Consider the single qubit unitary matrices commonly referred to as the Pauli matrices defined as:

$$I=\begin{pmatrix}1&0 \\ 0 &1\end{pmatrix},

X=\begin{pmatrix}0&1 \\ 1&0 \end{pmatrix},

Y=\begin{pmatrix}0&1 \\ -1&0 \end{pmatrix},

Z=\begin{pmatrix}1&0 \\ 0&-1 \end{pmatrix}.$$

It is worth noting that the operator $Y$ is commonly defined instead as the operator $\sigma_Y=iY$, but the definition of $Y$ introduced here will be convenient for our purposes. These operators satisfy the following properties:

\begin{align*}

X^2&=Z^2=-Y^2=I \\

XY&=-YX=Z, \\

YZ &=-ZY=X, \\

ZX &=-XZ=Y.

\end{align*}

These Properties give the set $P=\{\pm I,\pm X, \pm Y, \pm X\}$ a group structure under the usual matrix multiplication. Define $P_n:=\{U_1\otimes\dots\otimes U_n \ | \ U_j\in P, 0\leq j\leq n\}$, as the set of $n$-fold tensor products of Pauli operators from $P$. The set $P_n$ also forms a group structure under the natural multiplication and is called the

An important property about the Pauli operators is that they span the space of unitary operators acting on a single qubit. That is, any single qubit unitary $U$ can be expressed as

\[

U=c_II+c_XX+c_YY+c_ZZ,

\]

where the vector $(c_I,c_X,c_Y,c_Z)$ consists of complex numbers and is of unit norm. Similarly, any unitary operator acting on a $n$-qubit Hilbert space can be expressed in terms of elements of the Pauli group $P_n$.

Moreover, the Pauli Group $P_n$ also satisfies the following properties:

$$\Hil_S:=\{\ket{\psi}\in\Hil^{2^n} \ | \ M\ket{\psi}=\ket{\psi} \text{for all} M\in S \}$$

consists of the simultaneous eigenspace with eigenvalue $+1$ of elements of $S$. The space $\Hil_S$ is called the \emph{stabilizer code} associated with $S$, and $S$ is called the

A

Index the elements of a generating set of a stabilizer $S$ as $\{M_1,\dots,M_{n-k}\}$. The utility of the stabilizer formalism for quantum error correction comes from the fact that the elements of $S$ serve as operators for diagnosing possible errors that may occur to an encoded state of $\ket{\psi}\in\Hil_S$. In general, an error can be represented in terms elements $E_a\in P_n$. Then since every $E_a$ either commutes or anti-commutes with some generator $M_j\in S$, the following two cases may occur.

\[

M_jE_a=(-1)^{s_{a,j}}E_aM_j.

\]

If it is the case that for every $a\neq b$, with $s_{a,j}\neq s_{b,j}$ for all $j$, then the code is considered to be

Another condition which must be satisfied by the stabilizer $S$ in order to ensure complete error recovery due to arbitrary errors is that, for each possible error $E_a, E_b$ and any $\ket{\psi}\in\Hil_S$,

\[

\bra{\psi}E_a^\dagger E_b\ket{\psi}=C_{ab},

\]

such that the constants $C_{ab}$ are independent of $\ket{\psi}$. This condition can be equivalently shown to hold if one of the following holds for each possible pair of errors $E_a$ and $E_b$:

$$I=\begin{pmatrix}1&0 \\ 0 &1\end{pmatrix},

X=\begin{pmatrix}0&1 \\ 1&0 \end{pmatrix},

Y=\begin{pmatrix}0&1 \\ -1&0 \end{pmatrix},

Z=\begin{pmatrix}1&0 \\ 0&-1 \end{pmatrix}.$$

It is worth noting that the operator $Y$ is commonly defined instead as the operator $\sigma_Y=iY$, but the definition of $Y$ introduced here will be convenient for our purposes. These operators satisfy the following properties:

\begin{align*}

X^2&=Z^2=-Y^2=I \\

XY&=-YX=Z, \\

YZ &=-ZY=X, \\

ZX &=-XZ=Y.

\end{align*}

These Properties give the set $P=\{\pm I,\pm X, \pm Y, \pm X\}$ a group structure under the usual matrix multiplication. Define $P_n:=\{U_1\otimes\dots\otimes U_n \ | \ U_j\in P, 0\leq j\leq n\}$, as the set of $n$-fold tensor products of Pauli operators from $P$. The set $P_n$ also forms a group structure under the natural multiplication and is called the

*Pauli group*with order $|P_n|=2^{2n+1}$.An important property about the Pauli operators is that they span the space of unitary operators acting on a single qubit. That is, any single qubit unitary $U$ can be expressed as

\[

U=c_II+c_XX+c_YY+c_ZZ,

\]

where the vector $(c_I,c_X,c_Y,c_Z)$ consists of complex numbers and is of unit norm. Similarly, any unitary operator acting on a $n$-qubit Hilbert space can be expressed in terms of elements of the Pauli group $P_n$.

Moreover, the Pauli Group $P_n$ also satisfies the following properties:

- Every $M\in P_n$ in unitary: $M^\dagger=M^{-1}$.
- Every $M\in P_n$ satisfies $M^2=\pm I^{\otimes n}$.
- If $M^2=I^{\otimes n}$, then $M=M^\dagger$; if $M^2=-I$, then $M=-M^\dagger$.
- For any $M, N \in P_n$, either $MN=NM$ (they commute) or $MN=-NM$ (they anti-commute).

$$\Hil_S:=\{\ket{\psi}\in\Hil^{2^n} \ | \ M\ket{\psi}=\ket{\psi} \text{for all} M\in S \}$$

consists of the simultaneous eigenspace with eigenvalue $+1$ of elements of $S$. The space $\Hil_S$ is called the \emph{stabilizer code} associated with $S$, and $S$ is called the

*stabilizer*of the code.A

*generating set*of $S$ is a collection of elements of $S$ such that each element of $S$ can be expressed as some product of elements from the generating set. In addition, it is required that the elements of the generating set be*independent*, meaning that no element of the generating set can be expressed as a product of the other elements of the generating set. It can be shown \cite{Got} that if $S$ has $n-k$ generators, then the codes space $\Hil_S$ has dimension $2^k$ implying that it can effectively encode $k$ qubits.Index the elements of a generating set of a stabilizer $S$ as $\{M_1,\dots,M_{n-k}\}$. The utility of the stabilizer formalism for quantum error correction comes from the fact that the elements of $S$ serve as operators for diagnosing possible errors that may occur to an encoded state of $\ket{\psi}\in\Hil_S$. In general, an error can be represented in terms elements $E_a\in P_n$. Then since every $E_a$ either commutes or anti-commutes with some generator $M_j\in S$, the following two cases may occur.

- If $E_a$ anti-commutes with some $M_j$, then for $\ket{\psi}\in\Hil_S$,

\[ M_jE_a\ket{\psi}=-E_aM_j\ket{\psi}=-E_a\ket{\psi},\] which implies that the error can be detected if the the erred state $E_a\ket{\psi}$ is acted on by $M_j$.

- If $E_a$ commutes with some $M_j$, then for $\ket{\psi}\in\Hil_S$,

\[ M_jE_a\ket{\psi}=E_aM_j\ket{\psi}=E_a\ket{\psi}, \] and the error may go undetected when the erred state $E_a\ket{\psi}$ is acted on by $M_j$.

\[

M_jE_a=(-1)^{s_{a,j}}E_aM_j.

\]

If it is the case that for every $a\neq b$, with $s_{a,j}\neq s_{b,j}$ for all $j$, then the code is considered to be

*non degenerate*and there will be no ambiguity in what error occurred allowing for the error to be corrected by measuring the $n-k$ generators of $S$.Another condition which must be satisfied by the stabilizer $S$ in order to ensure complete error recovery due to arbitrary errors is that, for each possible error $E_a, E_b$ and any $\ket{\psi}\in\Hil_S$,

\[

\bra{\psi}E_a^\dagger E_b\ket{\psi}=C_{ab},

\]

such that the constants $C_{ab}$ are independent of $\ket{\psi}$. This condition can be equivalently shown to hold if one of the following holds for each possible pair of errors $E_a$ and $E_b$:

- $E_a^\dagger E_b\in S$,
- There exists an $M\in S$ that anti-commutes with $E_a^\dagger E_b$.

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