Fig. 1: The geometric approach to complexity provides a strong intuitive and physical basis for the complexity growth conjecture that we prove.
a, The complexity has been conjectured to grow linearly under random quantum circuits until times exponential in the number n of qubits4. b, The blue region depicts part of the space of n-qubit unitaries. A unitary U has a complexity that we define as the minimal number of two-qubit gates necessary to effect U (green jagged path; each path segment represents a gate). Nielsen’s complexity9,10,11,12, involved in ref. 4, attributes a high metric cost to directions associated with nonlocal operators. In this geometry, the unitary’s complexity is the shortest path that connects \({\mathbb{1}}\) to U (red line). Nielsen’s geometry suggests the toolbox of differential geometry, avoiding circuits’ discreteness. The circuit complexity upper-bounds Nielsen’s complexity; opposite bounds hold for approximate circuit complexity12.
