Stabilizer code#Definition

Quantum error-correcting codes restore a noisy,

decohered quantum state to a pure quantum state. A

stabilizer quantum error-correcting code appends ancilla qubits

to qubits that we want to protect. A unitary encoding circuit rotates the

global state into a subspace of a larger Hilbert space. This highly entangled,

encoded state corrects for local noisy errors. A quantum error-correcting code makes quantum computation

and quantum communication practical by providing a way for a sender and

receiver to simulate a noiseless qubit channel given a noisy qubit channel

whose noise conforms to a particular error model. The first quantum error-correcting codes are strikingly similar to classical block codes in their operation and performance.

The stabilizer theory of quantum error correction allows one to import some

classical binary or quaternary codes for use as a quantum code. However, when importing the

classical code, it must satisfy the dual-containing (or self-orthogonality)

constraint. Researchers have found many examples of classical codes satisfying

this constraint, but most classical codes do not. Nevertheless, it is still useful to import classical codes in this way (though, see how the entanglement-assisted stabilizer formalism overcomes this difficulty).

The stabilizer formalism exploits elements of

the Pauli group $\backslash Pi$ in formulating quantum error-correcting codes. The set

$\backslash Pi=\backslash left\backslash \{\; I,X,Y,Z\backslash right\backslash \}$ consists of the Pauli operators:

:$$

I\equiv

\begin{bmatrix}

1 & 0\\

0 & 1

\end{bmatrix}

,\ X\equiv

\begin{bmatrix}

0 & 1\\

1 & 0

\end{bmatrix}

,\ Y\equiv

\begin{bmatrix}

0 & -i\\

i & 0

\end{bmatrix}

,\ Z\equiv

\begin{bmatrix}

1 & 0\\

0 & -1

\end{bmatrix}

.

The above operators act on a single qubit – a state represented by a vector in a two-dimensional

Hilbert space. Operators in $\backslash Pi$ have eigenvalues $\backslash pm1$ and either commute

or anti-commute. The set $\backslash Pi^\{n\}$ consists of $n$-fold tensor products of

Pauli operators:

:$$

\Pi^{n}=\left\{

\begin{array}

[c]{c}

e^{i\phi}A_{1}\otimes\cdots\otimes A_{n}:\forall j\in\left\{ 1,\ldots

,n\right\} A_{j}\in\Pi,\ \ \phi\in\left\{ 0,\pi/2,\pi,3\pi/2\right\}

\end{array}

\right\} .

Elements of $\backslash Pi^\{n\}$ act on a quantum register of $n$ qubits. We

occasionally omit tensor product symbols in what follows so that

:$A\_\{1\}\backslash cdots\; A\_\{n\}\backslash equiv\; A\_\{1\}\backslash otimes\backslash cdots\backslash otimes\; A\_\{n\}.$

The $n$-fold Pauli group

$\backslash Pi^\{n\}$ plays an important role for both the encoding circuit and the

error-correction procedure of a quantum stabilizer code over $n$ qubits.

Let us define an $\backslash left[\; n,k\backslash right]$ stabilizer quantum error-correcting

code to encode $k$ logical qubits into $n$ physical qubits. The rate of such a

code is $k/n$. Its stabilizer $\backslash mathcal\{S\}$ is an abelian subgroup of the

$n$-fold Pauli group $\backslash Pi^\{n\}$. $\backslash mathcal\{S\}$

does not contain the operator $-I^\{\backslash otimes\; n\}$. The simultaneous

$+1$-eigenspace of the operators constitutes the codespace. The

codespace has dimension $2^\{k\}$ so that we can encode $k$ qubits into it. The

stabilizer $\backslash mathcal\{S\}$ has a minimal representation in terms of $n-k$

independent generators

:$\backslash left\backslash \{\; g\_\{1\},\backslash ldots,g\_\{n-k\}\backslash \; |\backslash \; \backslash forall\; i\backslash in\backslash left\backslash \{$

1,\ldots,n-k\right\} ,\ g_{i}\in\mathcal{S}\right\} .

The generators are

independent in the sense that none of them is a product of any other two (up

to a global phase). The operators $g\_\{1\},\backslash ldots,g\_\{n-k\}$ function in the same

way as a parity check matrix does for a classical linear block code.

One of the fundamental notions in quantum error correction theory is that it

suffices to correct a discrete error set with support in the Pauli group

$\backslash Pi^\{n\}$. Suppose that the errors affecting an

encoded quantum state are a subset $\backslash mathcal\{E\}$ of the Pauli group $\backslash Pi^\{n\}$:

:$\backslash mathcal\{E\}\backslash subset\backslash Pi^\{n\}.$

Because $\backslash mathcal\{E\}$ and $\backslash mathcal\{S\}$ are both subsets of $\backslash Pi^\{n\}$, an error $E\backslash in\backslash mathcal\{E\}$ that affects an

encoded quantum state either commutes or anticommutes with any particular

element $g$ in $\backslash mathcal\{S\}$. The error $E$ is correctable if it

anticommutes with an element $g$ in $\backslash mathcal\{S\}$. An anticommuting error

$E$ is detectable by measuring each element $g$ in $\backslash mathcal\{S\}$ and

computing a syndrome $\backslash mathbf\{r\}$ identifying $E$. The syndrome is a binary

vector $\backslash mathbf\{r\}$ with length $n-k$ whose elements identify whether the

error $E$ commutes or anticommutes with each $g\backslash in\backslash mathcal\{S\}$. An error

$E$ that commutes with every element $g$ in $\backslash mathcal\{S\}$ is correctable if

and only if it is in $\backslash mathcal\{S\}$. It corrupts the encoded state if it

commutes with every element of $\backslash mathcal\{S\}$ but does not lie in $\backslash mathcal\{S\}$

. So we compactly summarize the stabilizer error-correcting conditions: a

stabilizer code can correct any errors $E\_\{1\},E\_\{2\}$ in $\backslash mathcal\{E\}$ if

:$E\_\{1\}^\{\backslash dagger\}E\_\{2\}\backslash notin\backslash mathcal\{Z\}\backslash left(\; \backslash mathcal\{S\}\backslash right)$

or

:$E\_\{1\}^\{\backslash dagger\}E\_\{2\}\backslash in\backslash mathcal\{S\}$

where $\backslash mathcal\{Z\}\backslash left(\; \backslash mathcal\{S\}$

\right) is the centralizer of $\backslash mathcal\{S\}$ (i.e., the subgroup of elements that commute with all members of $\backslash mathcal\{S\}$, also known as the commutant).

A simple example of a stabilizer code is a three qubit

$\backslash left3,1,3\backslash right$ stabilizer code. It encodes $k=1$ logical qubit

into $n=3$ physical qubits and protects against a single-bit flip

error in the set $\backslash left\backslash \{$

X_{i}\right\}. This does not protect against other Pauli errors such as phase flip errors in the set $\backslash left\backslash \{$

Y_{i}\right\}.or $\backslash left\backslash \{$

Z_{i}\right\}. This has code distance $d=3$. Its stabilizer consists of $n-k=2$ Pauli operators:

:$$

\begin{array}

[c]{ccc}

g_{1} & = & Z & Z & I\\

g_{2} & = & I & Z & Z\\

\end{array}

If there are no bit-flip errors, both operators $g\_\{1\}$ and $g\_\{2\}$ commute, the syndrome is +1,+1, and no errors are detected.

If there is a bit-flip error on the first encoded qubit, operator $g\_\{1\}$ will anti-commute and $g\_\{2\}$ commute, the syndrome is -1,+1, and the error is detected. If there is a bit-flip error on the second encoded qubit, operator $g\_\{1\}$ will anti-commute and $g\_\{2\}$ anti-commute, the syndrome is -1,-1, and the error is detected. If there is a bit-flip error on the third encoded qubit, operator $g\_\{1\}$ will commute and $g\_\{2\}$ anti-commute, the syndrome is +1,-1, and the error is detected.

{{main|Five-qubit error correcting code}}

An example of a stabilizer code is the five qubit

$\backslash left5,1,3\backslash right$ stabilizer code. It encodes $k=1$ logical qubit

into $n=5$ physical qubits and protects against an arbitrary single-qubit

error. It has code distance $d=3$. Its stabilizer consists of $n-k=4$ Pauli operators:

:$$

\begin{array}

[c]{ccccccc}

g_{1} & = & X & Z & Z & X & I\\

g_{2} & = & I & X & Z & Z & X\\

g_{3} & = & X & I & X & Z & Z\\

g_{4} & = & Z & X & I & X & Z

\end{array}

The above operators commute. Therefore, the codespace is the simultaneous

+1-eigenspace of the above operators. Suppose a single-qubit error occurs on

the encoded quantum register. A single-qubit error is in the set $\backslash left\backslash \{$

X_{i},Y_{i},Z_{i}\right\} where $A\_\{i\}$ denotes a Pauli error on qubit $i$.

It is straightforward to verify that any arbitrary single-qubit error has a

unique syndrome. The receiver corrects any single-qubit error by identifying

the syndrome via a parity measurement and applying a corrective operation.

A simple but useful mapping exists between elements of $\backslash Pi$ and the binary

vector space $\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)\; ^\{2\}$. 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 $\backslash left[\; A\backslash right]$

is a set of equivalence classes of an operator $A$ that have the same phase:

:$$

\left[ A\right] =\left\{ \beta A\ |\ \beta\in\mathbb{C},\ \left\vert

\beta\right\vert =1\right\} .

Let $\backslash left[\; \backslash Pi\backslash right]$ be the set of phase-free Pauli operators where

$\backslash left[\; \backslash Pi\backslash right]\; =\backslash left\backslash \{\; \backslash left[\; A\backslash right]\; \backslash \; |\backslash \; A\backslash in\backslash Pi\backslash right\backslash \}$.

Define the map $N:\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)\; ^\{2\}\backslash rightarrow\backslash Pi$ as

:$$

00 \to I, \,\,

01 \to X, \,\,

11 \to Y, \,\,

10 \to Z

Suppose $u,v\backslash in\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)\; ^\{2\}$. Let us employ the

shorthand $u=\backslash left(\; z|x\backslash right)$ and $v=\backslash left(\; z^\{\backslash prime\}|x^\{\backslash prime$

}\right) where $z$, $x$, $z^\{\backslash prime\}$, $x^\{\backslash prime\}\backslash in\backslash mathbb\{Z\}\_\{2\}$. For

example, suppose $u=\backslash left(\; 0|1\backslash right)$. Then $N\backslash left(\; u\backslash right)\; =X$. The

map $N$ induces an isomorphism $\backslash left[\; N\backslash right]\; :\backslash left(\; \backslash mathbb\{Z\}$

_{2}\right) ^{2}\rightarrow\left[ \Pi\right] because addition of vectors

in $\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)\; ^\{2\}$ is equivalent to multiplication of

Pauli operators up to a global phase:

:$$

\left[ N\left( u+v\right) \right] =\left[ N\left( u\right) \right]

\left[ N\left( v\right) \right] .

Let $\backslash odot$ denote the symplectic product between two elements $u,v\backslash in\backslash left($

\mathbb{Z}_{2}\right) ^{2}:

:$$

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

The symplectic product $\backslash odot$ gives the commutation relations of elements of

$\backslash Pi$:

:$$

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 $N$ thus give a useful way to phrase

Pauli relations in terms of binary algebra.

The extension of the above definitions and mapping $N$ to multiple qubits is

straightforward. Let $\backslash mathbf\{A\}=A\_\{1\}\backslash otimes\backslash cdots\backslash otimes\; A\_\{n\}$ denote an

arbitrary element of $\backslash Pi^\{n\}$. We can similarly define the phase-free

$n$-qubit Pauli group $\backslash left[\; \backslash Pi^\{n\}\backslash right]\; =\backslash left\backslash \{\; \backslash left[$

\mathbf{A}\right] \ |\ \mathbf{A}\in\Pi^{n}\right\} where

:$$

\left[ \mathbf{A}\right] =\left\{ \beta\mathbf{A}\ |\ \beta\in

\mathbb{C},\ \left\vert \beta\right\vert =1\right\} .

The group operation $\backslash ast$ for the above equivalence class is as follows:

:$\backslash left[\; \backslash mathbf\{A\}\backslash right]\; \backslash ast\backslash left[\; \backslash mathbf\{B\}\backslash right]\; \backslash equiv\backslash left[$

A_{1}\right] \ast\left[ B_{1}\right] \otimes\cdots\otimes\left[

A_{n}\right] \ast\left[ B_{n}\right] =\left[ A_{1}B_{1}\right] \otimes\cdots\otimes\left[ A_{n}B_{n}\right]

=\left[ \mathbf{AB}\right] .

The equivalence class $\backslash left[\; \backslash Pi^\{n\}\backslash right]$ forms a commutative group

under operation $\backslash ast$. Consider the $2n$-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 $(\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)\; ^\{2n\},+)$ with

operation $+$ defined as binary vector addition. We employ the notation

$\backslash mathbf\{u\}=\backslash left(\; \backslash mathbf\{z\}|\backslash mathbf\{x\}\backslash right)\; ,\backslash mathbf\{v\}=\backslash left($

\mathbf{z}^{\prime}|\mathbf{x}^{\prime}\right) to represent any vectors

$\backslash mathbf\{u,v\}\backslash in\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)\; ^\{2n\}$ respectively. Each

vector $\backslash mathbf\{z\}$ and $\backslash mathbf\{x\}$ has elements $\backslash left(\; z\_\{1\},\backslash ldots$

,z_{n}\right) and $\backslash left(\; x\_\{1\},\backslash ldots,x\_\{n\}\backslash right)$ respectively with

similar representations for $\backslash mathbf\{z\}^\{\backslash prime\}$ and $\backslash mathbf\{x\}^\{\backslash prime\}$.

The symplectic product $\backslash odot$ of $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ is

:$$

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

z_{i}^{\prime},

or

:$$

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

where $u\_\{i\}=\backslash left(\; z\_\{i\}|x\_\{i\}\backslash right)$ and $v\_\{i\}=\backslash left(\; z\_\{i\}^\{\backslash prime$

}|x_{i}^{\prime}\right) . Let us define a map $\backslash mathbf\{N\}:\backslash left($

\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 $\backslash mathbf\{N\}\backslash left(\; \backslash mathbf\{u\}\backslash right)$ and $\backslash mathbf\{Z\}\backslash left($

\mathbf{z}\right) \mathbf{X}\left( \mathbf{x}\right) belong to the same

equivalence class:

:$$

\left[ \mathbf{N}\left( \mathbf{u}\right) \right] =\left[ \mathbf{Z}

\left( \mathbf{z}\right) \mathbf{X}\left( \mathbf{x}\right) \right] .

The map $\backslash left[\; \backslash mathbf\{N\}\backslash right]\; :\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)$

^{2n}\rightarrow\left[ \Pi^{n}\right] is an isomorphism for the same

reason given as in the previous case:

:$$

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

\mathbf{N}\left( \mathbf{u}\right) \right] \left[ \mathbf{N}\left(

\mathbf{v}\right) \right] ,

where $\backslash mathbf\{u,v\}\backslash in\backslash left(\; \backslash mathbb\{Z\}\_\{2\}\backslash right)\; ^\{2n\}$. The symplectic product

captures the commutation relations of any operators $\backslash mathbf\{N\}\backslash left($

\mathbf{u}\right) and $\backslash mathbf\{N\}\backslash left(\; \backslash mathbf\{v\}\backslash right)$:

:$$

\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.

By comparing quantum error correcting codes in this language to symplectic vector spaces, we can see the following. A symplectic subspace corresponds to a direct sum of Pauli algebras (i.e., encoded qubits), while an isotropic subspace corresponds to a set of stabilizers.

{{Reflist}}

• D. Gottesman, "Stabilizer codes and quantum error correction," quant-ph/9705052, Caltech Ph.D. thesis. https://arxiv.org/abs/quant-ph/9705052
• {{cite journal | last=Shor | first=Peter W. | title=Scheme for reducing decoherence in quantum computer memory | journal=Physical Review A | publisher=American Physical Society (APS) | volume=52 | issue=4 | date=1995-10-01 | issn=1050-2947 | doi=10.1103/physreva.52.r2493 | pages=R2493–R2496| pmid=9912632 | bibcode=1995PhRvA..52.2493S }}
• {{cite journal | last1=Calderbank | first1=A. R. | last2=Shor | first2=Peter W. | title=Good quantum error-correcting codes exist | journal=Physical Review A | publisher=American Physical Society (APS) | volume=54 | issue=2 | date=1996-08-01 | issn=1050-2947 | doi=10.1103/physreva.54.1098 | pages=1098–1105| pmid=9913578 |arxiv=quant-ph/9512032| bibcode=1996PhRvA..54.1098C | s2cid=11524969 }}
• {{cite journal | last=Steane | first=A. M. | title=Error Correcting Codes in Quantum Theory | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=77 | issue=5 | date=1996-07-29 | issn=0031-9007 | doi=10.1103/physrevlett.77.793 | pages=793–797| pmid=10062908 | bibcode=1996PhRvL..77..793S }}
• A. Calderbank, E. Rains, P. Shor, and N. Sloane, “Quantum error correction via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, pp. 1369–1387, 1998. Available at https://arxiv.org/abs/quant-ph/9608006

{{Quantum computing}}

Category:Linear algebra

Category:Quantum computing