Zhengwei Liuau Mathematical Science Center and Department of Mathematics, Tsinghua University, Beijing 100084, China
Mathematical PhysicsQuantum AlgebraarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.22008
Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.