Fundamental limitations for measurements in quantum many-body systems

TL;DR

This paper demonstrates a fundamental exponential scaling limitation in implementing measurements in quantum many-body systems using dynamical schemes, impacting quantum simulation capabilities.

Contribution

It introduces a theoretical framework showing exponential time scaling limits for measurement implementation via unitary evolution or Hamiltonian control.

Findings

Measurement realization time scales exponentially with system size.

Constructs epsilon-packings for observable manifolds and compares with quantum circuit coverings.

Highlights fundamental limitations in measurement schemes for quantum many-body systems.

Abstract

Dynamical measurement schemes are an important tool for the investigation of quantum many-body systems, especially in the age of quantum simulation. Here, we address the question whether generic measurements can be implemented efficiently if we have access to a certain set of experimentally realizable measurements and can extend it through time evolution. For the latter, two scenarios are considered (a) evolution according to unitary circuits and (b) evolution due to Hamiltonians that we can control in a time-dependent fashion. We find that the time needed to realize a certain measurement to a predefined accuracy scales in general exponentially with the system size -- posing a fundamental limitation. The argument is based, on the construction of $\varepsilon$-packings for manifolds of observables with identical spectra and a comparison of their cardinalities to those of $\varepsilon$-coverings for quantum circuits and unitary time-evolution operators. The former is related to the study of Grassmann manifolds.