Codes and Expansions (CodEx) Seminar

Xinyu Tan (Duke University)
Quadratic Form Diagonal Gates and Grassmannian Packings

We analyze Grassmannian packings where the individual subspaces are obtained by applying a family of diagonal operators to a quantum error correcting code. The diagonal operators are drawn from the Clifford hierarchy of unitary operators. The first level is the Pauli group. The second level is the Clifford group which consists of unitary operators that normalize the Pauli group. The \(l\)-th level consists of unitary operators that map Pauli operators to the \( (l-1) \)-th level under conjugation. Quadratic form diagonal (QFD) gates are a family of diagonal gates associated with quadratic forms connected to classical error correcting codes over rings. We focus on the Grassmannian packings obtained by conjugating a single Pauli projector by a group of QFD gates in the 3rd level of the Clifford hierarchy. We describe how each subspace is associated with a classical bent function and derive a formula for the distances between individual subspaces. More specifically, each QFD gate is determined by a symmetric matrix \( R \) over ring \( \mathbb{Z}_{2^l} \). When \( l = 3 \), we can expand \( R = R_0 + 2R_1 + 4R_2 \) to investigate how each \( R_i \) contributes to the distance analysis. We show that when \( R_0 \) is fixed, the possible distances between the subspaces obtained by conjugating the 3rd level QFD gates only take values from \( 0 \), \(2^{n-1} \), and \( 2^{n-2} \) where \( n \) is the number of physical qubits.