Stabilizer Code - Relation Between Pauli Group and Binary Vectors

Relation Between Pauli Group and Binary Vectors

A simple but useful mapping exists between elements of and the binary vector space . This mapping gives a simplification of quantum error correction theory. It represents quantum codes with binary vectors and binary operations rather than with Pauli operators and matrix operations respectively.

We first give the mapping for the one-qubit case. Suppose is a set of equivalence classes of an operator that have the same phase:

$\left =\left\{ \beta A\ |\ \beta\in\mathbb{C},\ \left\vert \beta\right\vert =1\right\} .$

Let be the set of phase-free Pauli operators where . Define the map as

$00 \to I, \,\, 01 \to X, \,\, 11 \to Y, \,\, 10 \to Z$

Suppose . Let us employ the shorthand and $v=\left( z^{\prime}|x^{\prime }\right)$ where, . For example, suppose . Then . The map induces an isomorphism $\left :\left( \mathbb{Z} _{2}\right) ^{2}\rightarrow\left$ because addition of vectors in is equivalent to multiplication of Pauli operators up to a global phase:

$\left =\left \left .$

Let denote the symplectic product between two elements $u,v\in\left( \mathbb{Z}_{2}\right) ^{2}$ :

$u\odot v\equiv zx^{\prime}-xz^{\prime}.$

The symplectic product gives the commutation relations of elements of :

$N\left( u\right) N\left( v\right) =\left( -1\right) ^{\left( u\odot v\right) }N\left( v\right) N\left( u\right) .$

The symplectic product and the mapping thus give a useful way to phrase Pauli relations in terms of binary algebra. The extension of the above definitions and mapping to multiple qubits is straightforward. Let denote an arbitrary element of . We can similarly define the phase-free -qubit Pauli group $\left =\left\{ \left[ \mathbf{A}\right] \ |\ \mathbf{A}\in\Pi^{n}\right\}$ where

$\left =\left\{ \beta\mathbf{A}\ |\ \beta\in \mathbb{C},\ \left\vert \beta\right\vert =1\right\} .$

The group operation for the above equivalence class is as follows:

$\left \ast\left \equiv\left[ A_{1}\right] \ast\left \otimes\cdots\otimes\left[ A_{n}\right] \ast\left =\left \otimes\cdots\otimes\left =\left .$

The equivalence class forms a commutative group under operation . Consider the -dimensional vector space

$\left( \mathbb{Z}_{2}\right) ^{2n}=\left\{ \left( \mathbf{z,x}\right) :\mathbf{z},\mathbf{x}\in\left( \mathbb{Z}_{2}\right) ^{n}\right\} .$

It forms the commutative group with operation defined as binary vector addition. We employ the notation $\mathbf{u}=\left( \mathbf{z}|\mathbf{x}\right) ,\mathbf{v}=\left( \mathbf{z}^{\prime}|\mathbf{x}^{\prime}\right)$ to represent any vectors respectively. Each vector and has elements $\left( z_{1},\ldots ,z_{n}\right)$ and respectively with similar representations for and . The \textit{symplectic product} of and is

$\mathbf{u}\odot\mathbf{v\equiv}\sum_{i=1}^{n}z_{i}x_{i}^{\prime}-x_{i} z_{i}^{\prime},$

$\mathbf{u}\odot\mathbf{v\equiv}\sum_{i=1}^{n}u_{i}\odot v_{i},$

where and $v_{i}=\left( z_{i}^{\prime }|x_{i}^{\prime}\right)$ . Let us define a map $\mathbf{N}:\left( \mathbb{Z}_{2}\right) ^{2n}\rightarrow\Pi^{n}$ as follows:

$\mathbf{N}\left( \mathbf{u}\right) \equiv N\left( u_{1}\right) \otimes\cdots\otimes N\left( u_{n}\right) .$

Let

$\mathbf{X}\left( \mathbf{x}\right) \equiv X^{x_{1}}\otimes\cdots\otimes X^{x_{n}}, \,\,\,\,\,\,\, \mathbf{Z}\left( \mathbf{z}\right) \equiv Z^{z_{1}}\otimes\cdots\otimes Z^{z_{n}},$

so that and $\mathbf{Z}\left( \mathbf{z}\right) \mathbf{X}\left( \mathbf{x}\right)$ belong to the same equivalence class:

$\left =\left[ \mathbf{Z} \left( \mathbf{z}\right) \mathbf{X}\left( \mathbf{x}\right) \right] .$

The map $\left :\left( \mathbb{Z}_{2}\right) ^{2n}\rightarrow\left$ is an isomorphism for the same reason given as the previous case:

$\left =\left[ \mathbf{N}\left( \mathbf{u}\right) \right] \left[ \mathbf{N}\left( \mathbf{v}\right) \right] ,$

where . The symplectic product captures the commutation relations of any operators $\mathbf{N}\left( \mathbf{u}\right)$ and :

$\mathbf{N\left( \mathbf{u}\right) N}\left( \mathbf{v}\right) =\left( -1\right) ^{\left( \mathbf{u}\odot\mathbf{v}\right) }\mathbf{N}\left( \mathbf{v}\right) \mathbf{N}\left( \mathbf{u}\right) .$

The above binary representation and symplectic algebra are useful in making the relation between classical linear error correction and quantum error correction more explicit.

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