Revision history [back]

I have a module (so-called stabilizer map) that can be expressed as a block diagonal matrix of polynomial rings over the field $Z_2$ (an example is defined in Chaper 3 here on page 54 of arxiv.1607.01387). How can I calculate the cokernel of this map or the image of this map in sagemath? Please let me know for more information.

I have a module (so-called stabilizer map) that can be expressed as a block diagonal matrix of polynomial rings over the field $Z_2$ (an example is defined in Chaper 3 here on page 54 of arxiv.1305.6973 and page 41 of arxiv.1607.01387). How can I calculate the cokernel of this map or the image of this map in sagemath? Please let me know for more information. information.

I have a module (so-called stabilizer map) map between two modules, that can be expressed as a block diagonal matrix of polynomial rings over the field $Z_2$ (an example is defined in Chaper 3 here on page 54 of arxiv.1305.6973 and page 41 of arxiv.1607.01387). How can I calculate the cokernel of this map or the image of this map in sagemath? Please let me know for more information.

I have a map between two modules, that can be expressed as a block diagonal matrix of polynomial rings over the field $Z_2$ (an example is defined in Chaper 3 here on page 54 of arxiv.1305.6973 and page 41 of arxiv.1607.01387). How can I calculate the cokernel of this map or the image of this map in sagemath? Please let me know for more information.

I have a map between two modules, that can be expressed as a block diagonal matrix of polynomial rings over the field $Z_2$ (an example is defined in Chaper 3 here on page 54 of arxiv.1305.6973 and page 41 of arxiv.1607.01387). How can I calculate the cokernel of this map or the image of this map in sagemath? Please let me know for more information.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have a two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. .The stabilizer map between two modules, that can be expressed written as a block diagonal matrix of
$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $ where for example, $y+xy$ is a polynomial rings over the field $Z_2$ (an example is defined in Chaper 3 here on page 54 of arxiv.1305.6973 and page 41 of arxiv.1607.01387). How can I calculate the cokernel of this map that specifies the action on the first two dimensional system as $y+xy=0x^{0}y^{0}+1x^{0}y^{1}+0x^{1}y^{0}+1x^{1}y^{1}$ where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or the image of this map in sagemath? Please let me know for more information.not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. .The stabilizer map can be written as
$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $ where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as $y+xy=0x^{0}y^{0}+1x^{0}y^{1}+0x^{1}y^{0}+1x^{1}y^{1}$ x^{1}y^{1}$
where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. .The stabilizer map can be written as
$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $ where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as $y+xy=0$ y+xy=0x^{0}y^{0}+1x^{0}y^{1}+0x^{1}y^{0}+1x^{1}y^{1}$ x^{1}y^{1} $
where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. .The stabilizer map can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $ $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+xy=0y+x y=0x^{0}y^{0}+1x^{0}y^{1}+0x^{1}y^{0}+1x^{1}y^{1} $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. .The stabilizer map can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0x^{0}y^{0}+1x^0 y^0+ 1x^{0}y^{1}+0x^0 y^1+0x^{1}y^{0}+1x^1 y^0 +1x^{1}y^{1} x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. .The stabilizer map can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0x^0 y^0+ 1x^0 y^1+0x^1 y^0 +1x^1 y^1 $ $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. .The stabilizer The map $\sigma$ can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0x^0 y^0+ 1x^0 y^1+0x^1 y^0 +1x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. The map $\sigma$ can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0\ x+xy & 0\ 0 & 1+y\ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0y=0 \hspace{1mm} x^0 y^0+ 11 \hspace{1mm} x^0 y^1+0y^1+0 \hspace{1mm} x^1 y^0 +1+1 \hspace{1mm} x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map. Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. The map $\sigma$ can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0\ 0 \ x+xy & 0\ 0 & 1+y\ 0 \ 0 & 1+y \ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0 \hspace{1mm} x^0 y^0+ 1 \hspace{1mm} x^0 y^1+0 \hspace{1mm} x^1 y^0 +1 \hspace{1mm} x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map.

Just for example (taken from page 54 of arxiv.1305.6973), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. The map $\sigma$ can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0 \ x+xy & 0 \ 0 & 1+y \ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0 \hspace{1mm} x^0 y^0+ 1 \hspace{1mm} x^0 y^1+0 \hspace{1mm} x^1 y^0 +1 \hspace{1mm} x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map.

Just for example (taken from page 54 of arxiv.1305.6973), arxiv.1305.6973 or page 41 of arxiv.1607.01387), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. The map $\sigma$ can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0 \ x+xy & 0 \ 0 & 1+y \ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0 \hspace{1mm} x^0 y^0+ 1 \hspace{1mm} x^0 y^1+0 \hspace{1mm} x^1 y^0 +1 \hspace{1mm} x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map.

Just for example (taken from page 54 of arxiv.1305.6973 or page 41 of arxiv.1607.01387), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. The map $\sigma$ can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0 \ x+xy & 0 \ 0 & 1+y \ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0 \hspace{1mm} x^0 y^0+ 1 \hspace{1mm} x^0 y^1+0 \hspace{1mm} x^1 y^0 +1 \hspace{1mm} x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map.

Just for example (taken from page 54 of arxiv.1305.6973 or page 41 of arxiv.1607.01387), though this is not essential for the question, I can have two ``interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $ II(0,0)-IX(0,1)-XI(1,0)-XX(1,1) $ and $ ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1) $ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. The map $\sigma$ can be written as

$ \sigma=\left(\begin{array}{cc} y+xy & 0 \ x+xy & 0 \ 0 & 1+y \ 0 & 1+x \end{array}\right) $

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$ y+x y=0 \hspace{1mm} x^0 y^0+ 1 \hspace{1mm} x^0 y^1+0 \hspace{1mm} x^1 y^0 +1 \hspace{1mm} x^1 y^1 $

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.