SYMPlectic AlgoriTHms sImulating QUantum procEsses – SYMPATHIQUE

The SYMPATHIQUE project proposes a novel representation of quantum data as algebraic variety with a symplectic structure. This approach is inspired by the compact tableau representation of stabilizer states. On the quantum circuit level this corresponds to adding the Toffoli gate to the stabilizer fragment, providing approximately universal quantum circuits. This point of view is supported by the recent development in diagrammatical approaches to quantum computing, linking the stabilizer ZX-calculus to symplectic spaces. The project will focus on the development of a stabilizer ZH-calculus with a complete equational theory allowing to reason on stabilizer+Toffoli circuits. On the algorithmic side, we aim to generalize the tableau algorithm, which amounts to manipulate linear subspaces using Gaussian elimination, by a symplectic version of the Buchberger's algorithm, allowing to handle algebraic variety described by Grobner bases. This should allow to apply familiar algorithms and techniques in computational algebra to simulate and verify quantum programs.