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I am looking to implement a quantum version of the arcsinus function. Such a problem is motivated by the HHL algorithm where $x\mapsto 1/x$ and $\arcsin$ can be used to get $1/x$ from the computational basis state into the amplitude.

My questions are based on the paper Optimizing Quantum Circuits for Arithmetic (arxiv link :https://arxiv.org/abs/1805.12445). Their idea is to use a polynomial approximation of the function $f$ and to partition the domain $\Omega$ of study of $f$ : $$ \Omega = \bigcup_{i=1}^M \Omega_i \quad \Omega_i\cap \Omega_j = \emptyset \, \forall i \neq j $$ and then perform a case distinction for each input, evaluating a different polynomial for $x\in \Omega_i$ and $y\in \Omega_j$, $i\neq j$. $M$ is chosen in order to achieve a certain precision and the degree of the polynomials are all bounded by a constant $d$.

Evaluating a single polynomial $P(x) = \sum_{i=0}^d a_ix^i$ can be done using the Horner scheme, where one iteratively performs a multiplication by $x$ and an addition by $a_i$ for $i\in \{d, d-1, \cdots 0\}$ :

$$ a_d \mapsto a_dx+a_{d-1} \mapsto a_dx^2+a_{d-1}x + a_{d-2} \mapsto \cdots \mapsto P(x)$$

At iteration $i$, the last iterate is added by ${a_i}$, while this does not represent any difficulty in classical computing, a register has to hold the set of coefficients ${a_i}$, and has to be changed at each iteration. In their paper, the authors assume that $\mathrm{NEXT}_a$ implements such an operation.

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My question : How can one implement efficiently the function $\mathrm{NEXT}_a$ ?

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