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tags: grover, amplitude estimation, quantum computing, 2020
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# [Modified Grover operator for quantum amplitude estimation](https://arxiv.org/pdf/2010.11656.pdf)
Shumpei Uno, Yohichi Suzuki, Keigo Hisanaga, Rudy Raymond, Tomoki Tanaka, Tamiya Onodera, Naoki Yamamoto

## Abstract
In this paper, we propose a quantum amplitude estimation method that uses a modified Grover operator and
quadratically improves the estimation accuracy in the ideal case, as in the conventional one using the standard
Grover operator. Under the depolarizing noise, the proposed method can outperform the conventional one in
the sense that it can in principle achieve the ultimate estimation accuracy characterized by the quantum Fisher
information in the limit of a large number of qubits, while the conventional one cannot achieve the same value of
ultimate accuracy. In general this superiority requires a sophisticated adaptive measurement, but we numerically
demonstrate that the proposed method can outperform the conventional one and approach to the ultimate accuracy,
even with a simple non-adaptive measurement strategy.

## 背景 (お気持ちを参照)

## お気持ち

Maximum likelyhood amplitude estimationで大事になってくるのは推定誤差だが、それはCramer-Raoの不等式から、Fisher情報量を用いて誤差の下限を評価することができる。
元々のMaximum likelyhood amplitude estimationのGrover Iteratorは
$$G = AU_0A^\dagger U_f$$
だが、depolarization noiseが入った時は、必ずしもそのノイズに強いIteratorとは言えない。
そこで、この論文ではdepolarization noiseが入った時により大きなFisher情報量を持つようなGrover Iterator $Q$ を考案した。
$$Q = U_0A^\dagger U_f A$$
ノイズがない時は$G$と$Q$のFisher情報量は同じ。