K-complexity and the Jordan-Wigner transformation - Pandit, Zayd (Department of Mathematics and Applied Mathematics, 2026)
Krylov complexity is a measure of operator growth that demonstrates universal properties and bounds a large class of complexities. One such measure from this bounded class is operator size. The relationship between operator size and operator growth has been conjectured to be non-trivial due to the existence of duality transformations such as the Jordan-Wigner (JW) transformation which map small operators to large, non-local operators. We investigate this claim directly in the case of the JW transformation which maps the XY Heisenberg chain to the Kitaev chain. We numerically calculate the complexity of dual operators, and analyse the early and late time behaviour and symmetries. We find that for Open Boundary Conditions (OBC) the early time behaviour of the K-Complexity correlates with operator size, but that large operators can have very low K-Complexity if dual to a small operator. We find that for Periodic Boundary Conditions (PBC) larger operators produce larger early time growth, but do not correlate to larger late-time complexity regardless of the size of the dual operator. The difference between the OBC and PBC results arise from an often overlooked break in translational symmetry across the PBC Jordan-Wigner transformation. We also find that state complexity is not sensitive to the break in translational symmetry.