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