Generalized Numerical Ranges and Quantum Error Correction
We describe how to use the generalized numerical ranges to help study problems in quantum error correction. Recent results and open problems will be mentioned.
Adiabatic Optimization and Dirichlet Graph Spectra
Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we pursue two particular questions: (1) do adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there is only one local minimum and (2) what characterizes the boundary between large- and small- gapped Hamiltonians?
On the Complexity of Commuting Quantum Circuits
On the Complexity of Commuting Quantum Circuits
Holographic Entanglement Entropy Inequalities
We initiate a systematic enumeration and classification of entropy inequalities satisfied by the Ryu-Takayanagi formula for conformal field theory states with smooth holographic dual geometries. For 2, 3, and 4 regions, we prove that the strong subadditivity and the monogamy of mutual information give the complete set of inequalities. This is in contrast to the situation for generic quantum systems, where a complete set of entropy inequalities is not known for 4 or more regions. We also find an infinite new family of inequalities applicable to 5 or more regions.