Symmetry Protected Quantum Computation

We consider a model of quantum computation using qubits where it is possible to measure whether given pair are in singlet (total spin 0) or triplet xmlns:mml="http://www.w3.org/1998/Math/MathML">1) state. The physical motivation that we can do these measurements way protected against revealing other information so long as all terms the Hamiltonian xmlns:mml="http://www.w3.org/1998/Math/MathML">SU(2)-invariant. conjecture this equivalent BQP. Towards goal, show: (1) capable universal with polylogarithmic overhead if supplemented by single qubit xmlns:mml="http://www.w3.org/1998/Math/MathML">X and xmlns:mml="http://www.w3.org/1998/Math/MathML">Z gates. (2) Without any additional gates, at least powerful weak "permutational computation" Jordan [14, 18]. (3) With postselection, PostBQP.