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Quantum Computation - Lecture 08 - Quantum Error Correction II Mateus de Oliveira Oliveira TCS-KTH January 20, 2013 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 1 / 20

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Quantum Computation - Lecture 08 - Quantum Error Correction II

Mateus de Oliveira Oliveira

TCS-KTH

January 20, 2013

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 1 / 20

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Stabilizer Codes

X = 1 1

  • Z =

1 −1

  • Mateus de Oliveira Oliveira (TCS-KTH)

Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 2 / 20

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Stabilizer Codes

X = 1 1

  • Z =

1 −1

  • |ψ = |00+|11

√ 2

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 2 / 20

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Stabilizer Codes

X = 1 1

  • Z =

1 −1

  • |ψ = |00+|11

√ 2

X1X2|ψ = |ψ

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 2 / 20

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Stabilizer Codes

X = 1 1

  • Z =

1 −1

  • |ψ = |00+|11

√ 2

X1X2|ψ = |ψ Z1Z2|ψ = |ψ

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 2 / 20

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Stabilizer Codes

X = 1 1

  • Z =

1 −1

  • |ψ = |00+|11

√ 2

X1X2|ψ = |ψ Z1Z2|ψ = |ψ We say that |ψ is stabilized by X1X2 and by Z1Z2.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 2 / 20

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Stabilizer Codes

X = 1 1

  • Z =

1 −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 by X1X2 and Z1Z2.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 2 / 20

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Stabilizer Codes

X = 1 1

  • Z =

1 −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 by X1X2 and Z1Z2. Quantum states with relevance for quantum error correction are often more compactly described by the stabilizer formalism.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 2 / 20

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Stabilizer Codes

Pauli Matrices: I = 1 1

  • X =

1 1

  • Y =

−i i

  • Z =

1 −1

  • Mateus de Oliveira Oliveira (TCS-KTH)

Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 3 / 20

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Stabilizer Codes

Pauli Matrices: I = 1 1

  • X =

1 1

  • Y =

−i i

  • Z =

1 −1

  • Pauli Group:

G1 = {±I, ±iI, ±X, ±iX, ±Y , ±iY , ±Z, ±iZ}

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 3 / 20

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Stabilizer Codes

Pauli Matrices: I = 1 1

  • X =

1 1

  • Y =

−i i

  • Z =

1 −1

  • Pauli Group:

G1 = {±I, ±iI, ±X, ±iX, ±Y , ±iY , ±Z, ±iZ} G1 forms a group under matrix multiplication.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 3 / 20

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Stabilizer Codes

Pauli Matrices: I = 1 1

  • X =

1 1

  • Y =

−i i

  • Z =

1 −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}

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 3 / 20

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Stabilizer Codes

Pauli Matrices: I = 1 1

  • X =

1 1

  • Y =

−i i

  • Z =

1 −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 II January 20, 2013 3 / 20

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Stabilizer Codes

Let S be a subgroup of Gn ❈ ❈

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 4 / 20

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Stabilizer Codes

Let S be a subgroup of Gn Define VS = {|ψ ∈ (❈2)⊗n|M|ψ = |ψ∀M ∈ Gn} ❈

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 4 / 20

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Stabilizer Codes

Let S be a subgroup of Gn Define VS = {|ψ ∈ (❈2)⊗n|M|ψ = |ψ∀M ∈ Gn} In other words VS is the set of n-qubit states that are stabilized by all matrices in S. ❈

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 4 / 20

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Stabilizer Codes

Let S be a subgroup of Gn Define VS = {|ψ ∈ (❈2)⊗n|M|ψ = |ψ∀M ∈ Gn} In other words VS is the set of n-qubit states that are stabilized by all matrices in S. Exercise: VS is a subspace of (❈2)⊗n

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 4 / 20

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Stabilizer Codes

Let S be a subgroup of Gn Define VS = {|ψ ∈ (❈2)⊗n|M|ψ = |ψ∀M ∈ Gn} In other words VS is the set of n-qubit states that are stabilized by all matrices in S. Exercise: VS is a subspace of (❈2)⊗n VS is the intersection of all Vx for x ∈ S

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 4 / 20

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Stabilizer Codes

Example:

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Example:

S = {I, Z1Z2, Z1Z3, Z2Z3}

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Example:

S = {I, Z1Z2, Z1Z3, Z2Z3}

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Example:

S = {I, Z1Z2, Z1Z3, Z2Z3}

◮ Z1Z2: |000, |001, |110, |111 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Example:

S = {I, Z1Z2, Z1Z3, Z2Z3}

◮ Z1Z2: |000, |001, |110, |111

Z1Z3: |000, |010, |101, |111

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Example:

S = {I, Z1Z2, Z1Z3, Z2Z3}

◮ Z1Z2: |000, |001, |110, |111

Z1Z3: |000, |010, |101, |111 Z2Z3: |000, |100, |011, |111

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Example:

S = {I, Z1Z2, Z1Z3, Z2Z3}

◮ Z1Z2: |000, |001, |110, |111

Z1Z3: |000, |010, |101, |111 Z2Z3: |000, |100, |011, |111 Then VS = {|000, |111}

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Example:

S = {I, Z1Z2, Z1Z3, Z2Z3}

◮ 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 II January 20, 2013 5 / 20

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Stabilizer Codes ◮ ◮ Let S be a subset of the Pauli group. VS is non trivial iff Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 6 / 20

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Stabilizer Codes

Let S be a subset of the Pauli group. VS 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 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 6 / 20

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Stabilizer Codes

Let S be a subset of the Pauli group. VS 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

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Stabilizer Codes

Let S be a subset of the Pauli group. VS 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 Gn generated by elements g1, ..., gl then 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 II January 20, 2013 6 / 20

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Examples of Stabilizer Codes

Action of a unitary on a stabilized set. Suppose VS is a subspace stabilized by a subgroup S generated by g1, g2, ..., gr.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20

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Examples of Stabilizer Codes

Action of a unitary on a stabilized set. Suppose VS is a subspace stabilized by a subgroup S generated by g1, g2, ..., gr. 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

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Examples of Stabilizer Codes

Action of a unitary on a stabilized set. Suppose VS is a subspace stabilized by a subgroup S generated by g1, g2, ..., gr. 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

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Examples of Stabilizer Codes

Action of a unitary on a stabilized set. Suppose VS is a subspace stabilized by a subgroup S generated by g1, 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}

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20

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Examples of Stabilizer Codes

Action of a unitary on a stabilized set. Suppose VS is a subspace stabilized by a subgroup S generated by g1, 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† generate USU†.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 7 / 20

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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

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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

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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

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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

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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

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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

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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

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20

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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

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 8 / 20

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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 II January 20, 2013 8 / 20

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Examples of Stabilizer Codes

Let U be the controlled-not. UX1U† = X1X2

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 9 / 20

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Examples of Stabilizer Codes

Let U be the controlled-not. UX1U† = X1X2 UX2U† = X2

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 9 / 20

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Examples of Stabilizer Codes

Let U be the controlled-not. UX1U† = X1X2 UX2U† = X2 UZ1U† = Z1

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 9 / 20

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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 II January 20, 2013 9 / 20

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Examples of Stabilizer Codes

Let S = 1 i

  • SXS† = Y

SZS† = Z (1) Any unitary U that UGnUn = Gn 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

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Examples of Stabilizer Codes

Let S = 1 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 called the normalizer of Gn.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 10 / 20

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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

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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 =

  • m

mPm

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20

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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 =

  • m

mPm where Pm 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

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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 =

  • m

mPm where Pm 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

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SLIDE 55

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 =

  • m

mPm where Pm 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

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SLIDE 56

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 =

  • m

mPm where Pm 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 Pm|ψ

  • p(m)

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 11 / 20

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Measurements

Let g ∈ Gn.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20

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Measurements

Let g ∈ Gn. 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

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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.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20

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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:

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20

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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:

◮ 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

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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:

◮ 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

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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:

◮ 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 g1 and

commutes with all the others g2, .., gn

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20

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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:

◮ 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 g1 and

commutes with all the others g2, .., gn

⋆ Suppose it anticommutes with g2. Then it commutes with g1g2. Then

replace g2 by g1g2.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 12 / 20

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Measurements

g commutes with all generators. g anticommutes with some generator, say g1.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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Measurements

g commutes with all generators.

◮ Then either g or −g is an element of the stabilizer

g anticommutes with some generator, say g1.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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Measurements

g commutes with all generators.

◮ Then either g or −g is an element of the stabilizer ◮ Since gjg|ψ = ggj|ψ = g|ψ for each stabilizer generator, g|ψ is in

VS and thus a multiple of |ψ.

g anticommutes with some generator, say g1.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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Measurements

g commutes with all generators.

◮ Then either g or −g is an element of the stabilizer ◮ Since gjg|ψ = ggj|ψ = g|ψ for each stabilizer generator, g|ψ is in

VS and thus a multiple of |ψ.

◮ Since g 2 = I, it follows that g|ψ = ±|ψ

g anticommutes with some generator, say g1.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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Measurements

g commutes with all generators.

◮ Then either g or −g is an element of the stabilizer ◮ Since gjg|ψ = ggj|ψ = g|ψ for each stabilizer generator, g|ψ is in

VS 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 g1.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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SLIDE 70

Measurements

g commutes with all generators.

◮ Then either g or −g is an element of the stabilizer ◮ Since gjg|ψ = ggj|ψ = g|ψ for each stabilizer generator, g|ψ is in

VS 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 g1.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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SLIDE 71

Measurements

g commutes with all generators.

◮ Then either g or −g is an element of the stabilizer ◮ Since gjg|ψ = ggj|ψ = g|ψ for each stabilizer generator, g|ψ is in

VS 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 g1.

◮ g has eigenvalue ±1 Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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SLIDE 72

Measurements

g commutes with all generators.

◮ Then either g or −g is an element of the stabilizer ◮ Since gjg|ψ = ggj|ψ = g|ψ for each stabilizer generator, g|ψ is in

VS 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 g1.

◮ 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 2(I + g)|ψψ|) p(−1) = tr(1 2(I − g)|ψψ|)

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 13 / 20

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SLIDE 73

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

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SLIDE 74

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.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 14 / 20

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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 II January 20, 2013 14 / 20

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SLIDE 76

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

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SLIDE 77

Stabilizer Codes

The stabilizer formalism is well suited for the description of error correcting codes. [n, k] stabilizer code: Vector space VS stabilized by a subgroup S of Gn such that −I / ∈ S and S has n − k independent and commuting generators, S = g1, ..., gn−k.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 15 / 20

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SLIDE 78

Stabilizer Codes

The stabilizer formalism is well suited for the description of error correcting codes. [n, k] stabilizer code: Vector space VS stabilized by a subgroup S of Gn such that −I / ∈ S and S has n − k independent and commuting generators, S = g1, ..., gn−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

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SLIDE 79

Stabilizer Codes

The stabilizer formalism is well suited for the description of error correcting codes. [n, k] stabilizer code: Vector space VS stabilized by a subgroup S of Gn such that −I / ∈ S and S has n − k independent and commuting generators, S = g1, ..., gn−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

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SLIDE 80

Stabilizer Codes

Encoding Qubits: Chose operators Z 1, ..., Z k such that g1, ..., gn−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

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Stabilizer Codes

Encoding Qubits: Chose operators Z 1, ..., Z k such that g1, ..., gn−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

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SLIDE 82

Stabilizer Codes

Encoding Qubits: Chose operators Z 1, ..., Z k such that g1, ..., gn−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 |x1, ..., xkL is defined to be the state with stabilizer g1, ..., gn−k, (−1)x1Z 1, ..., (−1)xkZ k

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20

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SLIDE 83

Stabilizer Codes

Encoding Qubits: Chose operators Z 1, ..., Z k such that g1, ..., gn−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 |x1, ..., xkL is defined to be the state with stabilizer 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.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20

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SLIDE 84

Stabilizer Codes

Encoding Qubits: Chose operators Z 1, ..., Z k such that g1, ..., gn−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 |x1, ..., xkL is defined to be the state with stabilizer 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 encoded qubit.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20

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SLIDE 85

Stabilizer Codes

Encoding Qubits: Chose operators Z 1, ..., Z k such that g1, ..., gn−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 |x1, ..., xkL is defined to be the state with stabilizer 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 encoded qubit. Since X jgkX

† j = gk, we have X jgk = gkX j

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 16 / 20

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SLIDE 86

Stabilizer Codes

Suppose C(S) is a stabilizer code corrupted by an error E ∈ Gn.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20

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SLIDE 87

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 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

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SLIDE 88

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 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

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SLIDE 89

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 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

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SLIDE 90

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 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

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SLIDE 91

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 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

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SLIDE 92

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 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 Gn.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 17 / 20

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SLIDE 93

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 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 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 II January 20, 2013 17 / 20

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SLIDE 94

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

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SLIDE 95

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 for all j, k.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

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SLIDE 96

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 for all j, k. Then {Ej} 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

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SLIDE 97

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 for all 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)

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

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SLIDE 98

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 for all 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:

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

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SLIDE 99

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 for all 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

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

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SLIDE 100

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 for all 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

⋆ Then PE † j EkP = 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

slide-101
SLIDE 101

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 for all 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

⋆ Then PE † j EkP = P since P is invariant under multiplication by

elements of S.

2

E †

j Ek in Gn − N(S)

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

slide-102
SLIDE 102

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 for all 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

⋆ Then PE † j EkP = P since P is invariant under multiplication by

elements of S.

2

E †

j Ek in Gn − N(S)

⋆ then E † j Ek must anticommute with some element gl of S Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

slide-103
SLIDE 103

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 for all 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

⋆ Then PE † j EkP = P since P is invariant under multiplication by

elements of S.

2

E †

j Ek in Gn − N(S)

⋆ then E † j Ek must anticommute with some element gl of S ⋆ Let g1, ..., gn−k be a set of generators of S so that P = Πn−k

l=1 (I+gl )

2n−k Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

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SLIDE 104

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 for all 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

⋆ Then PE † j EkP = P since P is invariant under multiplication by

elements of S.

2

E †

j Ek in Gn − N(S)

⋆ then E † j Ek must anticommute with some element gl of S ⋆ Let g1, ..., gn−k be a set of generators of S so that P = Πn−k

l=1 (I+gl )

2n−k ⋆ Using the anti-commutativity gives E † j EkP = (I − g1)E † j Ek Πn−k

l=2 (I+gl )

2n−k Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

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SLIDE 105

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 for all 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

⋆ Then PE † j EkP = P since P is invariant under multiplication by

elements of S.

2

E †

j Ek in Gn − N(S)

⋆ then E † j Ek must anticommute with some element gl of S ⋆ Let g1, ..., gn−k be a set of generators of S so that P = Πn−k

l=1 (I+gl )

2n−k ⋆ Using the anti-commutativity gives E † j EkP = (I − g1)E † j Ek Πn−k

l=2 (I+gl )

2n−k ⋆ But P(I − gl) = 0 since (I + g1)(I − g1) = 0. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 18 / 20

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SLIDE 106

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 for all 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

⋆ Then PE † j EkP = P since P is invariant under multiplication by

elements of S.

2

E †

j Ek in Gn − N(S)

⋆ then E † j Ek must anticommute with some element gl of S ⋆ Let g1, ..., gn−k be a set of generators of S so that P = Πn−k

l=1 (I+gl )

2n−k ⋆ Using the anti-commutativity gives E † j EkP = (I − g1)E † j Ek Πn−k

l=2 (I+gl )

2n−k ⋆ But P(I − gl) = 0 since (I + g1)(I − g1) = 0. ⋆ 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 II January 20, 2013 18 / 20

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SLIDE 107

Stabilizer Codes

Let g1, ..., gn−k be a set of generators for the stabilizer of an [n, k] stabilizer code.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 19 / 20

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SLIDE 108

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.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 19 / 20

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SLIDE 109

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:

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 19 / 20

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SLIDE 110

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:

◮ Measure the generators g1, ..., gn−k to obtain the syndrome. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 19 / 20

slide-111
SLIDE 111

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:

◮ Measure the generators g1, ..., gn−k to obtain the syndrome. ◮ The syndrome is simply the results β1, ..., βn−k of the measurements. Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 19 / 20

slide-112
SLIDE 112

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:

◮ Measure the generators g1, ..., gn−k to obtain the syndrome. ◮ The syndrome is simply the results β1, ..., βn−k of the measurements. ◮ if the error Ej occurred, then the the error syndrome is given by βl such

that EjglE †

j = βlgl.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 19 / 20

slide-113
SLIDE 113

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:

◮ Measure the generators g1, ..., gn−k to obtain the syndrome. ◮ The syndrome is simply the results β1, ..., βn−k of the measurements. ◮ if the error Ej occurred, then the the error syndrome is given by βl such

that EjglE †

j = βlgl.

◮ If Ej is the only error operator having this syndrome, then apply E †

j to

recover.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 19 / 20

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SLIDE 114

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:

◮ Measure the generators g1, ..., gn−k to obtain the syndrome. ◮ The syndrome is simply the results β1, ..., βn−k of the measurements. ◮ if the error Ej occurred, then the the error syndrome is given by βl such

that EjglE †

j = βlgl.

◮ If Ej is the only error operator having this syndrome, then apply E †

j to

recover.

◮ 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 II January 20, 2013 19 / 20

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SLIDE 115

Stabilizer Codes

Distance for a quantum Code: The weight of an error E ∈ Gn is the number of terms in the tensor product which are not equal to the identity.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 20 / 20

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SLIDE 116

Stabilizer Codes

Distance for a quantum Code: The weight of an error E ∈ Gn is the number of terms in the tensor product which are not equal to the identity. The distance of a stabilizer code C(S) is the minimum weight of an element of N(S) − S.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 20 / 20

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SLIDE 117

Stabilizer Codes

Distance for a quantum Code: The weight of an error E ∈ Gn is the number of terms in the tensor product which are not equal to the identity. The distance of a stabilizer code C(S) is the minimum weight of an element 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.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 20 / 20

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SLIDE 118

Stabilizer Codes

Distance for a quantum Code: The weight of an error E ∈ Gn is the number of terms in the tensor product which are not equal to the identity. The distance of a stabilizer code C(S) is the minimum weight of an element 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 errors

  • n any t qubits.

Mateus de Oliveira Oliveira (TCS-KTH) Quantum Computation - Lecture 08 - Quantum Error Correction II January 20, 2013 20 / 20