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).
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:
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.
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
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
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 and binary vectors
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.
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:
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
where and . Let us define a map as follows:
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 :
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 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 and applying a corrective operation.
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