Stabilizer code explained

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

Quantum error-correcting codes restore a noisy,decohered quantum state to a pure quantum state. Astabilizer quantum error-correcting code appends ancilla qubitsto qubits that we want to protect. A unitary encoding circuit rotates theglobal 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 computationand quantum communication practical by providing a way for a sender andreceiver to simulate a noiseless qubit channel given a noisy qubit channelwhose 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 someclassical binary or quaternary codes for use as a quantum code. However, when importing theclassical code, it must satisfy the dual-containing (or self-orthogonality)constraint. Researchers have found many examples of classical codes satisfyingthis 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

\Pi in formulating quantum error-correcting codes. The set\Pi = \left\{ I,X,Y,Z\right\} consists of the Pauli operators:I\equiv \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} , X\equiv \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , Y\equiv \begin{bmatrix} 0 & - i\\ i & 0 \end{bmatrix} , Z\equiv \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} . The above operators act on a single qubit – a state represented by a vector in a two-dimensionalHilbert space. Operators in \Pi have eigenvalues \pm 1 and either commuteor anti-commute. The set \Pi ^{n} consists of n -fold tensor products ofPauli operators:\Pi ^{n} = \left\{ \begin{array} [c]{c} e ^{i\phi} A _{1} ⊗ … ⊗ A _{n} :\forall j\in\left\{ 1 ,\ldots ,n\right\} A _{j} \in\Pi, \phi\in\left\{ 0 ,\pi/ 2 ,\pi, 3 \pi/ 2 \right\} \end{array} \right\} . Elements of \Pi ^{n} act on a quantum register of n qubits. Weoccasionally omit tensor product symbols in what follows so thatA _{1} … A _{n} \equiv A _{1} ⊗ … ⊗ A _{n} . The n -fold Pauli group\Pi ^{n} plays an important role for both the encoding circuit and theerror-correction procedure of a quantum stabilizer code over n qubits.

Definition

Let us define an

\left[ n,k\right] stabilizer quantum error-correctingcode to encode k logical qubits into n physical qubits. The rate of such acode is k/n . Its stabilizer l{S} is an abelian subgroup of then -fold Pauli group \Pi ^{n} . l{S} does not contain the operator - I ^{⊗} . The simultaneous+1 -eigenspace of the operators constitutes the codespace. Thecodespace has dimension 2 ^{k} so that we can encode k qubits into it. Thestabilizer l{S} has a minimal representation in terms of n - k independent generators\left\{ g _{1} ,\ldots,g _{n-k} | \forall i\in\left\{ 1 ,\ldots,n - k\right\} , g _{i} \in l{S}\right\} .

The generators areindependent in the sense that none of them is a product of any other two (upto a global phase). The operators

g _{1} ,\ldots,g _{n-k} function in the sameway 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 itsuffices to correct a discrete error set with support in the Pauli group

\Pi ^{n} . Suppose that the errors affecting anencoded quantum state are a subset l{E} of the Pauli group \Pi ^{n} :l{E}\subset\Pi ^{n} .

Because

l{E} and l{S} are both subsets of \Pi ^{n} , an error E\in l{E} that affects anencoded quantum state either commutes or anticommutes with any particularelement g in l{S} . The error E is correctable if itanticommutes with an element g in l{S} . An anticommuting errorE is detectable by measuring each element g in l{S} andcomputing a syndrome r identifying E . The syndrome is a binaryvector r with length n - k whose elements identify whether theerror E commutes or anticommutes with each g\in l{S} . An errorE that commutes with every element g in l{S} is correctable ifand only if it is in l{S} . It corrupts the encoded state if itcommutes with every element of l{S} but does not lie in l{S} . So we compactly summarize the stabilizer error-correcting conditions: astabilizer code can correct any errors E _{1} ,E _{2} in l{E} ifE ^{\dagger} _{1} E _{2} \notin l{Z}\left( l{S}\right)

or

E ^{\dagger} _{1} E _{2} \in l{S}

where

l{Z}\left( l{S} \right) is the centralizer of l{S} (i.e., the subgroup of elements that commute with all members of l{S} , also known as the commutant).

Simple example of a stabilizer code

A simple example of a stabilizer code is a three qubit

\left[[ 3 , 1 , 3 \right]] stabilizer code. It encodes k =1 logical qubitinto n =3 physical qubits and protects against a single-bit fliperror in the set \left\{ X _{i} \right\} . This does not protect against other Pauli errors such as phase flip errors in the set \left\{ Y _{i} \right\} .or \left\{ Z _{i} \right\} . This has code distance d =3 . Its stabilizer consists of n - k =2 Pauli operators:\begin{array} [c]{ccc} g _{1} & = & Z & Z & I\\ g _{2} & = & I & Z & Z\\ \end{array} If there are no bit-flip errors, both operators g _{1} and g _{2} commute, the syndrome is +1,+1, and no errors are detected.

If there is a bit-flip error on the first encoded qubit, operator

g _{1} will anti-commute and g _{2} commute, the syndrome is -1,+1, and the error is detected. If there is a bit-flip error on the second encoded qubit, operator g _{1} will anti-commute and g _{2} 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 g _{1} will commute and g _{2} anti-commute, the syndrome is +1,-1, and the error is detected.

Example of a stabilizer code

See main article: Five-qubit error correcting code. An example of a stabilizer code is the five qubit

\left[[ 5 , 1 , 3 \right]] stabilizer code. It encodes k =1 logical qubitinto n =5 physical qubits and protects against an arbitrary single-qubiterror. It has code distance d =3 . Its stabilizer consists of n - k =4 Pauli operators:\begin{array} [c]{ccccccc} g _{1} & = & X & Z & Z & X & I\\ g _{2} & = & I & X & Z & Z & X\\ g _{3} & = & X & I & X & Z & Z\\ g _{4} & = & Z & X & I & X & Z \end{array} The above operators commute. Therefore, the codespace is the simultaneous+1-eigenspace of the above operators. Suppose a single-qubit error occurs onthe encoded quantum register. A single-qubit error is in the set \left\{ X _{i} ,Y _{i} ,Z _{i} \right\} where A _{i} denotes a Pauli error on qubit i .It is straightforward to verify that any arbitrary single-qubit error has aunique syndrome. The receiver corrects any single-qubit error by identifyingthe syndrome via a parity measurement and applying a corrective operation.

Relation between Pauli group and binary vectors

A simple but useful mapping exists between elements of

\Pi and the binaryvector space \left( Z _{2} \right) ^{2} . This mapping gives asimplification of quantum error correction theory. It represents quantum codeswith binary vectors and binary operations rather than with Pauli operators andmatrix operations respectively.

We first give the mapping for the one-qubit case. Suppose

\left[ A\right] is a set of equivalence classes of an operator A that have the same phase:\left[ A\right] = \left\{ \beta A | \beta\in C , \left\vert \beta\right\vert =1 \right\} .

Let

\left[ \Pi\right] be the set of phase-free Pauli operators where\left[ \Pi\right] = \left\{ \left[ A\right] | A\in\Pi\right\} .Define the map N:\left( Z _{2} \right) ^{2} → \Pi as00 \to I, 01 \to X, 11 \to Y, 10 \to Z

Suppose

u,v\in\left( Z _{2} \right) ^{2} . Let us employ theshorthand u = \left( z|x\right) and v = \left( z ^{\prime} |x ^{\prime} \right) where z , x , z ^{\prime} , x ^{\prime} \in Z _{2} . Forexample, suppose u = \left( 0 | 1 \right) . Then N\left( u\right) = X . Themap N induces an isomorphism \left[ N\right] :\left( Z _{2} \right) ^{2} → \left[ \Pi\right] because addition of vectorsin \left( Z _{2} \right) ^{2} is equivalent to multiplication ofPauli 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 between two elements u,v\in\left( Z _{2} \right) ^{2} :u\odot v\equiv zx ^{\prime} - xz ^{\prime} . The symplectic product \odot gives the commutation relations of elements of\Pi N\left( u\right) N\left( v\right) = \left( -1 \right) ^{\left(} N\left( v\right) N\left( u\right) .

The symplectic product and the mapping

N thus give a useful way to phrasePauli relations in terms of binary algebra.The extension of the above definitions and mapping N to multiple qubits isstraightforward. Let A = A _{1} ⊗ … ⊗ A _{n} denote anarbitrary element of \Pi ^{n} . We can similarly define the phase-freen -qubit Pauli group \left[ \Pi ^{n} \right] = \left\{ \left[ A \right] | A \in\Pi ^{n} \right\} where\left[ A \right] = \left\{ \beta A | \beta\in C , \left\vert \beta\right\vert =1 \right\} . \ast for the above equivalence class is as follows:\left[ A \right] \ast\left[ B \right] \equiv\left[ A _{1} \right] \ast\left[ B _{1} \right] ⊗ … ⊗ \left[ A _{n} \right] \ast\left[ B _{n} \right] = \left[ A _{1} B _{1} \right] ⊗ … ⊗ \left[ A _{n} B _{n} \right] = \left[ AB \right] . The equivalence class \left[ \Pi ^{n} \right] forms a commutative groupunder operation \ast . Consider the 2 n -dimensional vector space\left( Z _{2} \right) ^{2n} = \left\{ \left( z,x \right) : z , x \in\left( Z _{2} \right) ^{n} \right\} . It forms the commutative group (\left( Z _{2} \right) ^{2n} , + ) withoperation + defined as binary vector addition. We employ the notationu = \left( z | x \right) , v = \left( z ^{\prime} | x ^{\prime} \right) to represent any vectorsu,v \in\left( Z _{2} \right) ^{2n} respectively. Eachvector z and x has elements \left( z _{1} ,\ldots ,z _{n} \right) and \left( x _{1} ,\ldots,x _{n} \right) respectively withsimilar representations for z ^{\prime} and x ^{\prime} .The symplectic product \odot of u and v isu \odot v\equiv \sum ^{n} _{i =1} z _{i} x ^{\prime} _{i} - x _{i} z ^{\prime} _{i} , oru \odot v\equiv \sum ^{n} _{i =1} u _{i} \odot v _{i} , where u _{i} = \left( z _{i} |x _{i} \right) and v _{i} = \left( z ^{\prime} _{i} |x ^{\prime} _{i} \right) . Let us define a map N :\left( Z _{2} \right) ^{2n} → \Pi ^{n} as follows:N \left( u \right) \equiv N\left( u _{1} \right) ⊗ … ⊗ N\left( u _{n} \right) . LetX \left( x \right) \equiv X ^{x _{1}} ⊗ … ⊗ X ^{x _{n}} , Z \left( z \right) \equiv Z ^{z _{1}} ⊗ … ⊗ Z ^{z _{n}} , so that N \left( u \right) and Z \left( z \right) X \left( x \right) belong to the sameequivalence class\left[ N \left( u \right) \right] = \left[ Z \left( z \right) X \left( x \right) \right] . The map \left[ N \right] :\left( Z _{2} \right) ^{2n} → \left[ \Pi ^{n} \right] is an isomorphism for the samereason given as in the previous case:\left[ N \left( u+v \right) \right] = \left[ N \left( u \right) \right] \left[ N \left( v \right) \right] , where u,v \in\left( Z _{2} \right) ^{2n} . The symplectic productcaptures the commutation relations of any operators N \left( u \right) and N \left( v \right) :N\left( u \right)N \left( v \right) = \left( -1 \right) ^{\left(} N \left( v \right) N \left( u \right) . The above binary representation and symplectic algebra are useful in makingthe 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

• D. Gottesman, "Stabilizer codes and quantum error correction," quant-ph/9705052, Caltech Ph.D. thesis. https://arxiv.org/abs/quant-ph/9705052
• Shor . Peter W. . Scheme for reducing decoherence in quantum computer memory . Physical Review A . American Physical Society (APS) . 52 . 4 . 1995-10-01 . 1050-2947 . 10.1103/physreva.52.r2493 . R2493–R2496. 9912632 . 1995PhRvA..52.2493S .
• Calderbank . A. R. . Shor . Peter W. . Good quantum error-correcting codes exist . Physical Review A . American Physical Society (APS) . 54 . 2 . 1996-08-01 . 1050-2947 . 10.1103/physreva.54.1098 . 1098–1105. 9913578 . quant-ph/9512032. 1996PhRvA..54.1098C . 11524969 .
• Steane . A. M. . Error Correcting Codes in Quantum Theory . Physical Review Letters . American Physical Society (APS) . 77 . 5 . 1996-07-29 . 0031-9007 . 10.1103/physrevlett.77.793 . 793–797. 10062908 . 1996PhRvL..77..793S .
• 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

Notes and References

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