In quantum computing and quantum communication, a stabilizer code is a class of quantum codes for performing quantum error correction. The toric code, and surface codes more generally,[1] are types of stabilizer codes considered very important for the practical realization of quantum information processing.

Conceptual background

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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).

Mathematical background

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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 – a state represented by a vector in a two-dimensional Hilbert space. Operators in   have eigenvalues   and either commute or anti-commute. The set   consists of  -fold tensor products of Pauli operators:

 

Elements of   act on a quantum register of   qubits. We occasionally omit 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   qubits.

Definition

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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 subgroup 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   qubits into it. The stabilizer   has a minimal representation 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

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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  . Suppose that the errors affecting an encoded quantum state are a subset   of the Pauli group  :

 

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 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).

Simple example of a stabilizer code

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A simple example of a stabilizer code is a three qubit   stabilizer code. It encodes   logical qubit into   physical qubits and protects against a single-bit flip error in the set  . This does not protect against other Pauli errors such as phase flip errors in the set  .or  . This has code distance  . Its stabilizer consists of   Pauli operators:

 

If there are no bit-flip errors, both operators   and   commute, the syndrome is +1,+1, and no errors are detected.

If there is a bit-flip error on the first encoded qubit, operator   will anti-commute and   commute, the syndrome is -1,+1, and the error is detected. If there is a bit-flip error on the second encoded qubit, operator   will anti-commute and   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   will commute and   anti-commute, the syndrome is +1,-1, and the error is detected.

Example of a stabilizer code

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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.

Relation between Pauli group and binary vectors

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A simple but useful mapping exists between elements of   and the binary vector space  . 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   is a set of equivalence classes of an operator   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   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 qubits 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

 

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:

 

The map   is an isomorphism for the same reason given as in 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 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.

References

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  1. ^ "What is the "surface code" in the context of quantum error correction?". Quantum Computing Stack Exchange. Retrieved 2024-01-12.