Linear growth of quantum circuit complexity

Quantifying quantum states' complexity is a key problem in various subfields of science, from computing to black-hole physics. We prove prominent conjecture by Brown and Susskind about how random circuits' increases. Consider constructing unitary Haar-random two-qubit gates. Implementing the exactly requires circuit some minimal number gates - unitary's exact complexity. that this grows linearly with gates, unit probability, until saturating after exponentially many Our proof surprisingly short, given established difficulty lower-bounding strategy combines differential topology elementary algebraic geometry an inductive construction Clifford circuits.