Stabilizer Formalism

The theory of quantum error correction plays a prominent role in the practical realization and engineering of
quantum computing and quantum communication devices. The first quantum
error-correcting codes are strikingly similar to classical block codes in their
operation and performance. 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 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).

Mathematical background

The stabilizer formalism exploits elements of
the Pauli group $\Pi$ in formulating quantum error-correcting codes. The set
$\Pi=\left\$ consists of the Pauli operators:
:$I\equiv \begin 1 & 0\\ 0 & 1 \end ,\ X\equiv \begin 0 & 1\\ 1 & 0 \end ,\ Y\equiv \begin 0 & -i\\ i & 0 \end ,\ Z\equiv \begin 1 & 0\\ 0 & -1 \end .$
The above operators act on a single qubit – a state represented by a vector in a two-dimensional
Hilbert space. Operators in $\Pi$ have eigenvalues $\pm1$ and either commute
or anti-commute. The set $\Pi^$ consists of $n$-fold tensor products of
Pauli operators:
:$\Pi^=\left\ .$
Elements of $\Pi^$ act on a quantum register of $n$ qubits. We
occasionally omit tensor product symbols in what follows so that
:$A_\cdots A_\equiv A_\otimes\cdots\otimes A_.$
The $n$-fold Pauli group
$\Pi^$ plays an important role for both the encoding circuit and the
error-correction procedure of a quantum stabilizer code over $n$ qubits.

Definition

Let us define an \left n,k\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 $\mathcal$ is an abelian subgroup of the
$n$-fold Pauli group $\Pi^$. $\mathcal$
does not contain the operator $-I^$. The simultaneous
$+1$- eigenspace of the operators constitutes the ''codespace''. The
codespace has dimension $2^$ so that we can encode $k$ qubits into it. The
stabilizer $\mathcal$ has a minimal representation in terms of $n-k$
independent generators
:$\left\ .$

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 $g_,\ldots,g_$ function in the same
way as a parity check matrix does for a classical linear block code.

Stabilizer error-correction conditions

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
$\Pi^$. Suppose that the errors affecting an
encoded quantum state are a subset $\mathcal$ of the Pauli group $\Pi^$:
:$\mathcal\subset\Pi^.$

Because $\mathcal$ and $\mathcal$ are both subsets of $\Pi^$, an error $E\in\mathcal$ that affects an
encoded quantum state either commutes or anticommutes with any particular
element $g$ in $\mathcal$. The error $E$ is correctable if it
anticommutes with an element $g$ in $\mathcal$. An anticommuting error
$E$ is detectable by measuring each element $g$ in $\mathcal$ and
computing a syndrome $\mathbf$ identifying $E$. The syndrome is a binary
vector $\mathbf$ with length $n-k$ whose elements identify whether the
error $E$ commutes or anticommutes with each $g\in\mathcal$. An error
$E$ that commutes with every element $g$ in $\mathcal$ is correctable if
and only if it is in $\mathcal$. It corrupts the encoded state if it
commutes with every element of $\mathcal$ but does not lie in $\mathcal$. So we compactly summarize the stabilizer error-correcting conditions: a
stabilizer code can correct any errors $E_,E_$ in $\mathcal$ if
:$E_^E_\notin\mathcal\left( \mathcal\right)$

or

:$E_^E_\in\mathcal$

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

Relation between Pauli group and binary vectors

A simple but useful mapping exists between elements of $\Pi$ and the binary
vector space $\left( \mathbb_\right) ^$. 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 \left A\right
is a set of equivalence classes of an operator $A$ that have the same phase:
:
\left A\right =\left\ .

Let \left \Pi\right be the set of phase-free Pauli operators where
\left \Pi\right =\left\ .
Define the map $N:\left( \mathbb_\right) ^\rightarrow\Pi$ as
:$00 \to I, \,\, 01 \to X, \,\, 11 \to Y, \,\, 10 \to Z$

Suppose $u,v\in\left( \mathbb_\right) ^$. Let us employ the
shorthand $u=\left( z, x\right)$ and $v=\left( z^, x^\right)$ where $z$, $x$, $z^$, $x^\in\mathbb_$. For
example, suppose $u=\left( 0, 1\right)$. Then $N\left( u\right) =X$. The
map $N$ induces an isomorphism \left N\right :\left( \mathbb
_\right) ^\rightarrow\left \Pi\right because addition of vectors
in $\left( \mathbb_\right) ^$ 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 $\odot$ denote the symplectic product between two elements $u,v\in\left( \mathbb_\right) ^$:
:$u\odot v\equiv zx^-xz^.$
The symplectic product $\odot$ gives the commutation relations of elements of
$\Pi$:
:$N\left( u\right) N\left( v\right) =\left( -1\right) ^N\left( v\right) N\left( u\right) .$

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 $\mathbf=A_\otimes\cdots\otimes A_$ denote an
arbitrary element of $\Pi^$. We can similarly define the phase-free
$n$-qubit Pauli group \left \Pi^\right =\left\ where
:
\left \mathbf\right =\left\ .

The group operation $\ast$ for the above equivalence class is as follows:
: \left \mathbf\right \ast\left \mathbf\right \equiv\left A_\right \ast\left B_\right \otimes\cdots\otimes\left A_\right \ast\left B_\right =\left A_B_\right \otimes\cdots\otimes\left A_B_\right=\left \mathbf\right .

The equivalence class \left \Pi^\right forms a commutative group
under operation $\ast$. Consider the $2n$-dimensional vector space
:$\left( \mathbb_\right) ^=\left\ .$
It forms the commutative group $(\left( \mathbb_\right) ^,+)$ with
operation $+$ defined as binary vector addition. We employ the notation
$\mathbf=\left( \mathbf, \mathbf\right) ,\mathbf=\left( \mathbf^, \mathbf^\right)$ to represent any vectors
$\mathbf\in\left( \mathbb_\right) ^$ respectively. Each
vector $\mathbf$ and $\mathbf$ has elements $\left( z_,\ldots ,z_\right)$ and $\left( x_,\ldots,x_\right)$ respectively with
similar representations for $\mathbf^$ and $\mathbf^$.
The ''symplectic product'' $\odot$ of $\mathbf$ and $\mathbf$ is
:$\mathbf\odot\mathbf\sum_^z_x_^-x_ z_^,$
or
:$\mathbf\odot\mathbf\sum_^u_\odot v_,$
where $u_=\left( z_, x_\right)$ and $v_=\left( z_^, x_^\right)$. Let us define a map $\mathbf:\left( \mathbb_\right) ^\rightarrow\Pi^$ as follows:
:$\mathbf\left( \mathbf\right) \equiv N\left( u_\right) \otimes\cdots\otimes N\left( u_\right) .$
Let
:$\mathbf\left( \mathbf\right) \equiv X^\otimes\cdots\otimes X^, \,\,\,\,\,\,\, \mathbf\left( \mathbf\right) \equiv Z^\otimes\cdots\otimes Z^,$
so that $\mathbf\left( \mathbf\right)$ and $\mathbf\left( \mathbf\right) \mathbf\left( \mathbf\right)$ belong to the same
equivalence class:
:
\left \mathbf\left( \mathbf\right) \right =\left \mathbf
\left( \mathbf\right) \mathbf\left( \mathbf\right) \right .

The map \left \mathbf\right :\left( \mathbb_\right)
^\rightarrow\left \Pi^\right is an isomorphism for the same
reason given as in the previous case:
:
\left \mathbf\left( \mathbf\right) \right =\left \mathbf\left( \mathbf\right) \right \left \mathbf\left(
\mathbf\right) \right ,

where $\mathbf\in\left( \mathbb_\right) ^$. The symplectic product
captures the commutation relations of any operators $\mathbf\left( \mathbf\right)$ and $\mathbf\left( \mathbf\right)$:
:$\mathbf\left( \mathbf\right) =\left( -1\right) ^\mathbf\left( \mathbf\right) \mathbf\left( \mathbf\right) .$
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.

Example of a stabilizer code

An example of a stabilizer code is the five qubit
\left 5,1,3\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
g_ & = & X & Z & Z & X & I\\
g_ & = & I & X & Z & Z & X\\
g_ & = & X & I & X & Z & Z\\
g_ & = & Z & X & I & X & Z
\end

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 $\left\$ where $A_$ 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.

References

* D. Gottesman, "Stabilizer codes and quantum error correction," quant-ph/9705052, Caltech Ph.D. thesis. https://arxiv.org/abs/quant-ph/9705052
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* 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

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