quantum error-correcting codes: discrete math meets physics · 2018. 7. 24. · we denote the...

Quantum Error-Correcting Codes LAWCI

Latin American Week on Coding and Information

UniCamp – Campinas, Brazil

2018, July 20–27

Quantum Error-Correcting Codes:

Discrete Math meets Physics

Markus Grassl

[email protected]

www.codetables.de

Markus Grassl – 1– 24.07.2018


Simple Quantum Error-Correcting Code

Repetition code: |0〉 7→ |000〉, |1〉 7→ |111〉

Encoding of one qubit:

α|0〉+ β|1〉 7→ α|000〉+ β|111〉.

This defines a two-dimensional subspace HC ≤ H2 ⊗H2 ⊗H2

bit-flip quantum state subspace

no error α|000〉+ β|111〉 (1⊗ 1⊗ 1)HC1st position α|100〉+ β|011〉 (X ⊗ 1⊗ 1)HC2nd position α|010〉+ β|101〉 (1⊗X ⊗ 1)HC3rd position α|001〉+ β|110〉 (1⊗ 1⊗X)HC

Hence we have an orthogonal decomposition of H2 ⊗H2 ⊗H2



Simple Quantum Error-Correcting Code

|ψ〉|0〉|0〉

sg

s

g

encoding

|0〉|0〉

s

ggsg

s

g

s

gsyndrome computation measurement

��q��qAK

|c1〉|c2〉|c3〉

s1

s2

Error X ⊗ I ⊗ I gives syndrome s1s2 = 10

Error I ⊗X ⊗ I gives syndrome s1s2 = 01

Error I ⊗ I ⊗X gives syndrome s1s2 = 11



Shor’s Nine-Qubit Code

Hadamard basis:

|+〉 = 1√2(|0〉+ |1〉), |−〉 = 1√

2(|0〉 − |1〉)

Bit-flip code: |0〉 7→ |000〉, |1〉 7→ |111〉.Phase-flip code: |0〉 7→ |+++〉, |1〉 7→ | − −−〉.

Concatenation with bit-flip code gives:

|0〉 7→ 1√23(|000〉+ |111〉)(|000〉+ |111〉)(|000〉+ |111〉)

|1〉 7→ 1√23(|000〉 − |111〉)(|000〉 − |111〉)(|000〉 − |111〉)

Claim: This code can correct one error, i. e., it is an [[n, k, d]]2 = [[9, 1, 3]]2.



Shor’s Nine-Qubit Code

Bit-flip code: |0〉 7→ |000〉, |1〉 7→ |111〉Effect of single-qubit errors:

• X-errors change the basis states, but can be corrected

• Z-errors at any of the three positions:

Z|000〉 = |000〉Z|111〉 = −|111〉

“encoded” Z-operator

=⇒ can be corrected by the second level of encoding



Knill-Laflamme Conditions Revisited

• code with ONB {|ci〉}, channel with Kraus operators {Ek}

• conditions:(i) 〈ci|E†kEℓ|cj〉 = 0 for i 6= j

(ii) 〈ci|E†kEℓ|ci〉 = 〈cj |E†kEℓ|cj〉 = αkl

• interpretation:(i) orthogonal states remain orthogonal under errors

(ii) errors “rotate” all basis states the same way

-|ci〉

6

|cj〉 errors=⇒

-XXXXXz��

�1

66

��

BBBM

Eℓ|cj〉

Eℓ|ci〉

Ek|cj〉

Ek|ci〉α

α



General Decoding Algorithm

E1C E2C · · · EkC · · ·V0 E1|c0〉 E2|c0〉 · · · Ek|c0〉 · · ·V1 E1|c1〉 E2|c1〉 · · · Ek|c1〉 · · ·...

......

. . ....

Vi E1|ci〉 E2|ci〉 · · · Ek|ci〉 · · ·...

......

.... . .

〈ci|E†kEl|cj〉 = δi,jαk,l (1)



Orthogonal Decomposition

E′1C E′2C · · · E′kC · · ·V0 E′1|c0〉 E′2|c0〉 · · · E′k|c0〉 · · ·V1 E′1|c1〉 E′2|c1〉 · · · E′k|c1〉 · · ·...

......

. . ....

Vi E′1|ci〉 E′2|ci〉 · · · E′k|ci〉 · · ·...

......

.... . .

rows are mutually

orthogonal

columns are mutually orthogonal

• new error operators E′k are linear combinations of the El• projection onto E′kC determines the error

• exponentially many orthogonal spaces E′kC



CSS Prelims: Linear Binary Codes

Generator matrix:

C = [n, k, d]2 is a k-dim. subspace of Fn2

=⇒ basis with k (row) vectors, G ∈ Fk×n2C = {iG : i ∈ Fk2}

Parity check matrix:

C = [n, k, d]2 is a k-dim. subspace of Fn2

=⇒ kernel of n− k homogeneous linear equations, H ∈ F(n−k)×n2C = {v : v ∈ Fn2 |vHt = 0}

Error syndrome:

erroneous codeword v = c+ e

=⇒ error syndrome vHt = cHt + eHt = eHt



CSS Prelims: Dual of a Classical Code

Let C ≤ Fn2 be a linear code. Then

C⊥ :=

{y ∈ Fn2 :

n∑

i=1

xiyi = 0 for all x ∈ C}

We denote the scalar product by x · y :=∑ni=1 xiyi.

Lemma: Let C be a linear binary code. Then

∑

c∈C(−1)x·c =

|C| for x ∈ C⊥,0 for x /∈ C⊥.



Bit-flips and Phase-flips

Let C ≤ Fn2 be a linear code. Then the image of the state1√|C|

∑

c∈C|c〉

under a bit-flip x ∈ Fn2 and a phase-flip z ∈ Fn2 is given by1√|C|

∑

c∈C(−1)z·c|c+ x〉.

Hadamard transform H ⊗ . . .⊗H maps this to(−1)xz√

|C⊥|∑

b∈C⊥(−1)x·b|b+ z〉



Proof:

Hadamard transform H⊗n =1√2n

∑

a,b∈Fn2

(−1)a·b|b〉〈a|

H⊗n(

1√|C|

∑

c∈C(−1)z·c|c+ x〉

)=

1√|C|2n

∑

a,b∈Fn2

∑

c∈C(−1)z·c+a·b|b〉〈a|c+ x〉

=1√|C|2n

∑

b∈Fn2

∑

c∈C(−1)z·c+(c+x)·b|b〉

=1√|C|2n

∑

b∈Fn2

(−1)x·b∑

c∈C(−1)c·(b+z)|b〉

=1√|C|2n

∑

b∈Fn2

(−1)x·(b+z)∑

c∈C(−1)c·b|b+ z〉

=(−1)x·z√

|C⊥|∑

b∈C⊥(−1)x·b|b+ z〉



CSS Codes

Introduced by R. Calderbank, P. Shor, and A. Steane

[Calderbank & Shor PRA, 54, 1098–1105, 1996]

[Steane, PRL 77, 793–797, 1996]

Construction: Let C1 = [n, k1, d1]q and C2 = [n, k2, d1]q be classical linear

codes with C⊥2 ≤ C1. Let {x1, . . . ,xK} be representatives for the cosetsC1/C

⊥2 . Define quantum states

|xi + C⊥2 〉 :=1√|C⊥2 |

∑

y∈C⊥2

|xi + y〉

Theorem: Then the vector space C spanned by these states is a quantum codewith parameters [[n, k1 + k2 − n, d]]q where d ≥ min(d1, d2).



CSS Codes — how they work

Basis states:

|xi + C⊥2 〉 =1√|C⊥2 |

∑

y∈C⊥2

|xi + y〉

Suppose a bit-flip error b happens to |xi + C⊥2 〉:1√|C⊥2 |

∑

y∈C⊥2

|xi + y + b〉

Now, we introduce an ancilla register initialized in |0〉 and compute thesyndrome.



CSS Codes — how they work

Let H1 be the parity check matrix of C1, i. e., xHt1 = 0 for all x ∈ C1.

1√|C⊥2 |

∑

y∈C⊥2

|xi + y + b〉|(xi + y + b)Ht1〉 =1√|C⊥2 |

∑

y∈C⊥2

|xi + y + b〉|bHt1〉

Then measure the ancilla to obtain s = bHt1. Use this to correct the error by a

conditional operation which flips the bits in b.

Phase-flips: Suppose we have the state

1√|C⊥2 |

∑

y∈C⊥2

(−1)(xi+y)·z|xi + y〉

Then H⊗n yields a superposition over a coset of C2 which has a bit-flip.

Correct it as before (with a parity check matrix for C2).



Encoding

Classical code

information 7−→ codeword

i = (i1, i2, . . . , ik) 7−→ i ·G =k∑

j=1

ijgj

Quantum code

information 7−→ cosets of the code C⊥2 in C1

|i〉 = |i1, i2, . . . , ik〉 7−→∑

c∈C⊥2

|c+ xi〉 =∑

c∈C⊥2

|c+k∑j=1

ijgj〉

=∑

x∈{0,1}k|x · G̃+

k∑j=1

ijgj〉

C⊥2 ⊆ C1, G̃ is generator matrix for C⊥2 ,the gj are linear independent coset representatives of C1/C

⊥2



Example: Seven Qubit Code

Given the dual C of the binary Hamming code, with generator matrix

G =

0 0 1 0 1 1 1

0 1 0 1 1 1 0

1 0 0 1 0 1 1

,

then C is a [7, 3, 4] and C ≤ C⊥. The dual code C⊥ is a [7, 4, 3] and hasgenerator matrix

G′ =

0 0 0 1 1 0 1

0 0 1 0 1 1 1

0 1 0 1 1 1 0

1 0 0 1 0 1 1



G t =

0 0 0 1

0 0 1 0

0 1 0 0

1 0 1 1

1 1 1 0

0 1 1 1

1 1 0 1

|0〉

|0〉

|0〉

|ψ〉

|0〉

|0〉

|0〉

H

H

H

•i

i

•

iii

•

iii

•

i

ii |c7〉

|c6〉

|c5〉

|c4〉

|c3〉

|c2〉

|c1〉

α|0〉+ β|1〉 7→ α 1√8

∑

i1,i2,i3∈{0,1}|0g4 + i3g3 + i2g2 + i1g1〉

+β1√8

∑

i1,i2,i3∈{0,1}|1g4 + i3g3 + i2g2 + i1g1〉



Computation of an Error Syndrome

Classical code:

received vector 7−→ syndrome

r = (r1, r2, . . . , rk) 7−→ r ·Ht =n−k∑

j=1

rjhj = (c+ e) ·Ht = e ·Ht

Quantum code:

quantum state 7−→ quantum state⊗ syndrome∑

c∈C⊥2|c+xi+e〉

7−→∑

c∈C⊥2

|c+ xi + e〉 ⊗ |(c+ xi + e) · H̃t〉

=

∑

c∈C⊥2

|c+ xi + e〉

⊗ |e · H̃t〉

H̃ is a parity check matrix of C1



Syndrome Computation of CSS Codes

|ψi〉=∑

c∈C⊥2

|c+ xi〉, H⊗N |ψi〉=∑

c∈C2(−1)c·xi |c〉

|c7〉|c6〉|c5〉|c4〉|c3〉|c2〉|c1〉

|0〉|0〉|0〉|0〉|0〉|0〉

•

d

•

d

•

d•

d

•

d

•

d

•

d

•

d

•

d

•

d

•

d

•

d

H

H

H

H

H

H

H

•

d

•

d

•

d

•

d

•

d

•

d

•

d

•

d

•

d

•

d

•

d

•

d

H

H

H

H

H

H

H

|c7〉|c6〉|c5〉|c4〉|c3〉|c2〉|c1〉

|s1〉|s2〉

bit–flips

|s3〉|s4〉|s5〉

sign–flips

|s6〉



Stabilizer Codes

Observables

C is a common eigenspace of the stabilizer group S

decomp. into

common eigenspaces

E1

E2

Ei

E2n-k−1

the orthogonal spaces are labeled by the eigenvalues

=⇒ operations that change the eigenvalues can be detected



Stabilizer Codes

Representation Theory

C is an eigenspace of S w.r.t. some irred. character χ1

decomp. into

irred. components

χ1

χ2

χi

χ2n-k−1

the orthogonal spaces are labeled by the character (values)

=⇒ operations that change the character (value) can be detected



The Stabilizer of a Quantum Code

Pauli group:

Gn ={± E1 ⊗ . . .⊗ En : Ei ∈ {I,X, Y, Z}

}

Let C ≤ C2n be a quantum code.

The stabilizer of C is defined to be the set

S =

{M ∈ Gn : M |v〉 = |v〉 for all |v〉 ∈ C

}.

S is an abelian (commutative) group!



The Stabilizer of the Repetition Code

Repetition code: Recall that C is the two-dimensional codespanned by

|0〉 = |000〉|1〉 = |111〉

What is the stabilizer of C?

S = {III, ZZI, ZIZ, IZZ}.



Stabilizer Codes

Let C be a quantum code with stabilizer S.

The code C is called a stabilizer code if and only if

M |v〉 = |v〉 for all M ∈ S

implies that |v〉 ∈ C.

In this case C is the joint +1-eigenspace of all M ∈ S.

Example: The repetition code is a stabilizer code.



Errors in Stabilizer Codes

Important identity:

ZX =

(1 0

0 −1

)(0 1

1 0

)= −XZ.

For error operators in the Pauli group Gn (i. e. tensor products ofI,X, Y, Z) there are two possibilities: either

• they commute: EF = FE

• or they anticommute: EF = −FE



Errors in Stabilizer Codes

Let S be the stabilizer of a quantum code C. Let PC be theprojector onto C.

If an error E anticommutes with some M ∈ S then E is detectableby C.

Indeed,

PCEPC = PCEMPC = −PCMEPC = −PCEPC.

Hence PCEPC = 0, i. e. the spaces C and EC are orthogonal.



Errors: the Good, the Bad, and the Ugly

Let S be the stabilizer of a stabilizer code C.

An error E is good if it does not affect the encoded information,

i. e., if E ∈ S.

An error E is bad if it is detectable, e. g., it anticommutes with

some M ∈ S.

An error E is ugly if it cannot be detected.



Examples of the Good, the Bad, and the Ugly

Let C the repetition code

Good: Z ⊗ Z ⊗ I since Z ⊗ Z ⊗ I|111〉 = |111〉Z ⊗ Z ⊗ I|000〉 = |000〉

Bad: X ⊗ I ⊗ I since X ⊗ I ⊗ I|111〉 = |011〉X ⊗ I ⊗ I|000〉 = |100〉

Ugly: X ⊗X ⊗X since X ⊗X ⊗X|111〉 = |000〉



Error Correction Capabilities

Let C be a stabilizer code with stabilizer S.Let N(S) be the normalizer of S in Gn. This is the set of allmatrices in G which commute with all of S. Clearly, we haveS ≤ N(S).All errors outside N(S) can be detected.

Good do not affect encoded information: x ∈ SBad anticommute with an element in S: x /∈ N(S)Ugly errors that cannot be detected: x ∈ N(S) \ SWe need to find stabilizers such that the minimum weight of an

element in N(S) \ S is as large as possible.Markus Grassl – 63– 24.07.2018


Error Correction Capabilities

Summary: Any two elements M1, M2 of S commute.

Detectable errors anticommute with some M ∈ S.

Task: Find a nice characterization of these properties.

Notation: Denote by Xa where a = (a1, . . . , an) ∈ Fn2 theoperator

Xa = Xa1 ⊗Xa2 ⊗ . . .⊗Xan .

For instance X110 = X ⊗X ⊗ I.Hence, any operator in Gn is of the form ±XaZb for somea, b ∈ Fn2 .



Symplectic Geometry

Consider M1 = XaZb, M2 = X

cZd. When do M1 and M2commute?

M1M2 = XaZbXcZd = (−1)b·cXa+cZb+d

M2M1 = XcZdXaZb = (−1)a·dXa+cZb+d

Hence, M1 and M2 commute iff a · d− b · c = 0 mod 2.Suppose that S is the stabilizer of a 2k-dimensional stabilizer code.

Then |S| = 2n−k. S can be generated by n− k operators XaZb.Let H = (X|Z) be an (n− k)× 2n matrix over F2. This matrix isused to describe the stabilizer. The rows of H contain the vectors

(a|b).Markus Grassl – 65– 24.07.2018


Short Description of a Stabilizer

Example: Let S = {III, ZZI, IZZ, ZIZ}Note that S is generated by ZZI and ZIZ.

H = (X|Z) =(

0 0 0 1 1 0

0 0 0 1 0 1

)

(a|b) = (000|110) and (c|d) = (000|101) are orthogonal:

a · d− b · c = 000 · 101 + 110 · 000 = 0

In this case we also write (a|b) ⊥ (c|d) with respect to thesymplectic inner product (a|b) ⋆ (c|d) := a · d− b · c.



Shor’s nine-qubit code

Hadamard basis:

|+〉 = 1√2(|0〉+ |1〉), |−〉 = 1√

2(|0〉 − |1〉)

Phase-flip code: |0〉 7→ |+++〉, |1〉 7→ | − −−〉.

Concatenation with bit-flip code gives:

|0〉 7→ 1√23(|000〉+ |111〉)(|000〉+ |111〉)(|000〉+ |111〉)

|1〉 7→ 1√23(|000〉 − |111〉)(|000〉 − |111〉)(|000〉 − |111〉)

This code can correct one error, i. e., it is a [[9, 1, 3]]2.



Stabilizer for the nine-qubit code

Z Z · · · · · · ·Z · Z · · · · · ·· · · Z Z · · · ·· · · Z · Z · · ·· · · · · · Z Z ·· · · · · · Z · ZX X X X X X · · ·X X X · · · X X X

Generates an abelian subgroup of G9. This generating system of eightgenerators is minimal. Each generator divides the space by 2.

Hence we obtain a [[9, 1, 3]]2 quantum code.



The Five Qubit Code

Stabilizer: S = 〈Y ZZY I, IY ZZY, Y IY ZZ, ZY IY Z〉(X|Z) matrix given by (recall: X = (1, 0), Y = (1, 1), Z = (0, 1)):

H =

1 0 0 1 0 1 1 1 1 0

0 1 0 0 1 0 1 1 1 1

1 0 1 0 0 1 0 1 1 1

0 1 0 1 0 1 1 0 1 1

C = [[5, 1, 3]]2shortest qubit code which encodes one qubit and can correct one error



Example: CSS-Code C = [[7, 1, 3]]2

Here H = (X|Z) is of a special form:

H =

1 0 0 1 0 1 1

0 1 0 1 1 1 0

0 0 1 0 1 1 1

1 0 0 1 0 1 1

0 1 0 1 1 1 0

0 0 1 0 1 1 1

• Can be applied for (binary) codes C⊥2 ⊆ C1. If C1 = [n, k1, d1] andC2 = [n, k2, d2], this gives a QECC [[n, k1 + k2 − n,min(d1, d2)]].

• Has been used to derive asymptotic good codes (explicitly).

• CSS codes have nice properties for fault-tolerant computing.



Bounds

Quantum Hamming Bound: Set of vectors obtained by applying each error

to each basis vector has to be linear independent. The subspace they span has

to fit into the Hilbert space of n qubits. For a purea QECC [[n, k, 2t+ 1]]2 this

implies:

2n ≥ 2kt∑

i=0

(22 − 1)i(ni

)

Quantum Singleton Bound/Knill–Laflamme Bound: For all quantum codes

[[n, k, d]]q we have that

n+ 2 ≥ k + 2d.Codes which meet the bound are called quantum MDS codes.

Conjecture: n ≤ q2 + 1 (or q2 + 2); d ≤ q + 1 (with few exceptions)

aall error-syndromes are different



Optimal Small Qubit-QECC

n/k 0 1 2 3 4 5 6

2 [[2, 0, 2]]2

3 [[3, 0, 2]]2

4 [[4, 0, 2]]2 [[4, 1, 2]]2 [[4, 2, 2]]2

5 [[5, 0, 3]]2 [[5, 1, 3]]2 [[5, 2, 2]]2

6 [[6, 0, 4]]2 [[6, 1, 3]]2 [[6, 2, 2]]2 [[6, 3, 2]]2 [[6, 4, 2]]2

7 [[7, 0, 3]]2 [[7, 1, 3]]2 [[7, 2, 2]]2 [[7, 3, 2]]2 [[7, 4, 2]]2

8 [[8, 0, 4]]2 [[8, 1, 3]]2 [[8, 2, 3]]2 [[8, 3, 3]]2 [[8, 4, 2]]2 [[8, 5, 2]]2 [[8, 6, 2]]2

9 [[9, 0, 4]]2 [[9, 1, 3]]2 [[9, 2, 3]]2 [[9, 3, 3]]2 [[9, 4, 2]]2 [[9, 5, 2]]2 [[9, 6, 2]]2

[[n, k, d]]2 means n is minimal for fixed (k, d).

[[n, k, d]]2 means d is maximal for fixed (n, k).

For more tables see http://www.codetables.de



Quantum Stabilizer Codes

[Gottesman, PRA 54 (1996); Calderbank, Rains, Shor, & Sloane, IEEE-IT 44 (1998)]

Basic Idea

Decomposition of the complex vector space into eigenspaces of operators.

Error Basis for Qudits

[A. Ashikhmin & E. Knill, Nonbinary quantum stabilizer codes, IEEE-IT 47 (2001)]

E = {XαZβ : α, β ∈ Fq},

where (you may think of Cq ∼= C[Fq])

Xα =∑

x∈Fq

|x+ α〉〈x| for α ∈ Fq

and Zβ =∑

z∈Fq

ωtr(βz)|z〉〈z| for β ∈ Fq (ω = ωp = exp(2πi/p))



Stabilizer Codes

common eigenspace of an Abelian subgroup S of the group Gn with elements

ωγ(Xα1Zβ1)⊗ (Xα2Zβ2)⊗ . . .⊗ (XαnZβn) =: ωγXαZβ,

where α,β ∈ Fnq , γ ∈ Fp.

quotient group:

Gn := Gn/〈ωI〉 ∼= (Fq × Fq)n ∼= Fnq × Fnq

S Abelian subgroup⇐⇒ (α,β) ⋆ (α′,β′) = 0 for all ωγ(XαZβ), ωγ′(Xα′Zβ′) ∈ S,

where ⋆ is a symplectic inner product on Fnq × Fnq .

Stabilizer codes correspond to symplectic codes over Fnq × Fnq .



Symplectic Codes

most general:

additive codes C ⊂ Fnq × Fnq that are self-orthogonal with respect to

(v,w) ⋆ (v′,w′) := tr(v ·w′ − v′ ·w) = tr(n∑

i=1

viw′i − v′iwi)

special cases:

Fq-linear codes C ⊂ Fnq × Fnq that are self-orthogonal with respect to

(v,w) ⋆ (v′,w′) := v ·w′ − v′ ·w =n∑

i=1

viw′i − v′iwi

Fq2 -linear Hermitian codes C ⊂ Fnq2 that are self-orthogonal with respect to

x ⋆ y :=

n∑

i=1

xqi yi



Quantum Codes from Classical Codes

Hermitian self-orthogonal code

linear code C = [n, k, d′]q2 ≤ Fnq2 that is self-orthogonal with respect to theHermitian inner product

x ⋆ y :=n∑

i=1

xqi yi,

i. e., C ≤ C⋆ = {x ∈ Fnq2 | ∀y ∈ C : x ⋆ y = 0}Theorem: (Hermitian construction)

Let C = [n, k, d′]q2 be a Hermitian self-orthogonal code and let

d := min{wgt(c) : c ∈ C⋆ \ C} ≥ dmin(C⋆).

Then there exists a quantum code C = [[n, n− 2k, d]]q.[Ketkar et al., Nonbinary stabilizer codes over finite fields, IEEE-IT 52, pp. 4892–4914 (2006)]



Shortening Quantum Codes

[E. Rains, Nonbinary Quantum Codes, IEEE-IT 45, pp. 1827–1832 (1999)]

• shortening of classical codes: C = [n, k, d] → Cs = [n− 1, k − 1, d]

• for stabilizer codes:shortening C⋆ → C⋆s =⇒ puncturing C → Cp =⇒ Cp 6⊂ (Cp)⋆ = C⋆s

General problem:

How to turn a non-symplectic code into a symplectic one?

Basic idea:

n∑

i=1

(viw′i − v′iwi) 6= 0 for some (v,w), (v′,w′) ∈ C




[E. Rains, Nonbinary Quantum Codes, IEEE-IT 45, pp. 1827–1832 (1999)]

• shortening of classical codes: C = [n, k, d] → Cs = [n− 1, k − 1, d]

• for stabilizer codes:shortening C⋆ → C⋆s =⇒ puncturing C → Cp =⇒ Cp 6⊂ (Cp)⋆ = C⋆s

General problem:

How to turn a non-symplectic code into a symplectic one?

Basic idea: find (α1, α2, . . . , αn) ∈ Fnq withn∑

i=1

(viw′i − v′iwi)αi = 0 for all (v,w), (v′,w′) ∈ C




puncture code of an Fq-linear code C over Fq × Fq:

P (C) :=〈{c, c′} : c, c′ ∈ C

〉⊥⊆ Fnq

with the vector valued bilinear form

{(v,w), (v′,w)} := (viw′i − v′iwi)ni=1 ∈ Fnq

α = (α1, α2, . . . , αn) ∈ P (C)

⇐⇒n∑

i=1

(viw′i − v′iwi)αi = 0 for all (v,w), (v′,w′) ∈ C

=⇒ symplectic code C̃ := {(v, (αiwi)ni=1) : (v,w) ∈ C}




α ∈ P (C) with wgtα = r:

• delete the positions with αi = 0

• C̃p is still a symplectic code

=⇒ code C̃ of length ñ = r with C̃ ⊆ C̃⋆

Theorem: (Rains)

Let C be a code over Fnq × Fnq with C⋆ = (n, qn+k, d).If α ∈ P (C) with wgt(α) = r, then there is a stabilizer codeC = [[r, k̃ ≥ r − (n− k), d̃ ≥ d]]q.In particular:

C = [[n, k, d]]q α→ C̃ = [[r, k̃ ≥ r − (n− k), d̃ ≥ d]]q



The Easy Case: CSS-like Construction

[Rötteler, Grassl, and Beth, ISIT 2004]

• start with a cyclic (constacyclic) MDS code C1 over Fq of length q + 1

• in general, C⊥1 6⊂ C1• compute P (C) for C = C⊥1 × C⊥1 :

P (C) =〈(cidi)

ni=1 : c,d ∈ C⊥1

〉⊥

• αi, αj roots of the generator polynomial of C1=⇒ αi+j is a root of the generator polynomial of P (C)

• P (C) is also a cyclic (constacyclic) MDS code which contains words of“all” weights

Quantum MDS codes C = [[n, n− 2d+ 2, d]]q exist for all 3 ≤ n ≤ q + 1 and1 ≤ d ≤ n/2 + 1.



The Harder Case: Hermitian-like Construction

• start with a cyclic (constacyclic) MDS code C over Fq2 of length q2 + 1

• in general, C is not a Hermitian self-orthogonal code

• P (C) =〈(cid

qi )ni=1 : c,d ∈ C

〉⊥∩ Fnq

=〈(cid

qi + c

qi di)

ni=1 : c,d ∈ C

〉⊥

• C is the dual of a code whose generator polynomial has roots αi, αj=⇒ αi+qj is a root of the generator polynomial of P (C)

• P (C) is also a cyclic (constacyclic) code, but in general no MDS code

• known so far [Beth, Grassl, Rötteler], [Klappenecker et al.]– QMDS codes exist for some n > q + 1 and d ≤ q + 1, including q2 − 1,q2, q2 + 1

– some other QMDS codes, e. g., derived from Reed-Muller codes



QMDS Codes: Computational Results q = 8

d6

n-

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

— quantum Singleton bound, — conjectured bound, 2 direct construction, + via P (C)



Special Cases

[Grassl & Rötteler, Quantum MDS Codes over Small Fields, ISIT 2015, pp. 1104–1108]

Theorem:

Our construction yields QMDS codes C = [[q2 + 1, q2 + 3− 2d, d]]qfor all 1 ≤ d ≤ q + 1 when q is odd, or when q is even and d is odd.Remark:

Our construction does not yield a QMDS code [[17, 11, 4]]4, but QMDS codes

[[4m + 1, 4m + 3− 2m+1, 2m]]2m for (at least) m = 3, 4, 5, 6, 7.

Theorem:

For q = 2m, there exist QMDS codes C = [[4m + 2, 4m − 4, 4]]2m .Proof: (main idea, see [Grassl & Rötteler arXiv:1502.05267 [quant-ph]])

Use the triple-extended Reed-Solomon code and show that P (C) contains a

word of weight q2 + 2.



QMDS of Distance Three

q2-ary simplex code C = [q2 + 1, 2, q2]q2 generated by

G =

1 0 1 1 1 . . . 1

0 1 1 ω ω2 . . . ωq2−2

=

g0

g1

,

where ω is a primitive element of GF (q2)

Considered as Fq-linear code generated by

{g0, g′0 = αg0, g1, g′1 = αg1}

where α ∈ GF (q2) \GF (q)



The Dual of the Puncture Code

P (C)⊥ =〈gq+10 , g0 ◦ g1q + gq0 ◦ g1, g0 ◦ αqgq1 + gq0 ◦ αg1, gq+11

〉

=〈f0, f1, f2, f3

〉,

using v ◦w = (v1w1, v2w2, . . . , vnwn) and vm = (vm1 , vm2 , . . . , vmn )We have

f0 = zq+1

f1 = xqz + xzq = homogenz(x+ x

q) = homogenz(tr(x))

f2 = αqxqz + αxzq = homogenz(αx+ α

qxq) = homogenz(tr(αx))

f3 = xq+1

Choosing α ∈ Fq2 \ Fq with −α2 = β1α+ 1 for some β1 ∈ Fq yields

f21 + β1f1f2 + f22 = (4− β21)f0f3.



Ovoid Code

Lemma: The dual of the puncture code is an ovoid code, i. e.

P (C)⊥ = [q2 + 1, 4, q2 − q]q.

(see, e. g., Example TF3 in [Calderbank & Kantor, 1986])

This code is a two-weight code with weights q2 − q and q2.

Ai =

1 for i = 0,

(q3 + q)(q − 1) for i = q2 − q,(q2 + 1)(q − 1) for i = q2,0 else.



The Dual of the Ovoid Code

Problem: We need the non-zero weights of the puncture code P (C).

The homogenized weight enumerator of P (C)⊥ is

WP (C)⊥ = Xq2+1 + (q3 + q)(q − 1)Xq+1Y q2−q + (q2 + 1)(q − 1)XY q2

MacWilliams transformation yields

WP (C)(X, Y ) = q−4WP (C)⊥(X + (q − 1)Y,X − Y )

=

q2+1∑

i=0

BiXq2+1−iY i

Conjecture: For q > 2, the code P (C) contains words of all weights

w = 4, . . . , q2 + 1, i. e., Bi > 0.

Confirmed for the first 50 prime powers as well as for small weights.



Geometric Proof

THANKS to Aart Blokhuis at ALCOMA 2010

Main idea: Show that we can find a linear combination of exactly w points of

the ovoid that is zero, corresponding to a word of weight w in the dual code.

• Choose 5 points Q0, . . . , Q4 of the ovoid O in a plane P (for q ≥ 4).

• Choose 2 points P1 and P2 of the ovoid outside of the plane.

• Any point in the plane can be expressed as linear combination of exactly 4points Qi.

• Choose m = w − 4 other points and consider their sum S.– If S ∈ P , use exactly 4 other points Qi to get zero.– Otherwise, consider the intersection of P with the line through S andP1 (or P2 if S = P1). Use exactly 3 other points Qi to get zero.


quantum error-correcting codes: discrete math meets physics · 2018. 7. 24. · we denote the...

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