Stabilizer code

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).
Mathematical background.
The Stabilizer formalism exploits elements of
the Pauli group formula_1 in formulating quantum error-correcting codes. The set
formula_2 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 formula_1 have eigenvalues formula_5 and either commute
or anti-commute. The set formula_6 consists of formula_7-fold tensor products of
Pauli operators:
Elements of formula_6 act on a quantum register of formula_7 qubits. We
occasionally omit tensor product symbols in what follows so that
The formula_7-fold Pauli group
formula_6 plays an important role for both the encoding circuit and the
error-correction procedure of a quantum stabilizer code over formula_7 qubits.
Definition.
Let us define an formula_15 stabilizer quantum error-correcting
code to encode formula_16 logical qubits into formula_7 physical qubits. The rate of such a
code is formula_18. Its stabilizer formula_19 is an abelian subgroup of the
formula_7-fold Pauli group formula_6. formula_19
does not contain the operator formula_23. The simultaneous
formula_24-eigenspace of the operators constitutes the "codespace". The
codespace has dimension formula_25 so that we can encode formula_16 qubits into it. The
stabilizer formula_19 has a minimal representation in terms of formula_28
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 formula_30 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
formula_6. Suppose that the errors affecting an
encoded quantum state are a subset formula_32 of the Pauli group formula_6:
Because formula_32 and formula_19 are both subsets of formula_6, an error formula_38 that affects an
encoded quantum state either commutes or anticommutes with any particular
element formula_39 in formula_19. The error formula_41 is correctable if it
anticommutes with an element formula_39 in formula_19. An anticommuting error
formula_41 is detectable by measuring each element formula_39 in formula_19 and
computing a syndrome formula_47 identifying formula_41. The syndrome is a binary
vector formula_47 with length formula_28 whose elements identify whether the
error formula_41 commutes or anticommutes with each formula_52. An error
formula_41 that commutes with every element formula_39 in formula_19 is correctable if
and only if it is in formula_19. It corrupts the encoded state if it
commutes with every element of formula_19 but does not lie in formula_58. So we compactly summarize the stabilizer error-correcting conditions: a
stabilizer code can correct any errors formula_59 in formula_32 if
or
where formula_63 is the centralizer of formula_19 (i.e., the subgroup of elements that commute with all members of formula_19, also known as the commutant).
Relation between Pauli group and binary vectors.
A simple but useful mapping exists between elements of formula_1 and the binary
vector space formula_67. 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 formula_68
is a set of equivalence classes of an operator formula_69 that have the same phase:
Let formula_71 be the set of phase-free Pauli operators where
formula_72.
Define the map formula_73 as
Suppose formula_75. Let us employ the
shorthand formula_76 and formula_77 where formula_78, formula_79, formula_80, formula_81. For
example, suppose formula_82. Then formula_83. The
map formula_84 induces an isomorphism formula_85 because addition of vectors
in formula_67 is equivalent to multiplication of
Pauli operators up to a global phase:
Let formula_88 denote the symplectic product between two elements formula_89:
The symplectic product formula_88 gives the commutation relations of elements of
formula_1:
The symplectic product and the mapping formula_84 thus give a useful way to phrase
Pauli relations in terms of binary algebra.
The extension of the above definitions and mapping formula_84 to multiple qubits is
straightforward. Let formula_96 denote an
arbitrary element of formula_6. We can similarly define the phase-free
formula_7-qubit Pauli group formula_99 where
The group operation formula_101 for the above equivalence class is as follows:
The equivalence class formula_103 forms a commutative group
under operation formula_101. Consider the formula_105-dimensional vector space
It forms the commutative group formula_107 with
operation formula_108 defined as binary vector addition. We employ the notation
formula_109 to represent any vectors
formula_110 respectively. Each
vector formula_111 and formula_112 has elements formula_113 and formula_114 respectively with
similar representations for formula_115 and formula_116.
The "symplectic product" formula_88 of formula_118 and formula_119 is
or
where formula_122 and formula_123. Let us define a map formula_124 as follows:
Let
so that formula_127 and formula_128 belong to the same
equivalence class:
The map formula_130 is an isomorphism for the same
reason given as in the previous case:
where formula_110. The symplectic product
captures the commutation relations of any operators formula_133 and formula_134:
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.
Example of a stabilizer code.
An example of a stabilizer code is the five qubit
formula_136 stabilizer code. It encodes formula_137 logical qubit
into formula_138 physical qubits and protects against an arbitrary single-qubit
error. It has code distance formula_139. Its stabilizer consists of formula_140 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 formula_142 where formula_143 denotes a Pauli error on qubit formula_144.
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.