How to find normalizer to a subgroup of the Pauli group?
$begingroup$
The Pauli operators are given by:
$X = left( begin{array} { c c } { 0 } & { 1 } \ { 1 } & { 0 } end{array} right) , quad Y = left( begin{array} { c c } { 0 } & { - i } \ { i } & { 0 } end{array} right) , quad Z = left( begin{array} { c c } { 1 } & { 0 } \ { 0 } & { - 1 } end{array} right),$
The Pauli group on n qubits, $G_{n}$, is the group generated by the operators described above applied to each of $n$ qubits in the tensor product Hilbert space $left( mathscr{H} right) ^ { otimes n }$, where $mathscr{H}$ is a 2D Hilbert space.
As an example for $n=1$ : $G _ { 1 } stackrel { mathrm { def } } { = } { pm I , pm i I , pm X , pm i X , pm Y , pm i Y , pm Z , pm i Z } equiv langle X , Y , Z rangle$. $G _{2}$ would of course contain tensor products of Pauli operatiors, such as $X otimes Z$, etc.
I am now given a subgroup of this group called the stabilizer $S_n$ of some state/vector. $S_n$ is defined such that this vector is the only common eigenvector of the elements of $S$ with eigenvalue +1.
The generators of $S_n$ have the following form: $K_i=X_i otimes_{jin N} Z_j$ with $iin {1,2,...,n}$ and N being the neighbourhood. You can imagine each 2-dimensional tensor space $mathscr{H}$ as a point on a sqare grid and $K_i$ acting with $X_i$ on the tensor space $mathscr{H}_i$ and $Z_j$ acing on the neighbouring grid points $mathscr{H}_j$. This means that $S_n$ depends on the structure of the grid and that each generator of $S_n$ acts (non-trivially) on 5 tensor spaces at the most. (e.g. $K_1=X_1 otimes Z_2 otimes Z_3 otimes Z_4 otimes Z_5$ if grid point 1 is surrounded by 4 neighbours.)
This is the general structure of $S_n$ and I would now like to find the normalizer of $S_n$ for arbitrary $n$. I would appreciate any help regarding this problem.
finite-groups quantum-groups
asked Dec 8 '18 at 12:01
HaddockHaddock
63
$endgroup$
add a comment |
$begingroup$
The Pauli operators are given by:
$X = left( begin{array} { c c } { 0 } & { 1 } \ { 1 } & { 0 } end{array} right) , quad Y = left( begin{array} { c c } { 0 } & { - i } \ { i } & { 0 } end{array} right) , quad Z = left( begin{array} { c c } { 1 } & { 0 } \ { 0 } & { - 1 } end{array} right),$
The Pauli group on n qubits, $G_{n}$, is the group generated by the operators described above applied to each of $n$ qubits in the tensor product Hilbert space $left( mathscr{H} right) ^ { otimes n }$, where $mathscr{H}$ is a 2D Hilbert space.
As an example for $n=1$ : $G _ { 1 } stackrel { mathrm { def } } { = } { pm I , pm i I , pm X , pm i X , pm Y , pm i Y , pm Z , pm i Z } equiv langle X , Y , Z rangle$. $G _{2}$ would of course contain tensor products of Pauli operatiors, such as $X otimes Z$, etc.
I am now given a subgroup of this group called the stabilizer $S_n$ of some state/vector. $S_n$ is defined such that this vector is the only common eigenvector of the elements of $S$ with eigenvalue +1.
The generators of $S_n$ have the following form: $K_i=X_i otimes_{jin N} Z_j$ with $iin {1,2,...,n}$ and N being the neighbourhood. You can imagine each 2-dimensional tensor space $mathscr{H}$ as a point on a sqare grid and $K_i$ acting with $X_i$ on the tensor space $mathscr{H}_i$ and $Z_j$ acing on the neighbouring grid points $mathscr{H}_j$. This means that $S_n$ depends on the structure of the grid and that each generator of $S_n$ acts (non-trivially) on 5 tensor spaces at the most. (e.g. $K_1=X_1 otimes Z_2 otimes Z_3 otimes Z_4 otimes Z_5$ if grid point 1 is surrounded by 4 neighbours.)
This is the general structure of $S_n$ and I would now like to find the normalizer of $S_n$ for arbitrary $n$. I would appreciate any help regarding this problem.
finite-groups quantum-groups
asked Dec 8 '18 at 12:01
HaddockHaddock
63
$endgroup$
add a comment |
$begingroup$
The Pauli operators are given by:
$X = left( begin{array} { c c } { 0 } & { 1 } \ { 1 } & { 0 } end{array} right) , quad Y = left( begin{array} { c c } { 0 } & { - i } \ { i } & { 0 } end{array} right) , quad Z = left( begin{array} { c c } { 1 } & { 0 } \ { 0 } & { - 1 } end{array} right),$
The Pauli group on n qubits, $G_{n}$, is the group generated by the operators described above applied to each of $n$ qubits in the tensor product Hilbert space $left( mathscr{H} right) ^ { otimes n }$, where $mathscr{H}$ is a 2D Hilbert space.
As an example for $n=1$ : $G _ { 1 } stackrel { mathrm { def } } { = } { pm I , pm i I , pm X , pm i X , pm Y , pm i Y , pm Z , pm i Z } equiv langle X , Y , Z rangle$. $G _{2}$ would of course contain tensor products of Pauli operatiors, such as $X otimes Z$, etc.
I am now given a subgroup of this group called the stabilizer $S_n$ of some state/vector. $S_n$ is defined such that this vector is the only common eigenvector of the elements of $S$ with eigenvalue +1.
The generators of $S_n$ have the following form: $K_i=X_i otimes_{jin N} Z_j$ with $iin {1,2,...,n}$ and N being the neighbourhood. You can imagine each 2-dimensional tensor space $mathscr{H}$ as a point on a sqare grid and $K_i$ acting with $X_i$ on the tensor space $mathscr{H}_i$ and $Z_j$ acing on the neighbouring grid points $mathscr{H}_j$. This means that $S_n$ depends on the structure of the grid and that each generator of $S_n$ acts (non-trivially) on 5 tensor spaces at the most. (e.g. $K_1=X_1 otimes Z_2 otimes Z_3 otimes Z_4 otimes Z_5$ if grid point 1 is surrounded by 4 neighbours.)
This is the general structure of $S_n$ and I would now like to find the normalizer of $S_n$ for arbitrary $n$. I would appreciate any help regarding this problem.
finite-groups quantum-groups
asked Dec 8 '18 at 12:01
HaddockHaddock
63
$endgroup$
The Pauli operators are given by:
$X = left( begin{array} { c c } { 0 } & { 1 } \ { 1 } & { 0 } end{array} right) , quad Y = left( begin{array} { c c } { 0 } & { - i } \ { i } & { 0 } end{array} right) , quad Z = left( begin{array} { c c } { 1 } & { 0 } \ { 0 } & { - 1 } end{array} right),$
The Pauli group on n qubits, $G_{n}$, is the group generated by the operators described above applied to each of $n$ qubits in the tensor product Hilbert space $left( mathscr{H} right) ^ { otimes n }$, where $mathscr{H}$ is a 2D Hilbert space.
As an example for $n=1$ : $G _ { 1 } stackrel { mathrm { def } } { = } { pm I , pm i I , pm X , pm i X , pm Y , pm i Y , pm Z , pm i Z } equiv langle X , Y , Z rangle$. $G _{2}$ would of course contain tensor products of Pauli operatiors, such as $X otimes Z$, etc.
I am now given a subgroup of this group called the stabilizer $S_n$ of some state/vector. $S_n$ is defined such that this vector is the only common eigenvector of the elements of $S$ with eigenvalue +1.
The generators of $S_n$ have the following form: $K_i=X_i otimes_{jin N} Z_j$ with $iin {1,2,...,n}$ and N being the neighbourhood. You can imagine each 2-dimensional tensor space $mathscr{H}$ as a point on a sqare grid and $K_i$ acting with $X_i$ on the tensor space $mathscr{H}_i$ and $Z_j$ acing on the neighbouring grid points $mathscr{H}_j$. This means that $S_n$ depends on the structure of the grid and that each generator of $S_n$ acts (non-trivially) on 5 tensor spaces at the most. (e.g. $K_1=X_1 otimes Z_2 otimes Z_3 otimes Z_4 otimes Z_5$ if grid point 1 is surrounded by 4 neighbours.)
This is the general structure of $S_n$ and I would now like to find the normalizer of $S_n$ for arbitrary $n$. I would appreciate any help regarding this problem.
finite-groups quantum-groups
finite-groups quantum-groups
asked Dec 8 '18 at 12:01
HaddockHaddock
63
asked Dec 8 '18 at 12:01
HaddockHaddock
63
asked Dec 8 '18 at 12:01
HaddockHaddock
63
asked Dec 8 '18 at 12:01
HaddockHaddock
63
asked Dec 8 '18 at 12:01
HaddockHaddock
63
63
add a comment |
add a comment |
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