges in practical quantum computation. Quantum error correction is fundamental for enabling fault-tolerant and universal quantum computing. One of the most promising strategies for fault-tolerant implementation of quantum gates involves transversal gates, which apply single-qubit unitaries independently across qubits. Their principal advantage lies in containing the spread of errors within code blocks, which simplifies error tracking and correction. The Clifford group, generated by gates such as CNOT, CZ, Phase, and Hadamard, is crucial for quantum circuits, and Clifford operations are often easy to implement fault-tolerantly. However, these are not sufficient for universal quantum computation. A universal gate set requires adding a non-Clifford gate—most commonly the T gate, represented as: $T=\begin{bmatrix} 1 & 0 \\ 0 & e^{i\pi/4} \end{bmatrix}.$ While self-dual CSS codes allow for transversal implementation of Clifford gates, they generally do not support transversal T gates. To overcome this, approaches like magic state distillation and state injection are used. These methods allow implementation of logical T gates by preparing and consuming special ancilla states, maintaining transversality for the remainder of the circuit. An important breakthrough was made by Bravyi and Haah, who introduced triorthogonal codes that support transversal T gates. Building on this, Rengaswamy et al. introduced CSS-T codes, a class of CSS codes that permit transversal implementation of T gates. These codes were shown to be optimal among non-degenerate stabilizer codes allowing such transversal T gate implementation. Formally, a pair of classical binary codes $(C_1, C_2)$ with parameters $[n,k_1,d_1]$ and $[n,k_2,d_2]$ is called a CSS-T code if: 1. $C_2 \subset C_1$, 2. $C_2$ is an even code, 3. For each $y \in C_2$, there exists a self-dual code supported on $y$ within the dual code $C_1^\perp$, with dimension $w_H(y)/2$, where $w_H(y)$ is the Hamming weight of $y$. These structural conditions ensure that such codes enable transversal T gate operations, facilitating more efficient fault-tolerant quantum computation. A major problem in quantum coding: the construction of high-rate quantum codes with increasing distance that support universal fault-tolerant operations. This project aims to contribute toward contributing this problem, especially in the context of CSS-T codes, by exploring their construction, optimization, and applications in universal quantum computation.