Stabilizer code - AbsoluteAstronomy.com

Quantum error correction

Quantum error correction is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum...

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

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

. 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
that has a particular error model.

The stabilizer theory of quantum error correction

Quantum error correction

allows one to import some
classical binary or quaternary codes for use as a quantum code. The only
"catch" when importing is that the
classical code 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

Entanglement-assisted stabilizer formalism

In the theory of quantum communication, the entanglement-assisted stabilizer formalism is a method for protecting quantum information with the help of entanglement shared between a sender and receiver before they transmit quantum data over a quantum communication channel...

overcomes this difficulty).

Mathematical background

The Stabilizer formalism exploits elements of
the Pauli group

in formulating quantum error-correcting codes. The set

consists of the Pauli operators:

The above operators act on a single qubit

Qubit

In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

---a state represented by a vector in a two-dimensional
Hilbert space

Hilbert space

. Operators in

have eigenvalues

and either commute

Commute

Commute, commutation or commutative may refer to:* Commuting, the process of travelling between a place of residence and a place of work* Commutative property, a property of a mathematical operation...

or anti-commute. The set

consists of

-fold tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

s of
Pauli operators:

Elements of

act on a quantum register of

qubit

Qubit

s. We
occasionally omit tensor product

Tensor product

symbols in what follows so that

The

-fold Pauli group

plays an important role for both the encoding circuit and the
error-correction procedure of a quantum stabilizer code over

qubit

Qubit

Definition

Let us define an

stabilizer quantum error-correcting
code to encode

logical qubits into

physical qubits. The rate of such a
code is

. Its stabilizer

is an abelian

Abelian

In mathematics, Abelian refers to any of number of different mathematical concepts named after Niels Henrik Abel:- Group theory :*Abelian group, a group in which the binary operation is commutative...

subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

of the

-fold Pauli group

does not contain the operator

. The simultaneous

-eigenspace of the operators constitutes the codespace. The
codespace has dimension

so that we can encode

qubit

Qubit

s into it. The
stabilizer

has a minimal representation

Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...

in terms of

independent generators

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

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

Discrete

Discrete in science is the opposite of continuous: something that is separate; distinct; individual.Discrete may refer to:*Discrete particle or quantum in physics, for example in quantum theory...

error set with support

Support (mathematics)

In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...

in the Pauli group

. Suppose that the errors affecting an
encoded quantum state are a subset

of the Pauli group

An error

that affects an
encoded quantum state either commute

Commute

s or anticommutes with any particular
element

. The error

is correctable if it
anticommutes with an element

. An anticommuting error

is detectable by measuring each element

and
computing a syndrome

Syndrome

In medicine and psychology, a syndrome is the association of several clinically recognizable features, signs , symptoms , phenomena or characteristics that often occur together, so that the presence of one or more features alerts the physician to the possible presence of the others...

identifying

. The syndrome is a binary
vector

with length

whose elements identify whether the
error

commute

Commute

s or anticommutes with each

. An error

that commute

Commute

s with every element

is correctable if
and only if it is in

. It corrupts the encoded state if it
commutes with every element of

but does not lie in

. So we compactly summarize the stabilizer error-correcting conditions: a
stabilizer code can correct any errors

where

is the centralizer of

Relation between Pauli group and binary vectors

A simple but useful mapping exists between elements of

and the binary
vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

. This mapping gives a
simplification of quantum error correction theory. It represents quantum codes
with binary vector

Binary vector

A T-DNA binary system is a pair of plasmids consisting of a binary plasmid and a helper plasmid. The two plasmids are used together to produce genetically modified plants. They are artificial vectors that have both been created from the naturally occurring Ti plasmid found in Agrobacterium...

s and binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

s 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

Operator (physics)

In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....

that have the same phase

Phase

-In physics:*Phase , a physically distinctive form of a substance, such as the solid, liquid, and gaseous states of ordinary matter**Phase transition is the transformation of a thermodynamic system from one phase to another*Phase...

Let

be the set of phase-free Pauli operators where

.
Define the map

Suppose

. Let us employ the
shorthand

and

where

. For
example, suppose

. Then

. The
map

induces an isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...

because addition of vectors
in

is equivalent to multiplication of
Pauli operators up to a global phase:

Let

denote the symplectic product between two elements

The symplectic product

gives the commutation relations of elements of

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 qubit

Qubit

s is
straightforward. Let

denote an
arbitrary element of

. We can similarly define the phase-free

-qubit Pauli group

where

The group operation

for the above equivalence class is as follows:

The equivalence class

forms a commutative group
under operation

. Consider the

-dimensional vector space

Vector space

It forms the commutative group

with
operation

defined as binary vector addition. We employ the notation

to represent any vectors

respectively. Each
vector

and

has elements

and

respectively with
similar representations for

and

.
The \textit{symplectic product}

and

where

and

. Let us define a map

as follows:

Let

so that

and

belong to the same
equivalence class:

The map

is an isomorphism

Isomorphism

for the same
reason given as the previous case:

where

. The symplectic product
captures the commutation relations of any operators

and

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

Quantum error correction

more explicit.

Example of a stabilizer code

An example of a stabilizer code is the five qubit

stabilizer code. It encodes

logical qubit
into

physical qubits and protects against an arbitrary single-qubit
error. Its stabilizer consists of

Pauli operators:

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

where

denotes a Pauli error on qubit

.
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 and applying a corrective operation.

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