Unitary k-designs from random number-conserving quantum circuits

Understand

Local random circuits scramble efficiently and accordingly have a range of applications in quantum information and quantum dynamics.

• With a global $U(1)$ charge however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries.
• We establish two results using the statistical mechanics of $k$-fold replicated circuits.
• First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries.