Tight Trotter error bound for general banded Toeplitz operators

Establish a tight Trotter error bound for general banded Toeplitz operators, including the exponential-in-agnitude factor in t times the coefficient 1-norm that can arise for non-contractive coefficient sets.

Background

The paper derives a QFT-based LieTrotter product formula for exponentiating a banded Toeplitz operator. Its stated error has a term of order t2 times the squared coefficient 1-norm divided by the number of Trotter steps, together with a truncation term controlled by the phase-approximation depth. The derivation assumes contractive diagonal components, meaning that their maximum real parts are nonpositive.

The authors note that for non-contractive coefficient sets, a more general Trotter estimate includes a factor exponential in t times the coefficient 1-norm. They leave unresolved the problem of obtaining a tight bound that applies to general banded Toeplitz operators without suppressing relevant constants or growth factors.

References

First, the Trotter bound of Theorem~\ref{thm:qft-trotter} is stated up to constants that, for non-contractive coefficient sets, include a factor exponential in $t\norm{\alpha}_1$; a tight statement for general banded Toeplitz operators is open.

— Quantum algorithms for the exponentiation of Toeplitz matrices and applications in partial differential equations  (2609.31372 - Gutiérrez et al., 25 Sep 2026) in Section Discussion and conclusions