Abstract
The authors study the estimation of a Hamiltonian parameter under Markovian noise with the help of fast quantum controls. When the sensing time is long, the paper shows that for a generic set of Markovian evolutions, there exists an approximate quantum error correction strategy which is the best among all types of quantum controls and minimizes the estimation error asymptotically.
Highlights
Quantum metrology is one of the most important state-ofthe-art quantum technologies, studying the precision limit of parameter estimation in quantum systems [1,2,3,4,5,6]
We propose an approximate quantum error correction (AQEC) strategy saturating the standard quantum limit (SQL) lower bound of precision and an efficient numerical algorithm solving the optimal AQEC codes for different noises
The algorithm runs as follows: (a) Solving minh,h,h|β=0 α using the semidefinite program (SDP) gives us an optimal α satisfying α = minh,h,h|β=0 α . (b) Suppose is the projection onto the subspace spanned by all eigenstates corresponding to the largest eigenvalue of α, we find an optimal C C †
Summary
Quantum metrology is one of the most important state-ofthe-art quantum technologies, studying the precision limit of parameter estimation in quantum systems [1,2,3,4,5,6]. A necessary and sufficient condition of achieving the HL under Markovian noise—the “Hamiltonian not in the Lindblad span” (HNLS) condition—was established using the channel extension method [29,30,31] These bounds were successful at showing the scaling limit of quantum strategies, only in several special cases, the saturability of these lower bounds was established, e.g., for dephasing and erasure noise [42] and for teleportation-covariant channels as a special type of programmable channels [45,46]. We describe our AQEC strategy consisting of both a twodimensional AQEC code and an optimal recovery channel This allows the original quantum channel to be reduced to an effective channel where a Z (the Pauli-Z operator) signal was sensed under dephasing noise—a special case where the precision lower bound was known to be saturable [11,39,42]. We optimize the achievable precision over all possible AQEC codes, which coincides with the precision lower bound, completing the proof
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