Abstract
We establish general limits on how precise a parameter, e.g. frequency or the strength of a magnetic field, can be estimated with the aid of full and fast quantum control. We consider uncorrelated noisy evolutions of N qubits and show that fast control allows to fully restore the Heisenberg scaling (~1/N^2) for all rank-one Pauli noise except dephasing. For all other types of noise the asymptotic quantum enhancement is unavoidably limited to a constant-factor improvement over the standard quantum limit (~1/N) even when allowing for the full power of fast control. The latter holds both in the single-shot and infinitely-many repetitions scenarios. However, even in this case allowing for fast quantum control helps to increase the improvement factor. Furthermore, for frequency estimation with finite resource we show how a parallel scheme utilizing any fixed number of entangled qubits but no fast quantum control can be outperformed by a simple, easily implementable, sequential scheme which only requires entanglement between one sensing and one auxiliary qubit.
Highlights
Precision measurements play a fundamental role in physics and beyond, as they constitute the main ingredient for many state-of-the-art applications and experiments [1]
We have considered the general limits of quantum metrology where one is, in principle, equipped with a full-scale quantum computer to assist in the sensing process
We have shown that one can use techniques from quantum error correction to detect or correct for certain kinds of errors while maintaining the sensing capabilities of the system
Summary
Precision measurements play a fundamental role in physics and beyond, as they constitute the main ingredient for many state-of-the-art applications and experiments [1]. In turn it was shown that quantum entanglement allows one to achieve the so-called Heisenberg scaling (HS) in precision, 1/N 2, a quadratic improvement as compared to classical approaches [4, 5] These precision limits apply to both single-shot protocols [6,7,8,9,10,11,12,13] as well as protocols utilizing many repetitions [3, 5, 14]. These lower bounds are often cumbersome to evaluate and hard to optimize, relying on educated guesses, numerical methods employing semi-definite programming or a combination of the two These bounds show that for typical uncorrelated noise processes the possible gain due to the usage of quantum resources is limited to a constant-factor improvement over the standard scaling, as opposed to a different scaling. A reader who is familiar with quantum metrology, or who is primarily interested in the results, can directly proceed to sections 4.1 and 4.2.4 where the first two of our main results ((i) and (ii)) are presented
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