The theory of
quantum error correction
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing tha ...
plays a prominent role in the practical realization and engineering of
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
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
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
. This highly
entangled,
encoded state corrects for local noisy errors. A quantum error-correcting code makes
quantum computation
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thoug ...
and
quantum communication practical by providing a way for a sender and
receiver to simulate a noiseless qubit channel given a
noisy qubit channel
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information ...
whose noise conforms to a particular error model.
The stabilizer theory of
quantum error correction
Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing tha ...
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
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\beg ...
in formulating quantum error-correcting codes. The set
consists of the
Pauli operators
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
:
:
The above operators act on a single
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
– a state represented by a vector in a two-dimensional
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
. Operators in
have
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
and either
commute
or
anti-commute
In mathematics, anticommutativity is a specific property of some non- commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
. The set
consists of
-fold
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
s of
Pauli operators:
:
Elements of
act on a
quantum register In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analogue of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.
Definition ...
of
qubits. We
occasionally omit
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
symbols in what follows so that
:
The
-fold
Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\beg ...
plays an important role for both the encoding circuit and the
error-correction procedure of a quantum stabilizer code over
qubits.
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 may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
subgroup
In group theory, a branch of mathematics, 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 subgrou ...
of the
-fold Pauli group
.
does not contain the operator
. The simultaneous
-
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of the operators constitutes the ''codespace''. The
codespace has dimension
so that we can encode
qubits into it. The
stabilizer
has a minimal
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
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 may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
error set with
support in the
Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\beg ...
. Suppose that the errors affecting an
encoded quantum state are a subset
of the
Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\beg ...
:
:
Because
and
are both subsets of
, an error
that affects an
encoded quantum state either
commutes or
anticommutes with any particular
element
in
. The error
is correctable if it
anticommutes with an element
in
. An anticommuting error
is detectable by
measuring
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
each element
in
and
computing a syndrome
identifying
. The syndrome is a binary
vector
with length
whose elements identify whether the
error
commutes or anticommutes with each
. An error
that commutes with every element
in
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
in
if
:
or
:
where
is the
centralizer of
(i.e., the subgroup of elements that commute with all members of
, also known as the commutant).
Relation between
Pauli group
In physics and mathematics, the Pauli group G_1 on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix I and all of the Pauli matrices
:X = \sigma_1 =
\begin
0&1\\
1&0
\end,\quad
Y = \sigma_2 =
\beg ...
and binary vectors
A simple but useful mapping exists between elements of
and the binary
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
. This mapping gives a
simplification of quantum error correction theory. It represents quantum codes
with
binary vectors and
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, an internal binary op ...
s rather than with
Pauli operators and
matrix operation
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begin ...
s respectively.
We first give the mapping for the one-qubit case. Suppose
is a set of
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
es of an
operator
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
that have the same
phase:
:
Let
be the set of phase-free Pauli operators where
.
Define the map
as
:
Suppose
. Let us employ the
shorthand
and
where
,
,
,
. For
example, suppose
. Then
. The
map
induces an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
because addition of vectors
in
is equivalent to multiplication of
Pauli operators up to a global phase:
:
Let
denote the
symplectic product In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument s ...
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 qubits is
straightforward. Let
denote an
arbitrary element of
. We can similarly define the phase-free
-qubit Pauli group
where
:
The
group operation
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Thes ...
for the above equivalence class is as follows:
:
The equivalence class
forms a
commutative group
under operation
. Consider the
-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
:
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 ''symplectic product''
of
and
is
:
or
:
where
and
. Let us define a map
as follows:
:
Let
:
so that
and
belong to the same
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
:
:
The map
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
for the same
reason given as in the previous case:
:
where
. The
symplectic product In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument s ...
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 (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is theorised as essential to achieve fault tolerant quantum computing tha ...
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
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...
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
stabilizer code. It encodes
logical qubit
into
physical qubits and protects against an arbitrary single-qubit
error. It has code distance
. 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 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
*
*
*
* 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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