π -Corrected Heisenberg Limit

Abstract

We show that for the most general adaptive noiseless estimation protocol, where Uϕ = e iϕΛ is a unitary map describing the elementary evolution of the probe system, the asymptotically saturable bound on the precision of estimating ϕ takes the form ∆ϕ ≥ π n(λ + −λ − ) , where n is the number of applications of Uϕ, while λ+, λ− are the extreme eigenvalues of the generator Λ. This differs by a factor of π from the conventional bound, which is derived directly from the properties of the quantum Fisher information. That is, the conventional bound is never saturable. This result applies both to idealized noiseless situations as well as to cases where noise can be effectively canceled out using variants of quantum-error correcting protocols.

Optimal probes and error-correction schemes in multi-parameter quantum metrology

“…Finally, we note that in this paper we have followed the frequentist estimation approach, and in principle more stringent HS bounds might be derived when following the Bayesian approach as was demonstrated recently in the single parameter case [75].…”

Dual-Frequency Quantum Phase Estimation Mitigates the Spectral Leakage of Quantum Algorithms

“…Observe that the sample-mean estimator based on the rectangular window exhibits an O(1/ √ N ) scaling, while the others exhibit O(1/N ) scaling. This is a phenomenon that has also been observed in [18], which suggests that the rectangular window does not provide a substantial quantum speedup in the sense of RMSE scaling, since the O(1/N ) scaling (i.e., the "Heisenberg limit [25]) is an important characteristic of quantum algorithms conceived for phase estimation.…”

Average number is an insufficient metric for interferometry

“…Even this finite improvement can be hard to achieve with better-than-Heisenberg strategies requiring either a very large number of repetitions [20,56] or a very large amount of prior knowledge [21,45]. Moreover, there is an extensive body of evidence supporting that a Heisenberg-scaling in the total number used across all repetitions cannot be surpassed [20,22,26,57,58].…”