The Krylov Constraint

Krylov complexity measures how an operator spreads across a basis of nested commutators — the Krylov basis — as it evolves in time. In chaotic systems, this spreading is fast and eventually saturates at a value set by the Hilbert space dimension. The saturation value and the approach to it encode information about the system’s complexity structure.

arXiv:2603.16291 computes Krylov complexity in supersymmetric large-N quantum mechanics. Supersymmetry pairs bosonic and fermionic degrees of freedom, and this pairing constrains the Lanczos coefficients — the parameters that govern how fast the operator spreads through the Krylov chain.

The constraint is structural: supersymmetry relates the Lanczos coefficients of the bosonic and fermionic sectors, reducing the number of independent parameters. The operator still spreads, still saturates, still exhibits the universal features of chaotic dynamics. But the spreading is not free — it is channeled by the supersymmetric pairing.

At large N, the system simplifies enough for exact computation. The Lanczos coefficients grow linearly (the signature of maximal chaos), but their slope is fixed by the supersymmetry algebra. The system is as chaotic as it can be given its symmetry — not more, not less.

The result connects two independent measures of quantum complexity: Krylov complexity (operator growth in the Lanczos basis) and the symmetry structure of the Hamiltonian. Supersymmetry does not suppress complexity — it constrains the channel through which complexity grows. The operator explores all of Hilbert space, but it explores it along a path that the symmetry has already shaped.