### 计算机代写|量子计算代写Quantum computing代考|General Notation for Operations on Transform Matrices

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## 计算机代写|量子计算代写Quantum computing代考|General Notation for Operations on Transform Matrices

The following notation describes the operations on a transform matrix M over $\mathrm{GF}(\mathrm{K})[5,12]$ :

$M_{D} \rightarrow M_{o}^{p_{0}}-p_{K-1} \mid q_{0} \quad-q_{K-1}$,
where $\mathrm{M}{\mathrm{D}}$ is the Derived (Modified) Matrix, $\mathrm{M}{\mathrm{O}}$ is the Original Matrix. The symbols $\mathrm{p}{0}, \mathrm{p}{1}, \ldots, \mathrm{p}{\mathrm{K}-1}$ are row multiplication numbers $\in \mathrm{GF}(\mathrm{K}),{0,1, \ldots, \mathrm{K}-1}$ are indices referring to row ${0}$, row ${ }{1}, \ldots$, and row $\mathrm{K}{-1}$. The symbols $\mathrm{q}{0}, \mathrm{q}{1}, \ldots, \mathrm{q}{\mathrm{K}-1}$ are column multiplication numbers $\in \mathrm{GF}(\mathrm{K})$, and ${0,1, \ldots, \mathrm{K}-1}$ are indices referring to column $n{0}$, column $n_{1}, \ldots$, and column $n_{K-1}$. The operations performed utilizing the upper notation are done through the multiplication of all the elements of row $w_{i}$ of the matrix $M_{0}$ by $p_{i}$ and then multiplying each resulting element of (row $\mathrm{i}{\mathrm{i}} \mathrm{p}{\mathrm{i}}$ ) by $\mathrm{q}{\mathrm{j}}$, where $\mathrm{i}, \mathrm{j}=0,1, \ldots, \mathrm{k}-1$ ( i.e., $\prod{i, \nabla_{j}}\left(p_{i} \cdot q_{j} \cdot\right.$ row $\left.{i}\right)$ ). The mathematical interpretation of this notation, in terms of matrices, is as follows: if $\mathrm{D}$ is a diagonal matrix $\Rightarrow \mathrm{D}=$ Diag $(\alpha, \beta, \ldots, \gamma)$, then $\mathrm{M}{\mathrm{D}}=\mathrm{D} \cdot \mathrm{M}{\mathrm{o}} \Rightarrow \mathrm{M}{\mathrm{D}}{ }^{-1}=\left(\mathrm{D} \cdot \mathrm{M}{\mathrm{o}}\right)^{-1}=$ $\mathrm{M}{0}^{-1} \mathrm{D}^{-1}$. The following Eq. can be applied to obtain the functional expansions for any modified transform matrix:
$$f=M_{s}^{-1} M \vec{F},$$
where $\mathrm{M}$ is the transform matrix, and $\vec{F}$ is the truth vector of the function $\mathrm{f}$.

## 计算机代写|量子计算代写Quantum computing代考|Invariant Families of Multi-Valued Spectral Transforms

To introduce Theorems $2.1,2.2$, and $2.3$, the following definitions are presented (where $\mathrm{p}$ is a prime number and $\mathrm{k}$ is a natural number of value $k \geq 1$ ).

Definition 2.1. The transform matrix that is generated by multiplying the rows of $G F\left(p^{k}\right)$ Shannon matrix by the numbers ${\alpha$, $\beta, \ldots, \gamma} \in \mathrm{GF}\left(\mathrm{p}^{\mathrm{k}}\right.$ ) respectively is called $\alpha \beta \ldots \gamma$ IS (invariant Shannon) matrix.

Definition 2.2. The transform matrix that is generated by multiplying the rows of $G F\left(p^{k}\right.$ ) Davio of type $t$ (denoted by $D_{t}$ ) matrix by the numbers ${\alpha, \beta, \ldots, \gamma} \in G F\left(\mathrm{p}^{\mathrm{k}}\right)$ respectively is called $\alpha \beta \ldots \gamma \mathrm{ID}_{\mathrm{t}}$ (invariant Davio of type $\mathrm{t}$ ) matrix, where $\mathrm{t} \in \mathrm{GF}\left(\mathrm{p}^{\mathrm{k}}\right.$ ).

Definition 2.3. The transform matrix that is generated by multiplying the rows of $G F\left(p^{k}\right)$ flipped Shannon matrix by the numbers ${\alpha, \beta, \ldots, \gamma} \in \operatorname{GF}\left(p^{\mathrm{k}}\right)$ respectively is called $\alpha \beta \ldots \gamma$ IfS (invariant flipped Shannon) matrix, where $t \in G F\left(p^{k}\right)$.

## 计算机代写|量子计算代写Quantum computing代考|Summary

In this Chapt. we introduced a systematic method to create and classify new multiple-valued invariant non-singular spectral transforms based on multi-valued fundamental Shannon expansions and Davio expansions over an arbitrary radix of Galois field.

The new spectral transforms will have an application in the construction of regular layout in three-dimensions as will be shown in Chapt. 4. The new spectral transforms have an important property: their basis functions are exactly the same as the basis functions of the fundamental Shannon and Davio expansions but scaled by constants (i.e., $\hat{\alpha}, \hat{\beta}, \ldots, \hat{\gamma}$ ) . Moreover, these constants are not generated arbitrarily; they are the multiplicative inverses of the corresponding constants that scale the rows of the corresponding basic multi-valued Shannon, Davio, and flipped Shannon transform matrices, and can be directly calculated according to the axioms of the Galois field. Due to the previously mentioned property, these transforms possess fast inverses and therefore are suitable for many applications including the fast computation of spectral transforms. All results in this Chapt. can be extended to an arbitrary $\mathrm{GF}\left(\mathrm{p}^{\mathrm{k}}\right)$ fields, where $p$ is a prime number and $k$ is a natural number $k \geq 1$. Also, although the new expansions that are developed in this Chapt. are for Galois field and 1-RPL, similar and analogous developments can be done for other complete algebraic structures and spectral transforms with different sorts of literals and operations.

Lattice structures based on the new ternary invariant Shannon and Davio expansions will be synthesized in Chapt. 4. The new 3-D lattice structures will be further extended to include reversible lattice structures in Chapt. 6, and their corresponding quantum circuits will be introduced in Chapt. 10. The new primitive in Fig. 2.4b will be extended to reversible logic in Chapt. 5 and will be used in Chapt. 6 to build reversible binary Shannon lattice structures. Also, the new families of multiple-valued invariant Shannon and Davio expansions that were introduced in this Chapt. will be fully generalized to include the reversible counterparts of such new expansions in Chapt. 5 , from which new reversible primitives and structures will be constructed in the following Chapts.

F=米s−1米F→,

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