A Quantum Algorithm for the Radical of a Lie Algebra: Kernel Projection and Conditioning

Abstract

Every finite-dimensional real or complex Lie algebra has a largest solvable ideal, its radical. In the compact dynamical Lie algebra (DLA) of a closed quantum system, this radical is the center; projecting onto it isolates directions that commute with the supplied algebra. With sparse Lie-bracket data and a known spectral gap, we construct a quantum circuit that approximately encodes this coefficient-space projector. The general construction combines the derived-algebra map with the Killing form, whereas compactness makes the Killing form alone sufficient. This distinction matters for numerical sensitivity, and our main theorem gives its exact law under an invariant-orthonormal basis adapted to the center and semisimple part. In a compact real algebra with a nonzero semisimple part, the condition number of the general operator is the three-halves power of the Killing-form condition number on that part. The compact projector in turn yields a bounded-error test for whether the center is trivial, provided that any nonzero center has at least a stated minimum dimension. This gives a controlled test for internal conserved directions in compact quantum dynamics.