Speaker
Description
In this talk, I present a unified view of quantum spectral probing in which structurelessness, in the form of unbiased mixing that avoids basis-dependent artifacts, is the key resource connecting density-of-states estimation and quantum topological data analysis. I will present DOS-QPE and its single-ancilla variant, which estimate global spectral features of Hamiltonians by probing them with mixed input states such as the maximally mixed state or mixtures over symmetry sectors. I will discuss how using the Acorn trick and its Haar/2-design randomness on a discarded purification register, we can realize these mixed probes in a way that is mathematically invariant yet more robust and verifiable, enabling randomness-based protocols while mitigating coherent errors. The same principle drives QTDA; to reliably extract Betti numbers from eigenspace degeneracies, we prepare structureless initial ensembles using efficient approximate unitary (t)-designs, replacing ideal Haar randomness with polynomial-depth circuits. Across these applications, controlled mixing and unbiasedness act as the common engine that turns spectral primitives into reliable geometric and topological tools.