Stabilizer Codes Let S be a subset of the Pauli group. V S is non trivial iff ◮ The elements of S commute ⋆ The elements of the Pauli Group either commute or anticommute. ⋆ Suppose elements M , N anticommute: MN = − NM ⋆ Then | ψ � = MN | ψ � = − NM | ψ � = | ψ � ◮ − I is not an element of S . ⋆ If − I ∈ S then − I | ψ � = | ψ � then | ψ � = 0. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 6 / 20
Stabilizer Codes Let S be a subset of the Pauli group. V S is non trivial iff ◮ The elements of S commute ⋆ The elements of the Pauli Group either commute or anticommute. ⋆ Suppose elements M , N anticommute: MN = − NM ⋆ Then | ψ � = MN | ψ � = − NM | ψ � = | ψ � ◮ − I is not an element of S . ⋆ If − I ∈ S then − I | ψ � = | ψ � then | ψ � = 0. Easy exercise: If S is a subgroup of G n generated by elements g 1 , ..., g l then all elements of S commute iff g i g j commute for every i , j . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 6 / 20
Examples of Stabilizer Codes Action of a unitary on a stabilized set. Suppose V S is a subspace stabilized by a subgroup S generated by g 1 , g 2 , ..., g r . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20
Examples of Stabilizer Codes Action of a unitary on a stabilized set. Suppose V S is a subspace stabilized by a subgroup S generated by g 1 , g 2 , ..., g r . We have that U | ψ � = Ug | ψ � = UgI | ψ � = UgU † U | ψ � . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20
Examples of Stabilizer Codes Action of a unitary on a stabilized set. Suppose V S is a subspace stabilized by a subgroup S generated by g 1 , g 2 , ..., g r . We have that U | ψ � = Ug | ψ � = UgI | ψ � = UgU † U | ψ � . Which means that UgU † stabilizes U | ψ � Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20
Examples of Stabilizer Codes Action of a unitary on a stabilized set. Suppose V S is a subspace stabilized by a subgroup S generated by g 1 , g 2 , ..., g r . We have that U | ψ � = Ug | ψ � = UgI | ψ � = UgU † U | ψ � . Which means that UgU † stabilizes U | ψ � The vector space V S is stabilized by the group { UgU † | g ∈ S } Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20
Examples of Stabilizer Codes Action of a unitary on a stabilized set. Suppose V S is a subspace stabilized by a subgroup S generated by g 1 , g 2 , ..., g r . We have that U | ψ � = Ug | ψ � = UgI | ψ � = UgU † U | ψ � . Which means that UgU † stabilizes U | ψ � The vector space V S is stabilized by the group { UgU † | g ∈ S } More: If g 1 , g 2 , ..., g k generate S then Ug 1 U † ... Ug k U † generate USU † . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20
Examples of Stabilizer Codes HXH † = Z Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y HZH † = X Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y HZH † = X | 0 � is the only 1-qubit state stabilized by Z Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y HZH † = X | 0 � is the only 1-qubit state stabilized by Z | + � is the only 1-qubit state stabilized by X Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y HZH † = X | 0 � is the only 1-qubit state stabilized by Z | + � is the only 1-qubit state stabilized by X We have that H | 0 � is stabilized by HZH † = | + � Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y HZH † = X | 0 � is the only 1-qubit state stabilized by Z | + � is the only 1-qubit state stabilized by X We have that H | 0 � is stabilized by HZH † = | + � � Z 1 , Z 2 , ..., Z n � stabilizes | 0 � ⊗ n Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y HZH † = X | 0 � is the only 1-qubit state stabilized by Z | + � is the only 1-qubit state stabilized by X We have that H | 0 � is stabilized by HZH † = | + � � Z 1 , Z 2 , ..., Z n � stabilizes | 0 � ⊗ n � X 1 , X 2 , ..., X n � stabilizes | + � ⊗ n Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes HXH † = Z HYH † = − Y HZH † = X | 0 � is the only 1-qubit state stabilized by Z | + � is the only 1-qubit state stabilized by X We have that H | 0 � is stabilized by HZH † = | + � � Z 1 , Z 2 , ..., Z n � stabilizes | 0 � ⊗ n � X 1 , X 2 , ..., X n � stabilizes | + � ⊗ n Observe that we need 2 n amplitudes to specify this last state Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20
Examples of Stabilizer Codes Let U be the controlled-not. UX 1 U † = X 1 X 2 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 9 / 20
Examples of Stabilizer Codes Let U be the controlled-not. UX 1 U † = X 1 X 2 UX 2 U † = X 2 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 9 / 20
Examples of Stabilizer Codes Let U be the controlled-not. UX 1 U † = X 1 X 2 UX 2 U † = X 2 UZ 1 U † = Z 1 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 9 / 20
Examples of Stabilizer Codes Let U be the controlled-not. UX 1 U † = X 1 X 2 UX 2 U † = X 2 UZ 1 U † = Z 1 UZ 2 U † = Z 1 Z 2 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 9 / 20
Examples of Stabilizer Codes � 1 � 0 Let S = 0 i SXS † = Y SZS † = Z (1) Any unitary U that UG n U n = G n can be composed from Hadamard, phase and C-NOT gates. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 10 / 20
Examples of Stabilizer Codes � 1 � 0 Let S = 0 i SXS † = Y SZS † = Z (1) Any unitary U that UG n U n = G n can be composed from Hadamard, phase and C-NOT gates. The set of all unitaries U such that UgU † ∈ G n for g ∈ G n is called the normalizer of G n . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 10 / 20
Measurements Recalling: An observable is an Hermitian Operator on the state space of the system being observed. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20
Measurements Recalling: An observable is an Hermitian Operator on the state space of the system being observed. A projective Measurement is described by an observable M whose spectral decomposition is � M = mP m m Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20
Measurements Recalling: An observable is an Hermitian Operator on the state space of the system being observed. A projective Measurement is described by an observable M whose spectral decomposition is � M = mP m m where P m is the projector onto the eigenspace of M with eigenvalue m . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20
Measurements Recalling: An observable is an Hermitian Operator on the state space of the system being observed. A projective Measurement is described by an observable M whose spectral decomposition is � M = mP m m where P m is the projector onto the eigenspace of M with eigenvalue m . The possible outcomes of the measurements correspond to the eigenvalues m of the observable. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20
Measurements Recalling: An observable is an Hermitian Operator on the state space of the system being observed. A projective Measurement is described by an observable M whose spectral decomposition is � M = mP m m where P m is the projector onto the eigenspace of M with eigenvalue m . The possible outcomes of the measurements correspond to the eigenvalues m of the observable. The probability of getting the result m is given by p ( m ) = � ψ | P | ψ � Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20
Measurements Recalling: An observable is an Hermitian Operator on the state space of the system being observed. A projective Measurement is described by an observable M whose spectral decomposition is � M = mP m m where P m is the projector onto the eigenspace of M with eigenvalue m . The possible outcomes of the measurements correspond to the eigenvalues m of the observable. The probability of getting the result m is given by p ( m ) = � ψ | P | ψ � Given that the outcome m occurred, the state of the quantum system immediately after the measurement is P m | ψ � � p ( m ) Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20
Measurements Let g ∈ G n . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements Let g ∈ G n . Since g is a Hermitian operator, it can be regarded as an observable. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements Let g ∈ G n . Since g is a Hermitian operator, it can be regarded as an observable. Assume the system is in state | ψ � with stabilizer � g 1 , ..., g n � . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements Let g ∈ G n . Since g is a Hermitian operator, it can be regarded as an observable. Assume the system is in state | ψ � with stabilizer � g 1 , ..., g n � . There are two possibilities for g ∈ G n : Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements Let g ∈ G n . Since g is a Hermitian operator, it can be regarded as an observable. Assume the system is in state | ψ � with stabilizer � g 1 , ..., g n � . There are two possibilities for g ∈ G n : ◮ g commutes with all the generators of the stabilizer Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements Let g ∈ G n . Since g is a Hermitian operator, it can be regarded as an observable. Assume the system is in state | ψ � with stabilizer � g 1 , ..., g n � . There are two possibilities for g ∈ G n : ◮ g commutes with all the generators of the stabilizer ◮ g anti-commutes with one or more of the generators of the stabilizer. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements Let g ∈ G n . Since g is a Hermitian operator, it can be regarded as an observable. Assume the system is in state | ψ � with stabilizer � g 1 , ..., g n � . There are two possibilities for g ∈ G n : ◮ g commutes with all the generators of the stabilizer ◮ g anti-commutes with one or more of the generators of the stabilizer. ⋆ In this case it anticommutes with a unique generator, say g 1 and commutes with all the others g 2 , .., g n Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements Let g ∈ G n . Since g is a Hermitian operator, it can be regarded as an observable. Assume the system is in state | ψ � with stabilizer � g 1 , ..., g n � . There are two possibilities for g ∈ G n : ◮ g commutes with all the generators of the stabilizer ◮ g anti-commutes with one or more of the generators of the stabilizer. ⋆ In this case it anticommutes with a unique generator, say g 1 and commutes with all the others g 2 , .., g n ⋆ Suppose it anticommutes with g 2 . Then it commutes with g 1 g 2 . Then replace g 2 by g 1 g 2 . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20
Measurements g commutes with all generators. g anticommutes with some generator, say g 1 . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements g commutes with all generators. ◮ Then either g or − g is an element of the stabilizer g anticommutes with some generator, say g 1 . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements g commutes with all generators. ◮ Then either g or − g is an element of the stabilizer ◮ Since g j g | ψ � = gg j | ψ � = g | ψ � for each stabilizer generator, g | ψ � is in V S and thus a multiple of | ψ � . g anticommutes with some generator, say g 1 . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements g commutes with all generators. ◮ Then either g or − g is an element of the stabilizer ◮ Since g j g | ψ � = gg j | ψ � = g | ψ � for each stabilizer generator, g | ψ � is in V S and thus a multiple of | ψ � . ◮ Since g 2 = I , it follows that g | ψ � = ±| ψ � g anticommutes with some generator, say g 1 . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements g commutes with all generators. ◮ Then either g or − g is an element of the stabilizer ◮ Since g j g | ψ � = gg j | ψ � = g | ψ � for each stabilizer generator, g | ψ � is in V S and thus a multiple of | ψ � . ◮ Since g 2 = I , it follows that g | ψ � = ±| ψ � ◮ Then either g or − g must be in the stabilizer. g anticommutes with some generator, say g 1 . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements g commutes with all generators. ◮ Then either g or − g is an element of the stabilizer ◮ Since g j g | ψ � = gg j | ψ � = g | ψ � for each stabilizer generator, g | ψ � is in V S and thus a multiple of | ψ � . ◮ Since g 2 = I , it follows that g | ψ � = ±| ψ � ◮ Then either g or − g must be in the stabilizer. ◮ Assume g ∈ S the same holds for − g ∈ S . Then g | ψ � = | ψ � , and thus measuring g gives the eigenvalue +1 with probability 1. g anticommutes with some generator, say g 1 . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements g commutes with all generators. ◮ Then either g or − g is an element of the stabilizer ◮ Since g j g | ψ � = gg j | ψ � = g | ψ � for each stabilizer generator, g | ψ � is in V S and thus a multiple of | ψ � . ◮ Since g 2 = I , it follows that g | ψ � = ±| ψ � ◮ Then either g or − g must be in the stabilizer. ◮ Assume g ∈ S the same holds for − g ∈ S . Then g | ψ � = | ψ � , and thus measuring g gives the eigenvalue +1 with probability 1. g anticommutes with some generator, say g 1 . ◮ g has eigenvalue ± 1 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements g commutes with all generators. ◮ Then either g or − g is an element of the stabilizer ◮ Since g j g | ψ � = gg j | ψ � = g | ψ � for each stabilizer generator, g | ψ � is in V S and thus a multiple of | ψ � . ◮ Since g 2 = I , it follows that g | ψ � = ±| ψ � ◮ Then either g or − g must be in the stabilizer. ◮ Assume g ∈ S the same holds for − g ∈ S . Then g | ψ � = | ψ � , and thus measuring g gives the eigenvalue +1 with probability 1. g anticommutes with some generator, say g 1 . ◮ g has eigenvalue ± 1 ◮ Thus the projectors for the measurement outcomes ± 1 are given by ( I ± g ) / 2, respectively and thus the measurement probabilities are given by p (+1) = tr (1 p ( − 1) = tr (1 2( I + g ) | ψ �� ψ | ) 2( I − g ) | ψ �� ψ | ) Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20
Measurements One can see that p (+1) = p ( − 1) = 1 / 2 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 14 / 20
Measurements One can see that p (+1) = p ( − 1) = 1 / 2 √ If the result +1 occurs, the result collapses to | ψ + � ≡ ( I + g ) | ψ � / 2, which has stabilizer � g 1 , g 2 , ..., g n � . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 14 / 20
Measurements One can see that p (+1) = p ( − 1) = 1 / 2 √ If the result +1 occurs, the result collapses to | ψ + � ≡ ( I + g ) | ψ � / 2, which has stabilizer � g 1 , g 2 , ..., g n � . If the result is − 1 then the posterior state is stabilized to �− g 1 , g 2 , ..., g n � Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 14 / 20
Stabilizer Codes The stabilizer formalism is well suited for the description of error correcting codes. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 15 / 20
Stabilizer Codes The stabilizer formalism is well suited for the description of error correcting codes. [ n , k ] stabilizer code: Vector space V S stabilized by a subgroup S of G n such that − I / ∈ S and S has n − k independent and commuting generators, S = � g 1 , ..., g n − k � . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 15 / 20
Stabilizer Codes The stabilizer formalism is well suited for the description of error correcting codes. [ n , k ] stabilizer code: Vector space V S stabilized by a subgroup S of G n such that − I / ∈ S and S has n − k independent and commuting generators, S = � g 1 , ..., g n − k � . By independent generators we mean that removing any of the g ′ i s makes the code shorter. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 15 / 20
Stabilizer Codes The stabilizer formalism is well suited for the description of error correcting codes. [ n , k ] stabilizer code: Vector space V S stabilized by a subgroup S of G n such that − I / ∈ S and S has n − k independent and commuting generators, S = � g 1 , ..., g n − k � . By independent generators we mean that removing any of the g ′ i s makes the code shorter. Denote this code by C ( S ) Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 15 / 20
Stabilizer Codes Encoding Qubits: Chose operators Z 1 , ..., Z k such that g 1 , ..., g n − k , Z 1 , ..., Z k forms and independet and commuting set. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20
Stabilizer Codes Encoding Qubits: Chose operators Z 1 , ..., Z k such that g 1 , ..., g n − k , Z 1 , ..., Z k forms and independet and commuting set. Z i play the role of a logical pauli Z operator on qubit i Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20
Stabilizer Codes Encoding Qubits: Chose operators Z 1 , ..., Z k such that g 1 , ..., g n − k , Z 1 , ..., Z k forms and independet and commuting set. Z i play the role of a logical pauli Z operator on qubit i The logical basis state | x 1 , ..., x k � L is defined to be the state with stabilizer � g 1 , ..., g n − k , ( − 1) x 1 Z 1 , ..., ( − 1) x k Z k � Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20
Stabilizer Codes Encoding Qubits: Chose operators Z 1 , ..., Z k such that g 1 , ..., g n − k , Z 1 , ..., Z k forms and independet and commuting set. Z i play the role of a logical pauli Z operator on qubit i The logical basis state | x 1 , ..., x k � L is defined to be the state with stabilizer � g 1 , ..., g n − k , ( − 1) x 1 Z 1 , ..., ( − 1) x k Z k � Choose operators X j which sends Z j to − Z j and leaves all other Z i and g i alone under conjugation. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20
Stabilizer Codes Encoding Qubits: Chose operators Z 1 , ..., Z k such that g 1 , ..., g n − k , Z 1 , ..., Z k forms and independet and commuting set. Z i play the role of a logical pauli Z operator on qubit i The logical basis state | x 1 , ..., x k � L is defined to be the state with stabilizer � g 1 , ..., g n − k , ( − 1) x 1 Z 1 , ..., ( − 1) x k Z k � Choose operators X j which sends Z j to − Z j and leaves all other Z i and g i alone under conjugation. X j has the effect of a quantum NOT gate acting on the j -th encoded qubit. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20
Stabilizer Codes Encoding Qubits: Chose operators Z 1 , ..., Z k such that g 1 , ..., g n − k , Z 1 , ..., Z k forms and independet and commuting set. Z i play the role of a logical pauli Z operator on qubit i The logical basis state | x 1 , ..., x k � L is defined to be the state with stabilizer � g 1 , ..., g n − k , ( − 1) x 1 Z 1 , ..., ( − 1) x k Z k � Choose operators X j which sends Z j to − Z j and leaves all other Z i and g i alone under conjugation. X j has the effect of a quantum NOT gate acting on the j -th encoded qubit. † Since X j g k X j = g k , we have X j g k = g k X j Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . If E anticommutes with an element of the stabilizer then E takes C ( S ) to an orthogonal subspace. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . If E anticommutes with an element of the stabilizer then E takes C ( S ) to an orthogonal subspace. Thus E can in principle be detected Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . If E anticommutes with an element of the stabilizer then E takes C ( S ) to an orthogonal subspace. Thus E can in principle be detected If E ∈ S then E does not corrupt the code. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . If E anticommutes with an element of the stabilizer then E takes C ( S ) to an orthogonal subspace. Thus E can in principle be detected If E ∈ S then E does not corrupt the code. If E commutes with all elements of S but it is not in S then nothing can be done. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . If E anticommutes with an element of the stabilizer then E takes C ( S ) to an orthogonal subspace. Thus E can in principle be detected If E ∈ S then E does not corrupt the code. If E commutes with all elements of S but it is not in S then nothing can be done. The set of all such E ’s that commutes with each element of S is called the centralizer of S , or Z ( S ), which in this case is equal to the normalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for all g ∈ S . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . If E anticommutes with an element of the stabilizer then E takes C ( S ) to an orthogonal subspace. Thus E can in principle be detected If E ∈ S then E does not corrupt the code. If E commutes with all elements of S but it is not in S then nothing can be done. The set of all such E ’s that commutes with each element of S is called the centralizer of S , or Z ( S ), which in this case is equal to the normalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for all g ∈ S . S ⊆ N ( S ) for any subgroup S of G n . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Suppose C ( S ) is a stabilizer code corrupted by an error E ∈ G n . If E anticommutes with an element of the stabilizer then E takes C ( S ) to an orthogonal subspace. Thus E can in principle be detected If E ∈ S then E does not corrupt the code. If E commutes with all elements of S but it is not in S then nothing can be done. The set of all such E ’s that commutes with each element of S is called the centralizer of S , or Z ( S ), which in this case is equal to the normalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for all g ∈ S . S ⊆ N ( S ) for any subgroup S of G n . N ( S ) = Z ( S ) for any subgroup S of G n not containing − I . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20
Stabilizer Codes Error correction conditions Let S be the stabilizer for a stabilizer code C ( S ) Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20
Stabilizer Codes Error correction conditions Let S be the stabilizer for a stabilizer code C ( S ) Let { E j } be a set of operation in G n such that E † j E k / ∈ N ( S ) − S for all j , k . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20
Stabilizer Codes Error correction conditions Let S be the stabilizer for a stabilizer code C ( S ) Let { E j } be a set of operation in G n such that E † j E k / ∈ N ( S ) − S for all j , k . Then { E j } is a correctable set of errors for the code C ( S ). Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20
Stabilizer Codes Error correction conditions Let S be the stabilizer for a stabilizer code C ( S ) Let { E j } be a set of operation in G n such that E † j E k / ∈ N ( S ) − S for all j , k . Then { E j } is a correctable set of errors for the code C ( S ). Let P be the projector onto the code space C ( S ) Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20
Stabilizer Codes Error correction conditions Let S be the stabilizer for a stabilizer code C ( S ) Let { E j } be a set of operation in G n such that E † j E k / ∈ N ( S ) − S for all j , k . Then { E j } is a correctable set of errors for the code C ( S ). Let P be the projector onto the code space C ( S ) For given j and k , there are two possibilities: Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20
Stabilizer Codes Error correction conditions Let S be the stabilizer for a stabilizer code C ( S ) Let { E j } be a set of operation in G n such that E † j E k / ∈ N ( S ) − S for all j , k . Then { E j } is a correctable set of errors for the code C ( S ). Let P be the projector onto the code space C ( S ) For given j and k , there are two possibilities: E † j E k ∈ S 1 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20
Stabilizer Codes Error correction conditions Let S be the stabilizer for a stabilizer code C ( S ) Let { E j } be a set of operation in G n such that E † j E k / ∈ N ( S ) − S for all j , k . Then { E j } is a correctable set of errors for the code C ( S ). Let P be the projector onto the code space C ( S ) For given j and k , there are two possibilities: E † j E k ∈ S 1 ⋆ Then PE † j E k P = P since P is invariant under multiplication by elements of S . Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20
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