Quantum Information Session - Serge Adonsou (July 29)

Title: New Entanglement-Assisted Codes and Mathematical Structures in Quantum Error Correction

Speaker: Serge Adonsou

Abstract: We develop new algebraic and analytical tools for quantum error correction (QEC) aimed at unifying entanglement-assisted coding theory with operator-algebra frameworks, while also providing mathematically rigorous insights for continuous-variable photonic implementations.

First, we introduce the framework of entanglement-assisted operator algebra quantum error correction (EA-OAQEC), extending entanglement-assisted QEC to the operator algebra setting. Built on a stabilizer-based formalism, EA-OAQEC unifies and generalizes subspace, subsystem, and hybrid classical–quantum entanglement-assisted codes. We establish a master error-correction theorem, define a generalized distance notion, and develop constructive techniques including gauge fixing, clean-qubit transformations, and entanglement- assisted gauge-fixing procedures.

Second, we study general logical operators: unitaries that preserve a stabilizer codespace beyond the traditional Pauli normalizer. We derive explicit structure theorems for the centralizer and normalizer in the full unitary setting and show how normalizer transformations can be parameterized via classical bit matrix multiply complement (BMMC) permutations.

Finally, motivated by photonic quantum computing, we analyze finite-rank compressions of the continuous position and momentum operators relevant to Gottesman-Kitaev-Preskill (GKP) codes. We prove that the spectra of these compressions are governed by the roots of classical Hermite polynomials and connect the compressed operators to physically realizable approximations of displacement operations and finite-energy GKP states.