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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 ...
\Pi in formulating quantum error-correcting codes. The set \Pi=\left\ 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 ...
: : 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 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 \Pi 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 ...
\pm1 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 \Pi^ consists of n-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: : \Pi^=\left\ . Elements of \Pi^ 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 n 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 :A_\cdots A_\equiv A_\otimes\cdots\otimes A_. The n-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 ...
\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 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 n-fold Pauli group \Pi^. \mathcal does not contain the operator -I^. The simultaneous +1-
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 2^ so that we can encode k qubits into it. The stabilizer \mathcal 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 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 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 ...
\Pi^. Suppose that the errors affecting an encoded quantum state are a subset \mathcal 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 ...
\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 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 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 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 \Pi 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 ...
\left( \mathbb_\right) ^. 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 \left A\right 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 ...
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 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 ...
\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 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 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 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 ...
\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 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 ...
: \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 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 ...
: : \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 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: : \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 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 \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 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 \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 * * * * 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 Linear algebra Quantum computing