Properties recoverable from non-controlled time evolution

Determine what properties of an eigenvalue transform f(H) can be obtained from non-controlled time evolution of a matrix A, thereby characterizing the information accessible without controlled time-evolution gates in the general matrix-function setting.

Background

The paper develops Randomized Extrapolation Trotterization for estimating scalar quantities, observables, and measurement distributions associated with matrix functions f(H). The algorithms explicitly use controlled time-evolution operations, typically within Hadamard-test circuits.

The authors note related work studying phase estimation and related properties using direct time evolution without control gates, or with limited control. They then pose the unresolved question of identifying the properties of a matrix function that remain accessible when controlled time evolution is unavailable.

References

Recent study has also asked what properties of $A$ can be determined from direct time evolution and no control gates at all , or limited control . We frame the following interesting open question in the general context of matrix functions: what properties can be gleaned from $f(H)$ given non-controlled time evolution of $A$?

— Resource-efficient quantum eigenvalue transform with commutator scaling  (2608.13862 - Mazumder et al., 14 Aug 2026) in Section 1, Discussion subsection