This paper presents constructions of approximate unitary k-designs using shallow, low-communication quantum circuits. The authors show that it is possible to generate approximate k-designs with sublinear depth in the system size, which answers an open question in quantum information theory. The constructions use the alternating projection method to analyze overlapping Haar twirls and derive bounds on the convergence speed to the full twirl in the 2-norm. By replacing system dimension with von Neumann subalgebra indices, the 2-norm distance is converted to relative error without introducing additional dimension dependence. The results show that these constructions can achieve relative error designs in depth $ O((k \log k + \log m + \log(1/\epsilon))k \text{ polylog}(k)) $, where $ m $ is the number of qudits in the complete system and $ \epsilon $ is the approximation error. The constructions also show that entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size. The paper also discusses the implications of these results for quantum communication and entanglement bounds, and highlights the importance of area-law entanglement in condensed matter and high-energy physics. The authors conclude that their results provide a new record for scaling of circuit depth in the system size and suggest further investigation into potential efficiencies of recursive constructions.This paper presents constructions of approximate unitary k-designs using shallow, low-communication quantum circuits. The authors show that it is possible to generate approximate k-designs with sublinear depth in the system size, which answers an open question in quantum information theory. The constructions use the alternating projection method to analyze overlapping Haar twirls and derive bounds on the convergence speed to the full twirl in the 2-norm. By replacing system dimension with von Neumann subalgebra indices, the 2-norm distance is converted to relative error without introducing additional dimension dependence. The results show that these constructions can achieve relative error designs in depth $ O((k \log k + \log m + \log(1/\epsilon))k \text{ polylog}(k)) $, where $ m $ is the number of qudits in the complete system and $ \epsilon $ is the approximation error. The constructions also show that entanglement generated by the sublinear depth scheme follows area laws on spatial lattices up to corrections logarithmic in the full system size. The paper also discusses the implications of these results for quantum communication and entanglement bounds, and highlights the importance of area-law entanglement in condensed matter and high-energy physics. The authors conclude that their results provide a new record for scaling of circuit depth in the system size and suggest further investigation into potential efficiencies of recursive constructions.