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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.


Stabilizer Codes

X =

[0 11 0

]Z =

[1 00 −1

]|ψ〉 = |00〉+|11〉√

2





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.




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.




Stabilizer Codes

X =

[0 11 0

]Z =

[1 00 −1

]|ψ〉 = |00〉+|11〉√

2





Stabilizer Codes

X =

[0 11 0

]Z =

[1 00 −1

]|ψ〉 = |00〉+|11〉√

2





Stabilizer Codes

X =

[0 11 0

]Z =

[1 00 −1

]|ψ〉 = |00〉+|11〉√

2





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.


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}




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}




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}


Gn = {P1 ⊗ P2 ⊗ ...⊗ Pn|Pi ∈ G1}

Gn also forms a group under matrix multiplication.


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}




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


Stabilizer Codes


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 .




Stabilizer Codes






Stabilizer Codes






Stabilizer Codes






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 |


Stabilizer Codes

I I I I Example:

S = {I ,Z1Z2,Z1Z3,Z2Z3}



Stabilizer Codes

I I I I Example:

S = {I ,Z1Z2,Z1Z3,Z2Z3}



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 |


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 |


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 |


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 |


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 |


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 .


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〉




Stabilizer Codes






Stabilizer Codes






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†.





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}


† generateUSU†.





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}


† generateUSU†.






{UgU†|g ∈ S}


† generateUSU†.






{UgU†|g ∈ S}


† generateUSU†.



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



HXH† = Z

HYH† = −Y

HZH† = X








HXH† = Z









HXH† = Z









HXH† = Z









HXH† = Z




We have that H|0〉 is stabilized by HZH† = |+〉

〈Z1,Z2, ...,Zn〉 stabilizes |0〉⊗n





HXH† = Z









HXH† = Z









HXH† = Z









Let U be the controlled-not.

UX1U† = X1X2

UX2U† = X2

UZ1U† = Z1

UZ2U† = Z1Z2




UX1U† = X1X2

UX2U† = X2

UZ1U† = Z1

UZ2U† = Z1Z2




UX1U† = X1X2

UX2U† = X2

UZ1U† = Z1

UZ2U† = Z1Z2




UX1U† = X1X2

UX2U† = X2

UZ1U† = Z1

UZ2U† = Z1Z2



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.



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.


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)


Measurements

Recalling:



M =∑m

mPm




Pm|ψ〉√p(m)


Measurements

Recalling:



M =∑m

mPm




Pm|ψ〉√p(m)


Measurements

Recalling:



M =∑m

mPm




Pm|ψ〉√p(m)


Measurements

Recalling:



M =∑m

mPm



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)


Measurements

Recalling:



M =∑m

mPm




Pm|ψ〉√p(m)


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.


Measurements

Let g ∈ Gn.







Measurements

Let g ∈ Gn.


Assume the system is in state |ψ〉 with stabilizer 〈g1, ..., gn〉.

There are two possibilities for g ∈ Gn:





Measurements

Let g ∈ Gn.







Measurements

Let 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.




Measurements

Let g ∈ Gn.







Measurements

Let g ∈ Gn.







Measurements

Let g ∈ Gn.







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|)


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





p(+1) = tr(1

2(I + g)|ψ〉〈ψ|) p(−1) = tr(

1

2(I − g)|ψ〉〈ψ|)


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





p(+1) = tr(1

2(I + g)|ψ〉〈ψ|) p(−1) = tr(

1

2(I − g)|ψ〉〈ψ|)


Measurements


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





p(+1) = tr(1

2(I + g)|ψ〉〈ψ|) p(−1) = tr(

1

2(I − g)|ψ〉〈ψ|)


Measurements


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.




p(+1) = tr(1

2(I + g)|ψ〉〈ψ|) p(−1) = tr(

1

2(I − g)|ψ〉〈ψ|)


Measurements







p(+1) = tr(1

2(I + g)|ψ〉〈ψ|) p(−1) = tr(

1

2(I − g)|ψ〉〈ψ|)


Measurements




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|)


Measurements




g anticommutes with some generator, say g1.I g has eigenvalue ±1I Thus the projectors for the measurement outcomes ±1 are given by


p(+1) = tr(1

2(I + g)|ψ〉〈ψ|) p(−1) = tr(

1

2(I − g)|ψ〉〈ψ|)


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〉


Measurements



2,which has stabilizer 〈g1, g2, ..., gn〉.

If the result is −1 then the posterior state is stabilized to〈−g1, g2, ..., gn〉


Measurements



2,which has stabilizer 〈g1, g2, ..., gn〉.If the result is −1 then the posterior state is stabilized to〈−g1, g2, ..., gn〉


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)


Stabilizer 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.



Stabilizer Codes





Stabilizer Codes





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


Stabilizer Codes

Encoding Qubits:




〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉






Stabilizer Codes

Encoding Qubits:




〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉






Stabilizer Codes

Encoding Qubits:




〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉






Stabilizer Codes

Encoding Qubits:




〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉






Stabilizer Codes

Encoding Qubits:




〈g1, ..., gn−k , (−1)x1Z 1, ..., (−1)xkZ k〉






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 .


Stabilizer Codes










Stabilizer Codes










Stabilizer Codes










Stabilizer Codes










Stabilizer Codes










Stabilizer Codes










Stabilizer Codes










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)


Stabilizer Codes







1 E †j Ek ∈ S





l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes







1 E †j Ek ∈ S





l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes







1 E †j Ek ∈ S





l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes







1 E †j Ek ∈ S





l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes






For given j and k , there are two possibilities:1 E †j Ek ∈ S





l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes











l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes











l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes











l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes











l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes











l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



Stabilizer Codes











l=1(I+gl )

2n−k


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)


Stabilizer Codes











l=1(I+gl )

2n−k


Πn−kl=2

(I+gl )

2n−k



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.


Stabilizer Codes



Error-detection:





†j′ ,





Stabilizer Codes



Error-detection:





†j′ ,





Stabilizer Codes



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




†j′ ,





Stabilizer Codes



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




†j′ ,





Stabilizer Codes



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




†j′ ,





Stabilizer Codes







†j′ ,





Stabilizer Codes







†j′ ,





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.


Stabilizer Codes







Stabilizer Codes







Stabilizer Codes







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