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Quantum Computation - Lecture 08 - Quantum ErrorCorrection II
Mateus de Oliveira Oliveira
TCS-KTH
January 20, 2013
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 1 / 20
Stabilizer Codes
X =
[0 11 0
]Z =
[1 00 −1
]
|ψ〉 = |00〉+|11〉√2
X1X2|ψ〉 = |ψ〉Z1Z2|ψ〉 = |ψ〉We say that |ψ〉 is stabilized by X1X2 and by Z1Z2.
|ψ〉 is the only state that up to a global phase that is stabilized byX1X2 and Z1Z2.
Quantum states with relevance for quantum error correction are oftenmore compactly described by the stabilizer formalism.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 2 / 20
Stabilizer Codes
X =
[0 11 0
]Z =
[1 00 −1
]|ψ〉 = |00〉+|11〉√
2
X1X2|ψ〉 = |ψ〉Z1Z2|ψ〉 = |ψ〉We say that |ψ〉 is stabilized by X1X2 and by Z1Z2.
|ψ〉 is the only state that up to a global phase that is stabilized byX1X2 and Z1Z2.
Quantum states with relevance for quantum error correction are oftenmore compactly described by the stabilizer formalism.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 2 / 20
Stabilizer Codes
X =
[0 11 0
]Z =
[1 00 −1
]|ψ〉 = |00〉+|11〉√
2
X1X2|ψ〉 = |ψ〉
Z1Z2|ψ〉 = |ψ〉We say that |ψ〉 is stabilized by X1X2 and by Z1Z2.
|ψ〉 is the only state that up to a global phase that is stabilized byX1X2 and Z1Z2.
Quantum states with relevance for quantum error correction are oftenmore compactly described by the stabilizer formalism.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 2 / 20
Stabilizer Codes
X =
[0 11 0
]Z =
[1 00 −1
]|ψ〉 = |00〉+|11〉√
2
X1X2|ψ〉 = |ψ〉Z1Z2|ψ〉 = |ψ〉
We say that |ψ〉 is stabilized by X1X2 and by Z1Z2.
|ψ〉 is the only state that up to a global phase that is stabilized byX1X2 and Z1Z2.
Quantum states with relevance for quantum error correction are oftenmore compactly described by the stabilizer formalism.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 2 / 20
Stabilizer Codes
X =
[0 11 0
]Z =
[1 00 −1
]|ψ〉 = |00〉+|11〉√
2
X1X2|ψ〉 = |ψ〉Z1Z2|ψ〉 = |ψ〉We say that |ψ〉 is stabilized by X1X2 and by Z1Z2.
|ψ〉 is the only state that up to a global phase that is stabilized byX1X2 and Z1Z2.
Quantum states with relevance for quantum error correction are oftenmore compactly described by the stabilizer formalism.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 2 / 20
Stabilizer Codes
X =
[0 11 0
]Z =
[1 00 −1
]|ψ〉 = |00〉+|11〉√
2
X1X2|ψ〉 = |ψ〉Z1Z2|ψ〉 = |ψ〉We say that |ψ〉 is stabilized by X1X2 and by Z1Z2.
|ψ〉 is the only state that up to a global phase that is stabilized byX1X2 and Z1Z2.
Quantum states with relevance for quantum error correction are oftenmore compactly described by the stabilizer formalism.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 2 / 20
Stabilizer Codes
X =
[0 11 0
]Z =
[1 00 −1
]|ψ〉 = |00〉+|11〉√
2
X1X2|ψ〉 = |ψ〉Z1Z2|ψ〉 = |ψ〉We say that |ψ〉 is stabilized by X1X2 and by Z1Z2.
|ψ〉 is the only state that up to a global phase that is stabilized byX1X2 and Z1Z2.
Quantum states with relevance for quantum error correction are oftenmore compactly described by the stabilizer formalism.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 2 / 20
Stabilizer Codes
Pauli Matrices:
I =
[1 00 1
]X =
[0 11 0
]Y =
[0 −ii 0
]Z =
[1 00 −1
]
Pauli Group:
G1 = {±I ,±iI ,±X ,±iX ,±Y ,±iY ,±Z ,±iZ}
G1 forms a group under matrix multiplication.
Gn = {P1 ⊗ P2 ⊗ ...⊗ Pn|Pi ∈ G1}Gn also forms a group under matrix multiplication.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 3 / 20
Stabilizer Codes
Pauli Matrices:
I =
[1 00 1
]X =
[0 11 0
]Y =
[0 −ii 0
]Z =
[1 00 −1
]Pauli Group:
G1 = {±I ,±iI ,±X ,±iX ,±Y ,±iY ,±Z ,±iZ}
G1 forms a group under matrix multiplication.
Gn = {P1 ⊗ P2 ⊗ ...⊗ Pn|Pi ∈ G1}Gn also forms a group under matrix multiplication.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 3 / 20
Stabilizer Codes
Pauli Matrices:
I =
[1 00 1
]X =
[0 11 0
]Y =
[0 −ii 0
]Z =
[1 00 −1
]Pauli Group:
G1 = {±I ,±iI ,±X ,±iX ,±Y ,±iY ,±Z ,±iZ}
G1 forms a group under matrix multiplication.
Gn = {P1 ⊗ P2 ⊗ ...⊗ Pn|Pi ∈ G1}Gn also forms a group under matrix multiplication.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 3 / 20
Stabilizer Codes
Pauli Matrices:
I =
[1 00 1
]X =
[0 11 0
]Y =
[0 −ii 0
]Z =
[1 00 −1
]Pauli Group:
G1 = {±I ,±iI ,±X ,±iX ,±Y ,±iY ,±Z ,±iZ}
G1 forms a group under matrix multiplication.
Gn = {P1 ⊗ P2 ⊗ ...⊗ Pn|Pi ∈ G1}
Gn also forms a group under matrix multiplication.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 3 / 20
Stabilizer Codes
Pauli Matrices:
I =
[1 00 1
]X =
[0 11 0
]Y =
[0 −ii 0
]Z =
[1 00 −1
]Pauli Group:
G1 = {±I ,±iI ,±X ,±iX ,±Y ,±iY ,±Z ,±iZ}
G1 forms a group under matrix multiplication.
Gn = {P1 ⊗ P2 ⊗ ...⊗ Pn|Pi ∈ G1}Gn also forms a group under matrix multiplication.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 3 / 20
Stabilizer Codes
Let S be a subgroup of Gn
Define VS = {|ψ〉 ∈ (C2)⊗n|M|ψ〉 = |ψ〉∀M ∈ Gn}In other words VS is the set of n-qubit states that are stabilized by allmatrices in S .
Exercise: VS is a subspace of (C2)⊗n
VS is the intersection of all Vx for x ∈ S
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 4 / 20
Stabilizer Codes
Let S be a subgroup of Gn
Define VS = {|ψ〉 ∈ (C2)⊗n|M|ψ〉 = |ψ〉∀M ∈ Gn}
In other words VS is the set of n-qubit states that are stabilized by allmatrices in S .
Exercise: VS is a subspace of (C2)⊗n
VS is the intersection of all Vx for x ∈ S
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 4 / 20
Stabilizer Codes
Let S be a subgroup of Gn
Define VS = {|ψ〉 ∈ (C2)⊗n|M|ψ〉 = |ψ〉∀M ∈ Gn}In other words VS is the set of n-qubit states that are stabilized by allmatrices in S .
Exercise: VS is a subspace of (C2)⊗n
VS is the intersection of all Vx for x ∈ S
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 4 / 20
Stabilizer Codes
Let S be a subgroup of Gn
Define VS = {|ψ〉 ∈ (C2)⊗n|M|ψ〉 = |ψ〉∀M ∈ Gn}In other words VS is the set of n-qubit states that are stabilized by allmatrices in S .
Exercise: VS is a subspace of (C2)⊗n
VS is the intersection of all Vx for x ∈ S
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 4 / 20
Stabilizer Codes
Let S be a subgroup of Gn
Define VS = {|ψ〉 ∈ (C2)⊗n|M|ψ〉 = |ψ〉∀M ∈ Gn}In other words VS is the set of n-qubit states that are stabilized by allmatrices in S .
Exercise: VS is a subspace of (C2)⊗n
VS is the intersection of all Vx for x ∈ S
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 4 / 20
Stabilizer Codes
Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}
I Z1Z2: |000〉, |001〉, |110〉, |111〉Z1Z3: |000〉, |010〉, |101〉, |111〉Z2Z3: |000〉, |100〉, |011〉, |111〉Then VS = {|000〉, |111〉}Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}
I Z1Z2: |000〉, |001〉, |110〉, |111〉Z1Z3: |000〉, |010〉, |101〉, |111〉Z2Z3: |000〉, |100〉, |011〉, |111〉Then VS = {|000〉, |111〉}Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}
I Z1Z2: |000〉, |001〉, |110〉, |111〉Z1Z3: |000〉, |010〉, |101〉, |111〉Z2Z3: |000〉, |100〉, |011〉, |111〉Then VS = {|000〉, |111〉}Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}I Z1Z2: |000〉, |001〉, |110〉, |111〉
Z1Z3: |000〉, |010〉, |101〉, |111〉Z2Z3: |000〉, |100〉, |011〉, |111〉Then VS = {|000〉, |111〉}Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}I Z1Z2: |000〉, |001〉, |110〉, |111〉
Z1Z3: |000〉, |010〉, |101〉, |111〉
Z2Z3: |000〉, |100〉, |011〉, |111〉Then VS = {|000〉, |111〉}Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}I Z1Z2: |000〉, |001〉, |110〉, |111〉
Z1Z3: |000〉, |010〉, |101〉, |111〉Z2Z3: |000〉, |100〉, |011〉, |111〉
Then VS = {|000〉, |111〉}Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}I Z1Z2: |000〉, |001〉, |110〉, |111〉
Z1Z3: |000〉, |010〉, |101〉, |111〉Z2Z3: |000〉, |100〉, |011〉, |111〉Then VS = {|000〉, |111〉}
Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Example:
S = {I ,Z1Z2,Z1Z3,Z2Z3}I Z1Z2: |000〉, |001〉, |110〉, |111〉
Z1Z3: |000〉, |010〉, |101〉, |111〉Z2Z3: |000〉, |100〉, |011〉, |111〉Then VS = {|000〉, |111〉}Obs: Any group can be generated by log |G |
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 5 / 20
Stabilizer Codes
I I I I Let S be a subset of the Pauli group. VS is non trivial iff
I The elements of S commuteF The elements of the Pauli Group either commute or anticommute.F Suppose elements M,N anticommute: MN = −NMF Then |ψ〉 = MN|ψ〉 = −NM|ψ〉 = |ψ〉
I −I is not an element of S .F If −I ∈ S then −I |ψ〉 = |ψ〉 then |ψ〉 = 0.
Easy exercise: If S is a subgroup of Gn generated by elements g1, ..., glthen all elements of S commute iff gigj commute for every i , j .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 6 / 20
Stabilizer Codes
Let S be a subset of the Pauli group. VS is non trivial iffI The elements of S commute
F The elements of the Pauli Group either commute or anticommute.F Suppose elements M,N anticommute: MN = −NMF Then |ψ〉 = MN|ψ〉 = −NM|ψ〉 = |ψ〉
I −I is not an element of S .F If −I ∈ S then −I |ψ〉 = |ψ〉 then |ψ〉 = 0.
Easy exercise: If S is a subgroup of Gn generated by elements g1, ..., glthen all elements of S commute iff gigj commute for every i , j .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 6 / 20
Stabilizer Codes
Let S be a subset of the Pauli group. VS is non trivial iffI The elements of S commute
F The elements of the Pauli Group either commute or anticommute.F Suppose elements M,N anticommute: MN = −NMF Then |ψ〉 = MN|ψ〉 = −NM|ψ〉 = |ψ〉
I −I is not an element of S .F If −I ∈ S then −I |ψ〉 = |ψ〉 then |ψ〉 = 0.
Easy exercise: If S is a subgroup of Gn generated by elements g1, ..., glthen all elements of S commute iff gigj commute for every i , j .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 6 / 20
Stabilizer Codes
Let S be a subset of the Pauli group. VS is non trivial iffI The elements of S commute
F The elements of the Pauli Group either commute or anticommute.F Suppose elements M,N anticommute: MN = −NMF Then |ψ〉 = MN|ψ〉 = −NM|ψ〉 = |ψ〉
I −I is not an element of S .F If −I ∈ S then −I |ψ〉 = |ψ〉 then |ψ〉 = 0.
Easy exercise: If S is a subgroup of Gn generated by elements g1, ..., glthen all elements of S commute iff gigj commute for every i , j .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 6 / 20
Examples of Stabilizer Codes
Action of a unitary on a stabilized set.
Suppose VS is a subspace stabilized by a subgroup S generated byg1, g2, ..., gr .
We have that U|ψ〉 = Ug |ψ〉 = UgI |ψ〉 = UgU†U|ψ〉.Which means that UgU† stabilizes U|ψ〉The vector space VS is stabilized by the group
{UgU†|g ∈ S}
More: If g1, g2, ..., gk generate S then Ug1U†... UgkU
† generateUSU†.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 7 / 20
Examples of Stabilizer Codes
Action of a unitary on a stabilized set.
Suppose VS is a subspace stabilized by a subgroup S generated byg1, g2, ..., gr .
We have that U|ψ〉 = Ug |ψ〉 = UgI |ψ〉 = UgU†U|ψ〉.
Which means that UgU† stabilizes U|ψ〉The vector space VS is stabilized by the group
{UgU†|g ∈ S}
More: If g1, g2, ..., gk generate S then Ug1U†... UgkU
† generateUSU†.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 7 / 20
Examples of Stabilizer Codes
Action of a unitary on a stabilized set.
Suppose VS is a subspace stabilized by a subgroup S generated byg1, g2, ..., gr .
We have that U|ψ〉 = Ug |ψ〉 = UgI |ψ〉 = UgU†U|ψ〉.Which means that UgU† stabilizes U|ψ〉
The vector space VS is stabilized by the group
{UgU†|g ∈ S}
More: If g1, g2, ..., gk generate S then Ug1U†... UgkU
† generateUSU†.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 7 / 20
Examples of Stabilizer Codes
Action of a unitary on a stabilized set.
Suppose VS is a subspace stabilized by a subgroup S generated byg1, g2, ..., gr .
We have that U|ψ〉 = Ug |ψ〉 = UgI |ψ〉 = UgU†U|ψ〉.Which means that UgU† stabilizes U|ψ〉The vector space VS is stabilized by the group
{UgU†|g ∈ S}
More: If g1, g2, ..., gk generate S then Ug1U†... UgkU
† generateUSU†.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 7 / 20
Examples of Stabilizer Codes
Action of a unitary on a stabilized set.
Suppose VS is a subspace stabilized by a subgroup S generated byg1, g2, ..., gr .
We have that U|ψ〉 = Ug |ψ〉 = UgI |ψ〉 = UgU†U|ψ〉.Which means that UgU† stabilizes U|ψ〉The vector space VS is stabilized by the group
{UgU†|g ∈ S}
More: If g1, g2, ..., gk generate S then Ug1U†... UgkU
† generateUSU†.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 7 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉
〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
HXH† = Z
HYH† = −YHZH† = 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† = |+〉〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n
〈X1,X2, ...,Xn〉 stabilizes |+〉⊗n
Observe that we need 2n amplitudes to specify this last state
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 8 / 20
Examples of Stabilizer Codes
Let U be the controlled-not.
UX1U† = X1X2
UX2U† = X2
UZ1U† = Z1
UZ2U† = Z1Z2
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 9 / 20
Examples of Stabilizer Codes
Let U be the controlled-not.
UX1U† = X1X2
UX2U† = X2
UZ1U† = Z1
UZ2U† = Z1Z2
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 9 / 20
Examples of Stabilizer Codes
Let U be the controlled-not.
UX1U† = X1X2
UX2U† = X2
UZ1U† = Z1
UZ2U† = Z1Z2
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 9 / 20
Examples of Stabilizer Codes
Let U be the controlled-not.
UX1U† = X1X2
UX2U† = X2
UZ1U† = Z1
UZ2U† = Z1Z2
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 9 / 20
Examples of Stabilizer Codes
Let S =
[1 00 i
]SXS† = Y SZS† = Z (1)
Any unitary U that UGnUn = Gn can be composed from Hadamard,
phase and C-NOT gates.
The set of all unitaries U such that UgU† ∈ Gn for g ∈ Gn is calledthe normalizer of Gn.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 10 / 20
Examples of Stabilizer Codes
Let S =
[1 00 i
]SXS† = Y SZS† = Z (1)
Any unitary U that UGnUn = Gn can be composed from Hadamard,
phase and C-NOT gates.
The set of all unitaries U such that UgU† ∈ Gn for g ∈ Gn is calledthe normalizer of Gn.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 10 / 20
Measurements
Recalling:
An observable is an Hermitian Operator on the state space of thesystem being observed.
A projective Measurement is described by an observable M whosespectral decomposition is
M =∑m
mPm
where Pm is the projector onto the eigenspace of M with eigenvaluem.
The possible outcomes of the measurements correspond to theeigenvalues 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 systemimmediately after the measurement is
Pm|ψ〉√p(m)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 11 / 20
Measurements
Recalling:
An observable is an Hermitian Operator on the state space of thesystem being observed.
A projective Measurement is described by an observable M whosespectral decomposition is
M =∑m
mPm
where Pm is the projector onto the eigenspace of M with eigenvaluem.
The possible outcomes of the measurements correspond to theeigenvalues 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 systemimmediately after the measurement is
Pm|ψ〉√p(m)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 11 / 20
Measurements
Recalling:
An observable is an Hermitian Operator on the state space of thesystem being observed.
A projective Measurement is described by an observable M whosespectral decomposition is
M =∑m
mPm
where Pm is the projector onto the eigenspace of M with eigenvaluem.
The possible outcomes of the measurements correspond to theeigenvalues 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 systemimmediately after the measurement is
Pm|ψ〉√p(m)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 11 / 20
Measurements
Recalling:
An observable is an Hermitian Operator on the state space of thesystem being observed.
A projective Measurement is described by an observable M whosespectral decomposition is
M =∑m
mPm
where Pm is the projector onto the eigenspace of M with eigenvaluem.
The possible outcomes of the measurements correspond to theeigenvalues 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 systemimmediately after the measurement is
Pm|ψ〉√p(m)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 11 / 20
Measurements
Recalling:
An observable is an Hermitian Operator on the state space of thesystem being observed.
A projective Measurement is described by an observable M whosespectral decomposition is
M =∑m
mPm
where Pm is the projector onto the eigenspace of M with eigenvaluem.
The possible outcomes of the measurements correspond to theeigenvalues 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 systemimmediately after the measurement is
Pm|ψ〉√p(m)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 11 / 20
Measurements
Recalling:
An observable is an Hermitian Operator on the state space of thesystem being observed.
A projective Measurement is described by an observable M whosespectral decomposition is
M =∑m
mPm
where Pm is the projector onto the eigenspace of M with eigenvaluem.
The possible outcomes of the measurements correspond to theeigenvalues 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 systemimmediately after the measurement is
Pm|ψ〉√p(m)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 11 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizerI g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizerI g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.
There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizerI g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizerI g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizer
I g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizerI g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizerI g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
Let g ∈ Gn.
Since g is a Hermitian operator, it can be regarded as an observable.
Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.There are two possibilities for g ∈ Gn:
I g commutes with all the generators of the stabilizerI g anti-commutes with one or more of the generators of the stabilizer.
F In this case it anticommutes with a unique generator, say g1 andcommutes with all the others g2, .., gn
F Suppose it anticommutes with g2. Then it commutes with g1g2. Thenreplace g2 by g1g2.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 12 / 20
Measurements
g commutes with all generators.
I Then either g or −g is an element of the stabilizerI Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is in
VS and thus a multiple of |ψ〉.I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉I Then either g or −g must be in the stabilizer.I 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 g1.
I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by
(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 20
Measurements
g commutes with all generators.I Then either g or −g is an element of the stabilizer
I Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is inVS and thus a multiple of |ψ〉.
I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉I Then either g or −g must be in the stabilizer.I 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 g1.
I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by
(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 20
Measurements
g commutes with all generators.I Then either g or −g is an element of the stabilizerI Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is in
VS and thus a multiple of |ψ〉.
I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉I Then either g or −g must be in the stabilizer.I 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 g1.
I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by
(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 20
Measurements
g commutes with all generators.I Then either g or −g is an element of the stabilizerI Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is in
VS and thus a multiple of |ψ〉.I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉
I Then either g or −g must be in the stabilizer.I 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 g1.
I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by
(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 20
Measurements
g commutes with all generators.I Then either g or −g is an element of the stabilizerI Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is in
VS and thus a multiple of |ψ〉.I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉I Then either g or −g must be in the stabilizer.
I Assume g ∈ S the same holds for −g ∈ S . Then g |ψ〉 = |ψ〉, and thusmeasuring g gives the eigenvalue +1 with probability 1.
g anticommutes with some generator, say g1.
I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by
(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 20
Measurements
g commutes with all generators.I Then either g or −g is an element of the stabilizerI Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is in
VS and thus a multiple of |ψ〉.I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉I Then either g or −g must be in the stabilizer.I 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 g1.
I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by
(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 20
Measurements
g commutes with all generators.I Then either g or −g is an element of the stabilizerI Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is in
VS and thus a multiple of |ψ〉.I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉I Then either g or −g must be in the stabilizer.I 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 g1.I g has eigenvalue ±1
I Thus the projectors for the measurement outcomes ±1 are given by(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 20
Measurements
g commutes with all generators.I Then either g or −g is an element of the stabilizerI Since gjg |ψ〉 = ggj |ψ〉 = g |ψ〉 for each stabilizer generator, g |ψ〉 is in
VS and thus a multiple of |ψ〉.I Since g2 = I , it follows that g |ψ〉 = ±|ψ〉I Then either g or −g must be in the stabilizer.I 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 g1.I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by
(I ± g)/2, respectively and thus the measurement probabilities aregiven by
p(+1) = tr(1
2(I + g)|ψ〉〈ψ|) p(−1) = tr(
1
2(I − g)|ψ〉〈ψ|)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 13 / 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 〈g1, g2, ..., gn〉.If the result is −1 then the posterior state is stabilized to〈−g1, g2, ..., gn〉
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 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 〈g1, g2, ..., gn〉.
If the result is −1 then the posterior state is stabilized to〈−g1, g2, ..., gn〉
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 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 〈g1, g2, ..., gn〉.If the result is −1 then the posterior state is stabilized to〈−g1, g2, ..., gn〉
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 14 / 20
Stabilizer Codes
The stabilizer formalism is well suited for the description of errorcorrecting codes.
[n, k] stabilizer code: Vector space VS stabilized by a subgroup S ofGn such that −I /∈ S and S has n − k independent and commutinggenerators, S = 〈g1, ..., gn−k〉.By independent generators we mean that removing any of the g ′i smakes the code shorter.
Denote this code by C (S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 15 / 20
Stabilizer Codes
The stabilizer formalism is well suited for the description of errorcorrecting codes.
[n, k] stabilizer code: Vector space VS stabilized by a subgroup S ofGn such that −I /∈ S and S has n − k independent and commutinggenerators, S = 〈g1, ..., gn−k〉.
By independent generators we mean that removing any of the g ′i smakes the code shorter.
Denote this code by C (S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 15 / 20
Stabilizer Codes
The stabilizer formalism is well suited for the description of errorcorrecting codes.
[n, k] stabilizer code: Vector space VS stabilized by a subgroup S ofGn such that −I /∈ S and S has n − k independent and commutinggenerators, S = 〈g1, ..., gn−k〉.By independent generators we mean that removing any of the g ′i smakes the code shorter.
Denote this code by C (S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 15 / 20
Stabilizer Codes
The stabilizer formalism is well suited for the description of errorcorrecting codes.
[n, k] stabilizer code: Vector space VS stabilized by a subgroup S ofGn such that −I /∈ S and S has n − k independent and commutinggenerators, S = 〈g1, ..., gn−k〉.By independent generators we mean that removing any of the g ′i smakes the code shorter.
Denote this code by C (S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 15 / 20
Stabilizer Codes
Encoding Qubits:
Chose operators Z 1, ...,Z k such that g1, ..., gn−k ,Z 1, ...,Z k forms andindependet and commuting set.
Z i play the role of a logical pauli Z operator on qubit i
The logical basis state |x1, ..., xk〉L is defined to be the state withstabilizer
〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉
Choose operators X j which sends Z j to −Z j and leaves all other Zi
and gi alone under conjugation.
X j has the effect of a quantum NOT gate acting on the j-th encodedqubit.
Since X jgkX†j = gk , we have X jgk = gkX j
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 16 / 20
Stabilizer Codes
Encoding Qubits:
Chose operators Z 1, ...,Z k such that g1, ..., gn−k ,Z 1, ...,Z k forms andindependet and commuting set.
Z i play the role of a logical pauli Z operator on qubit i
The logical basis state |x1, ..., xk〉L is defined to be the state withstabilizer
〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉
Choose operators X j which sends Z j to −Z j and leaves all other Zi
and gi alone under conjugation.
X j has the effect of a quantum NOT gate acting on the j-th encodedqubit.
Since X jgkX†j = gk , we have X jgk = gkX j
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 16 / 20
Stabilizer Codes
Encoding Qubits:
Chose operators Z 1, ...,Z k such that g1, ..., gn−k ,Z 1, ...,Z k forms andindependet and commuting set.
Z i play the role of a logical pauli Z operator on qubit i
The logical basis state |x1, ..., xk〉L is defined to be the state withstabilizer
〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉
Choose operators X j which sends Z j to −Z j and leaves all other Zi
and gi alone under conjugation.
X j has the effect of a quantum NOT gate acting on the j-th encodedqubit.
Since X jgkX†j = gk , we have X jgk = gkX j
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 16 / 20
Stabilizer Codes
Encoding Qubits:
Chose operators Z 1, ...,Z k such that g1, ..., gn−k ,Z 1, ...,Z k forms andindependet and commuting set.
Z i play the role of a logical pauli Z operator on qubit i
The logical basis state |x1, ..., xk〉L is defined to be the state withstabilizer
〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉
Choose operators X j which sends Z j to −Z j and leaves all other Zi
and gi alone under conjugation.
X j has the effect of a quantum NOT gate acting on the j-th encodedqubit.
Since X jgkX†j = gk , we have X jgk = gkX j
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 16 / 20
Stabilizer Codes
Encoding Qubits:
Chose operators Z 1, ...,Z k such that g1, ..., gn−k ,Z 1, ...,Z k forms andindependet and commuting set.
Z i play the role of a logical pauli Z operator on qubit i
The logical basis state |x1, ..., xk〉L is defined to be the state withstabilizer
〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉
Choose operators X j which sends Z j to −Z j and leaves all other Zi
and gi alone under conjugation.
X j has the effect of a quantum NOT gate acting on the j-th encodedqubit.
Since X jgkX†j = gk , we have X jgk = gkX j
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 16 / 20
Stabilizer Codes
Encoding Qubits:
Chose operators Z 1, ...,Z k such that g1, ..., gn−k ,Z 1, ...,Z k forms andindependet and commuting set.
Z i play the role of a logical pauli Z operator on qubit i
The logical basis state |x1, ..., xk〉L is defined to be the state withstabilizer
〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉
Choose operators X j which sends Z j to −Z j and leaves all other Zi
and gi alone under conjugation.
X j has the effect of a quantum NOT gate acting on the j-th encodedqubit.
Since X jgkX†j = gk , we have X jgk = gkX j
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 16 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Suppose C (S) is a stabilizer code corrupted by an error E ∈ Gn.
If E anticommutes with an element of the stabilizer then E takesC (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 nothingcan be done.
The set of all such E ’s that commutes with each element of S iscalled the centralizer of S , or Z (S), which in this case is equal to thenormalizer of S , i.e., the set of all E ’s such that EgE † ∈ S for allg ∈ S .
S ⊆ N(S) for any subgroup S of Gn.
N(S) = Z (S) for any subgroup S of Gn not containing −I .
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 17 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:
1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:
1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:
1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:
1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:
1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.
F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Error correction conditions
Let S be the stabilizer for a stabilizer code C (S)
Let {Ej} be a set of operation in Gn such that E †j Ek /∈ N(S)− S forall j , k .
Then {Ej} 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:1 E †j Ek ∈ S
F Then PE †j EkP = P since P is invariant under multiplication byelements of S .
2 E †j Ek in Gn − N(S)
F then E †j Ek must anticommute with some element gl of S
F Let g1, ..., gn−k be a set of generators of S so that P =Πn−k
l=1(I+gl )
2n−k
F Using the anti-commutativity gives E †j EkP = (I − g1)E†j Ek
Πn−kl=2
(I+gl )
2n−k
F But P(I − gl) = 0 since (I + g1)(I − g1) = 0.F Then PE †j EkP = 0 whenever E †j Ek ∈ Gn − N(S)
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 18 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:
I Measure the generators g1, ..., gn−k to obtain the syndrome.I The syndrome is simply the results β1, ..., βn−k of the measurements.I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:
I Measure the generators g1, ..., gn−k to obtain the syndrome.I The syndrome is simply the results β1, ..., βn−k of the measurements.I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:
I Measure the generators g1, ..., gn−k to obtain the syndrome.I The syndrome is simply the results β1, ..., βn−k of the measurements.I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:I Measure the generators g1, ..., gn−k to obtain the syndrome.
I The syndrome is simply the results β1, ..., βn−k of the measurements.I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:I Measure the generators g1, ..., gn−k to obtain the syndrome.I The syndrome is simply the results β1, ..., βn−k of the measurements.
I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:I Measure the generators g1, ..., gn−k to obtain the syndrome.I The syndrome is simply the results β1, ..., βn−k of the measurements.I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:I Measure the generators g1, ..., gn−k to obtain the syndrome.I The syndrome is simply the results β1, ..., βn−k of the measurements.I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k]stabilizer code.
Let {Ej} be a set of correctable errors.
Error-detection:I Measure the generators g1, ..., gn−k to obtain the syndrome.I The syndrome is simply the results β1, ..., βn−k of the measurements.I if the error Ej occurred, then the the error syndrome is given by βl such
that EjglE†j = βlgl .
I If Ej is the only error operator having this syndrome, then apply E †j torecover.
I If there distinct errors Ej and Ej′ such that EjglE†j = βlgl = Ej′glE
†j′ ,
then EjPE†j = Ej′PE
†j′ , where P is the projector onto the code space,
so E †j Ej′PE†j′Ej = P.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 19 / 20
Stabilizer Codes
Distance for a quantum Code:
The weight of an error E ∈ Gn is the number of terms in the tensorproduct which are not equal to the identity.
The distance of a stabilizer code C (S) is the minimum weight of anelement of N(S)− S .
If C (S) is an [n, k] code with distance d then we say that C (S) is an[n, k, d ] stabilizer code.
A code with distance at least 2t + 1 is able to correct arbitrary errorson any t qubits.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 20 / 20
Stabilizer Codes
Distance for a quantum Code:
The weight of an error E ∈ Gn is the number of terms in the tensorproduct which are not equal to the identity.
The distance of a stabilizer code C (S) is the minimum weight of anelement of N(S)− S .
If C (S) is an [n, k] code with distance d then we say that C (S) is an[n, k, d ] stabilizer code.
A code with distance at least 2t + 1 is able to correct arbitrary errorson any t qubits.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 20 / 20
Stabilizer Codes
Distance for a quantum Code:
The weight of an error E ∈ Gn is the number of terms in the tensorproduct which are not equal to the identity.
The distance of a stabilizer code C (S) is the minimum weight of anelement of N(S)− S .
If C (S) is an [n, k] code with distance d then we say that C (S) is an[n, k, d ] stabilizer code.
A code with distance at least 2t + 1 is able to correct arbitrary errorson any t qubits.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 20 / 20
Stabilizer Codes
Distance for a quantum Code:
The weight of an error E ∈ Gn is the number of terms in the tensorproduct which are not equal to the identity.
The distance of a stabilizer code C (S) is the minimum weight of anelement of N(S)− S .
If C (S) is an [n, k] code with distance d then we say that C (S) is an[n, k, d ] stabilizer code.
A code with distance at least 2t + 1 is able to correct arbitrary errorson any t qubits.
Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction IIJanuary 20, 2013 20 / 20