Structured-ansatz Lyapunov symmetrizer construction

Develop a structured-ansatz Lyapunov construction that solves for the symmetrizer W of the convolutional perfectly matched layer generator within a polynomially parametrised operator family, with the aim of obtaining a quantum-implementable symmetrizer without dense classical precomputation.

Background

The paper introduces a positive-definite Lyapunov matrix W satisfying a shifted Lyapunov equation for the two-dimensional convolutional perfectly matched layer (CPML) generator. Its square root S transforms the indefinite Hermitian part of the memory-form generator into a dissipative one, thereby replacing the exponentially growing Schrödingerisation recovery threshold with a conditioning-dependent cost.

Numerical tests show that the resulting symmetrizer is dense on its active support and that its correction to a scalar multiple of the identity is essentially full rank. Low-rank, narrow-band, and other highly sparse approximations generally fail to enforce dissipativity, although moderate density thresholding and hybrid approximations can work after absorbing residual indefiniteness into an additional classical shift. The unresolved problem is therefore to construct W or S directly in a structured, polynomially parametrised family that preserves dissipativity while enabling an efficient quantum implementation and avoiding the current dense O(n^3) classical construction.