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This research establishes a definitive mathematical limit where Krylov complexity saturates slightly beyond the Heisenberg length in low-energy quantum states. By proving that orthogonal polynomials crowd away from regions of low state density at large indices, the study provides concrete evidence for this saturation mechanism across potentially infinite Hilbert spaces. These findings offer significant constraints on theoretical models linking computational difficulty to spacetime geometry and the volume conjecture in quantum gravity research.
Krylov Complexity Saturated Beyond Heisenberg Length in Quantum Systems
This research demonstrates Krylov complexity saturation in quantum systems, proving it stops growing just past the Heisenberg length in low-energy states.
Andrew Lucas and Amit Vikram of the University of Colorado, Boulder have demonstrated that Krylov complexity, a measure of how difficult a quantum state is to create, demonstrably stops growing a little past the Heisenberg length in any low-energy state, including the thermofield double state. The researchers found that orthogonal polynomials, essential to calculating this complexity on a discrete energy spectrum, crowd away from regions where the density of states is low at large Krylov index, revealing a specific…
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