Linear growth of quantum circuit complexity

TL;DR

This paper proves that the exact circuit complexity of Haar-random quantum unitaries grows linearly with the number of gates until saturation, confirming a key conjecture in quantum complexity theory.

Contribution

It provides a rigorous proof that the complexity of random quantum circuits increases linearly, using a novel combination of differential topology and algebraic geometry.

Findings

Complexity grows linearly with the number of gates

Growth continues until exponential saturation point

Proof employs differential topology and algebraic geometry techniques

Abstract

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