Universal Solutions to the Constant Quantum Yang-Baxter Equation
- John Preskill, Richard P. Feynman Professor of Theoretical Physics , California Institute of Technology
- Gorjan Alagic, Postdoctoral Scholar, California Institute of Technology
The Quantum Yang-Baxter Equation (QYBE) is the algebraic version of the Yang-Baxter relation, one of the generating relations of the braid group. The aim of this project is to look for unitary solutions to the constant QYBE that are also universal as quantum gates. Results in this direction would help us understand the relationship between topological and quantum entanglement. The braiding nature of Yang-Baxter gates could also allow for useful properties such as circuit obfuscation. Two main directions were explored in this project: a search for unitary solutions to the QYBE and, determining whether any known solutions are universal as quantum gates. Since braid representations are naturally solutions to the QYBE, certain special representations such as the Fibonacci representation were also investigated. An existing proof that the 4-by-4 solutions to the QYBE are not universal was generalized, when the matrix in consideration is a permutation matrix with arbitrary phase factors. We also found two families of solutions for general dimensions. Overall, this work seems to indicate that Yang-Baxter gates by themselves might not be powerful enough for universal quantum computation.