Erratum to: Int J Theor Phys (2016) 55:3234–3250

DOI 10.1007/s10773-016-2954-6

The original version of this article unfortunately contained mistakes that were introduced during production processing. The introduced mistakes are listed and corrected below:1. Eq. (7)

$$={\otimes}^{7}_{i=0}\left( T^{i}_{YX}\left.\left|C^{i}_{YX} \right\rangle\right.\right)= {\otimes}^{7}_{i=0}\left.\left|C^{i}_{YX}\otimes h^{i}_{YX}\right\rangle\right.=\left.\left|f\left( Y,X\right)\right\rangle\right. $$

should be

$$ ={\otimes}^{7}_{i=0}\left( T^{i}_{YX}\left.\left|C^{i}_{YX} \right\rangle\right.\right)= {\otimes}^{7}_{i=0}\left.\left|C^{i}_{YX}\oplus h^{i}_{YX}\right\rangle\right.=\left.\left|f\left( Y,X\right)\right\rangle\right. \!. $$
(7)

2. The sentence below Eq. (7), “UYX” should be “ U Y X ”.3. The last line in Eq. (9)

$$\begin{array}{@{}rcl@{}} =\frac{1}{2^{n}}\left( \sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{\underset{yx\neq YX}{x=0}} \left.\left.\left.{\otimes}^{7}_{i=0}\right|C^{i}_{yx} \right\rangle\right|\left.\left.\left.\left.\left.\!\!\!\!{\vphantom{{\otimes}^{7}_{i=0}}}yx \right\rangle + {\otimes}^{7}_{i=0}\right| C^{i}_{YX} h^{i}_{YX}\right\rangle\right|XY\right\rangle\right) \end{array} $$

should be

$$\begin{array}{@{}rcl@{}} =\frac{1}{2^{n}}\left( \sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{\underset{yx\neq YX}{x=0}} \left.\left.\left.{\otimes}^{7}_{i=0}\right|C^{i}_{yx} \right\rangle\right|\left.\left.\left.\left.\left.\!\!\!\!{\vphantom{{\otimes}^{7}_{i=0}}}yx\right\rangle + {\otimes}^{7}_{i=0}\right| C^{i}_{YX} \oplus h^{i}_{YX}\right\rangle\right|YX\right\rangle\right). \end{array} $$
(9)

4. Eq. (10)

$$\begin{array}{@{}rcl@{}} \left.\left.B_{Y_{1}{X}_{1}}B_{YX}\right|{M}\right\rangle &=&B_{Y_{1} X_{1}} \left( B_{YX}\frac{1}{2^{n}}\sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{y=0}\left.\left.\left.\left.{\otimes}^{7}_{i=0}\right|C^{i}_{yx}\right\rangle\right|yx\right\rangle\right)\\ &=&\left.\!\!B_{Y_{1}X_{1}}\frac{1}{2^{n}}\!\left( \!\!\sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{\underset{yx\neq YX}{x=0}}\!\!\left.\left.\left.{\otimes}^{7}_{i=0}\right|\! C^{i}_{yx}\!\right\rangle\right|\!yx\!\right\rangle+ \!\!\left.\left.{\otimes}^{7}_{i=0}\right|C^{i}_{YX}\left.\left.h^{i}_{YX}\right\rangle\right|XY\right\rangle\right)\\ &=&\frac{1}{2^{n}}\!\left( \!\sum\limits^{2^{n}-1}_{y=0}\!\!\sum\limits^{2^{n}-1}_{\underset{yx\neq YX,Y_{1}X_{1}}{x=0}}\!\!\!\!\left.\left.{\otimes}^{7}_{i=0}\right|\left.\left.\!C^{i}_{yx}\right\rangle\right|yx\right\rangle+\left.{\otimes}^{7}_{i=0}\right|\left.\left.\left.\!\!C^{i}_{YX} h^{i}_{YX}\right\rangle\right| YX\right\rangle+\left.\!{\otimes}^{7}_{i=0}\right|C^{i}_{Y_{1}X_{1}}\left.\left.\!\!\otimes h^{i}_{Y_{1}X_{1}}\right\rangle\right|Y_{1}X_{1}\right) \end{array} $$

should be

$$\begin{array}{@{}rcl@{}} \!\!\!\!\left.\left.B_{Y_{1}{X}_{1}}B_{YX}\right|{M}\right\rangle &=&\!B_{Y_{1} X_{1}} \left( B_{YX}\frac{1}{2^{n}}\sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{x=0}\left.\left.\left.\left.{\otimes}^{7}_{i=0}\right|C^{i}_{yx}\right\rangle\right|yx\right\rangle\right)\\ &=&\!\left.\!\!B_{Y_{1}X_{1}}\frac{1}{2^{n}}\!\left( \!\!\sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{\underset{yx\neq YX}{x=0}}\!\!\left.\left.\left.{\otimes}^{7}_{i=0}\right|\! C^{i}_{yx}\!\right\rangle\right|\!yx\!\right\rangle+ \!\!\left.\left.{\otimes}^{7}_{i=0}\right|C^{i}_{YX}\left.\left.\oplus h^{i}_{YX}\right\rangle\right|YX\right\rangle\right)\\ &=&\!\frac{1}{2^{n}}\!\left( \!\sum\limits^{2^{n}-1}_{y=0}\!\!\sum\limits^{2^{n}-1}_{\underset{yx\neq YX,Y_{1}X_{1}}{x=0}}\!\!\!\!\left.\left.{\otimes}^{7}_{i=0}\right|\left.\left.\!C^{i}_{yx}\right\rangle\right|yx\right\rangle+\left.{\otimes}^{7}_{i=0}\right|\left.\left.\left.\!\!C^{i}_{YX}\oplus h^{i}_{YX}\right\rangle\right| YX\right\rangle+\left.\!{\otimes}^{7}_{i=0}\right|C^{i}_{Y_{1}X_{1}}\left.\left.\!\!\oplus h^{i}_{Y_{1}X_{1}}\right\rangle\right|Y_{1}X_{1}\!\right). \end{array} $$
(10)

5. First line 1 below section 4.1 where it says,

$$\left|M\right.\rangle=\frac{1}{2^{n}}\sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{y=0}|g(y,x)\rangle|yx\rangle$$

should be

$$\left|M\right\rangle=\frac{1}{2^{n}} \sum\limits^{2^{n}-1}_{y=0}\sum\limits^{2^{n}-1}_{x=0}|g(y,x)\rangle|yx\rangle.$$

The publisher regrets that the following errors were introduced during the production process.