New methods for quantum control and circuit synthesis with symmetry-respecting interactions

Universality of 2-qubit unitary transformations is one of the cornerstones of quantum computing, with many applications and implications extending beyond this field. However, it has been shown that this universality does not hold in the presence of global continuous symmetries such as U(1) and SU(2): generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems [I. Marvian, Nature Physics (2022)]. Further investigations have revealed that the restrictions imposed by the locality of interactions vary significantly for different symmetry groups. While there is currently no comprehensive theory for general symmetry groups, recent work has developed the theory of symmetric quantum circuits for the case of Abelian symmetries. In this talk, first, I will give an overview of this ongoing project. In the second part, I will focus on the special case of energy-conserving unitaries, i.e., those that conserve the sum of Pauli Z operators on all qubits, corresponding to a global U(1) symmetry. I will present explicit circuit synthesis methods for realizing all such unitaries with XY interaction alone, using 2 ancilla qubits. In particular, I will discuss the properties of circuits containing only the square-root-of-iSWAP gates.