# How to create this feature map?

In this paper, the following feature map is used:

$$x \to \vert\phi(x)\rangle = \frac{1}{\sqrt{2^k}}\sum_{i=0}^{2^k-1}\vert x\cdot g^i\rangle$$

But no circuit is provided. A theoretical description of the circuit is provided in the supplementary information on page 17 (the image is provided below).

The steps require:

1. Modular multiplication and exponentiation: $$C_{y,k}\vert i \rangle \vert 0^n \rangle= \vert i \rangle \vert (y\cdot g^i) \% p \rangle$$ But they don't describe how to create this.
2. Discrete log: $$U_y \vert (y\cdot g^i) \% p \rangle \vert0\rangle = \vert i \rangle\vert (y\cdot g^i) \% p \rangle$$

The overall process is described as: $$\vert0^n\rangle \overset{H^{\otimes k}}{\to}\frac{1}{\sqrt{2^k}}\sum_{i \in\{0,1\}^k}\vert i\rangle \overset{C_{y,k}}{\to}\frac{1}{\sqrt{2^k}}\sum_{i \in\{0,1\}^k}\vert i\rangle \vert (y\cdot g^i) \% p \rangle \overset{U_y^\dagger}{\to}\frac{1}{\sqrt{2^k}}\sum_{i \in\{0,1\}^k}\vert (y\cdot g^i) \% p \rangle$$

How to create this feature map?

• Welcome to QCSE! Please do not use images for text and formulas. For text, use text. For formulas, use mathjax. Using images makes the question unsearchable and hinders proper rendering. Also, instead of pasting a page from the paper, you should mention page number and write up your own understanding of the feature map construction. Please edit your question to improve it. Dec 14 '21 at 17:56