Spin-Liquids in Controllable Quantum Matter and non-Abelian Topological Orders from Measurement

In contrast to symmetry breaking phases of matter, unconventional phases like spin liquids feature long-range quantum entanglement and relate to the deconfined phases of gauge theories. However, the realization of these states  in materials has proved challenging.  New avenues to create and probe such phases have arisen with the rapid development of synthetic quantum platforms, ranging from Rydberg atom arrays to Google's quantum processors. I will describe recent theoretical proposals towards this goal, in particular how measurement can be used to efficiently "gauge" a symmetry and  create nontrivial states including fractons and non-Abelian topological orders. The number of measurement layers needed to prepare a state imposes a unexpected `hardness of preparation' hierarchy - in particular we find that certain non Abelian states  based on solvable gauge groups can be efficiently prepared, establishing an link to Galois' characterization of solvable polynomial equations. A subset of solvable gauge groups turns out to be no harder to prepare than Abelian states and can be realized on existing quantum platforms