quantum algebraic-geometric codes
TRANSCRIPT
Quantum Algebraic-Geometric Codes
Chiu Fan LeeClare HallCambridge
Part III essay in theDepartment of Applied Mathematics and Theoretical Physics
University of Cambridge
May 2001
Abstract
This is a Part III essay aiming to discuss the contruction of quantum error-correctingcodes through the use of the theory of algebraic function fields, which produces codeswith asymptotically good parameters. This exposition emphasises constructibility andseveral applications of the main theorem (theorem 6.2) are given. Prerequisites are thefirst 12 lectures of the Part III course Quantum Information Theory [16] or chapter 7of Preskill’s Lecture Notes [15], and the first two chapters of Stichtenoth’s AlgebraicFunction Fields and Codes [19].
Contents
1 Introduction 2
2 Review on coding theory 32.1 Review on classical coding theory . . . . . . . . . . . . . . . . . . . . . . 32.2 Review on quantum coding theory . . . . . . . . . . . . . . . . . . . . . . 42.3 From non-binary to binary codes . . . . . . . . . . . . . . . . . . . . . . 5
3 Stabilizer formalism and encoding procedure 73.1 Stabilizer formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Converting stabilizer matrix into standard form . . . . . . . . . . . . . . 83.3 Quantum circuit for encoding . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Calderbank-Shor-Steane construction 124.1 CSS construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Enlargement of CSS quantum codes . . . . . . . . . . . . . . . . . . . . . 12
5 Algebraic function fields and codes 145.1 Review on algebraic function fields . . . . . . . . . . . . . . . . . . . . . 145.2 Algebraic geometric codes . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 Quantum algebraic geometric codes 18
7 Examples 247.1 Rational function field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2 Elliptic function field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257.3 Hermitian function field . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8 Conclusion 29
1
Chapter 1
Introduction
As of today, scientists widely speculate that quantum computers could offer more com-putational power than conventional computers are able to. But due to the delicatenature of quantum quantities, the introduction of error during computation seems in-evitable. Therefore, protections of information either by encoding or other methodswere introduced. This essay aims to discuss the application of algebraic geometry onquantum error-correcting codes. The motivation behind is the potential error-correctingpower of this method of construction. For instance, quantum algebraic-geometric codesare the only known quantum error-correcting codes that are asymptotically good, a no-tion which will be explained later.
The essay is organised in eight chapters. The next chapter offers a brief reviewon general coding theory in both classical and quantum cases. Chapter three investi-gates the stabilizer formalism, which corresponds to a very large subset of all knownquantum error-correcting codes. Chapter four discusses the Calderbank-Shor-Steaneconstruction of quantum codes, which makes use of the self-duality property of classicalcodes to produce quantum codes. Chapter five serves as a review and a quick referenceguide on the theory of algebraic-geometric codes. Chapter six describes how algebraicgeometry is used to construct asymptotically good quantum codes. The machinery de-veloped is then applied to some specific algebraic function fields in chapter seven.
To make the exposition more transparent, all algorithms, definitions, examples andremarks will end with the symbol ‘4’, while all proofs will end with the symbol ‘♦’throughout the essay.
2
Chapter 2
Review on coding theory
2.1 Review on classical coding theory
In this section, we will review some important notions in classical coding theory.
Let GFq denote the finite field with q elements. We consider the n-dimensionalvector space GF n
q .
Definition 2.1 For a = (a1, . . . , an) and b = (b1, . . . , bn) ∈ GF nq , let
d(a, b) := #i | ai 6= bi.
The weight of an element in GF nq is defined as
wt(a) := d(a, 0) = #i | ai 6= 0.
4
Definition 2.2 A code C is a linear subspace of GF nq ; the elements of C are called
codewords. We call n the length of C and denote dimC by k. An [n, k] code is a code oflength n and dimension k. The minimum distance d(C) is defined to be
d(C) := mind(a, b) | a, b ∈ C and a 6= b = minwt(c) | 0 6= c ∈ C.
We refer to such a code C as an [n, k, d] code. 4
Definition 2.3 Let C be an [n, k] code over GF nq . A generator matrix G of C is a k×n
matrix whose rows are a basis of C. 4
Definition 2.4 If C ⊆ GF nq is a code, then
C⊥ := u ∈ GF nq |
n∑i=1
uici = 0, ∀c ∈ C
is called the dual of C. 4
3
Definition 2.5 A generator matrix H of C⊥ is said to be a parity-check matrix for C.4
Note that Hut = 0 for all u ∈ C where ut is the transpose of u.
2.2 Review on quantum coding theory
So much for classical coding theory, we will now introduce some important elements inthe theory of quantum error-correcting code. See [15] for the physical significance of thestatements in this section.
Let C2 be a 2-dimensional complex Hilbert space. An element of C2 is called aqubit. The space (C2)⊗n is the space of quantum words of length n.
Definition 2.6 A quantum code Q is a K-dimensional linear subspace of (C2)⊗n. Werefer to Q as an ((n,K)) quantum code. Note the double brackets used to distinguishquantum codes from classical codes. 4
Definition 2.7 An error is a linear operator E acting on (C2)⊗n. 4
Definition 2.8 We define the support SuppE in the following way. If E can be writtenas Ii ⊗ E ′, where Ii is the identity operator acting on the i-th tensor component andE ′ acts on other tensor components only, then we say i 6∈ SuppE. The weight of E isdefined to be
wt(E) = #SuppE.
4
Definition 2.9 A quantum code Q is called d-error correcting if for all errors E,F withwt(E), wt(F ) ≤ d and for all pair of vectors u, v ∈ Q,
〈F (v) | E(u)〉 = cE,F 〈v | u〉,
where cE,F is a complex number depending only on E and F and 〈 | 〉 is the standardHermitian inner product. Furthermore, if cE,F is 0 unless E = F , then we say thequantum code is nondegenerate.
Similar to the classical case, we will refer to Q as an ((n,K, d)) quantum code. 4
For long quantum codes to be useful, we need to know their asymptotic behaviour.Letting kQ = log2 KQ, we set
RQ =kQn, (2.1)
δQ =dQn, (2.2)
R(δ) = lim supn→∞
RQ, (2.3)
4
where the limit is taken over all quantum codes with δQ ≥ δ. The best known noncon-structive lower bound (see lecture 9 of [16]) on R(δ) is
R(δ) ≥ 1− δ log2 3−H(δ), (2.4)
where H(x) = −x log2 x− (1− x) log2(1− x) is the binary entropy function.
2.3 From non-binary to binary codes
In order to utilize the potential of algebraic construction, one needs to consider algebraicfunctions over finite fields other then GF2. This section is thus aimed to estimate theparameters of the code obtained by binary expanding a code over GF2m .
Definition 2.10 The trace of a ∈ GF2m is defined by
Tr(a) =m−1∑i=0
a2i .
4
Theorem 2.11 For all natural number m, there exists a self-dual basis u1, . . . , um ofGF2m over GF2, i.e.,
Tr(uiuj) = δij.
Proof. See [12]. ♦
The following theorem is due to T. Kasami and S. Lin [11].
Theorem 2.12 Let C be a code over GF2m, C ⊃ C⊥ and ui, i = 1, . . . ,m be a self-dual basis of GF2m over GF2. Let D and E be codes obtained by the symbolwise binaryexpansion of codes C and C⊥ in the basis ui. Then (1) D ⊃ E and (2) E = D⊥.
Proof. (1) The statement is obvious since D,E are obtained from C,C⊥ by binaryexpansion.
(2) Let a = (a1, . . . , an) ∈ C and b = (b1, . . . , bn) ∈ C⊥. Let
aj =m∑i=1
a(j)i ui and bj =
m∑i=1
b(j)i ui.
Then
0 =n∑j=1
ajbj = Tr(n∑j=1
ajbj) (2.5)
5
= Tr(n∑j=1
m∑i,k=1
a(j)i b
(j)k uiuk) (2.6)
=n∑j=1
m∑i,k=1
a(j)i b
(j)k Tr(uiuk) (2.7)
=n∑j=1
m∑i=1
a(j)i b
(j)i (2.8)
So D⊥ ⊇ E. Now as the dimensions of D and E are complementary, we have E = D⊥.♦
Remark 2.13 It is obvious that if we start with a triple C ′ ⊃ C ⊇ C⊥ of codes overGF2m , the binary expansion again gives us a triple D′ ⊃ D ⊇ D⊥ of binary codes. 4
6
Chapter 3
Stabilizer formalism and encodingprocedure
3.1 Stabilizer formalism
Definition 3.1 For (a | b), (u | v) ∈ 0, 12n we define a symplectic scalar product on0, 12n by
(a | b) (u | v) = a · v ⊕ u · b,
where ⊕ denotes mod-2 addition. 4
Definition 3.2 A stabilizer matrix
H = (HX | HZ)
is a (n− k)× 2n matrix of rank (n− k) such that for any rows (a | b), (u | v) in H,
(a | b) (u | v) = 0
and
a · b :=n∑i=1
aibi = 0.
4
Definition 3.3 Let S be the vector space generated by the rows of H, we define theweight of an element (a | b) ∈ S as
wt(a | b) =∑i
(ai ∨ bi),
where ai ∨ bi = maxai, bi. 4
7
Theorem 3.4 For any stabilizer matrix H, and S the vector space generated by the rowsof H. If d is an integer such that for all (a | b) ∈ 0, 12n with weight =
∑i(ai ∨ bi) < d,
either (a | b) lies in S or lies outside of the -orthogonal space S⊥ . Then there isa quantum code Q associated to H with length n and dimension 2k which is d-errorcorrecting.
Furthermore, if for all vectors (a | b) with wt(a | b) < d, they all lie outside of S⊥ ,then the quantum code is nondegenerate.
Proof. See lecture 11 of [16] or chapter 7 of [15]. ♦
Definition 3.5 Given a stabilitzer matrix H = (HX | HZ) with (n− k) rows, we referto the corresponding quantum code Q as an [[n, k, d]] if Q is d-error correcting. 4
3.2 Converting stabilizer matrix into standard form
In this following two sections, we will investigate the procedure for encoding. For moredetailed treatment, we refer the readers to [4, 7, 14].
Given a stabilizer matrix H, by Gaussian elimination (which correspond to mul-tiplying two generators and switching qubits enumerations and hence do not change thecorresponding quantum code), we can transform H into the following form:
rn− k − r
( r︷︸︸︷I0
n−r︷︸︸︷A0
∣∣∣r︷︸︸︷BD
n−r︷︸︸︷CE
)where I denotes the r × r identity matrix and r is the rank of HX .
Then, by doing Gaussian elimination on E, we can have H transformed to:
rn− k − r − s
s
( r︷︸︸︷I00
n−k−r−s︷ ︸︸ ︷A1
00
k+s︷ ︸︸ ︷A2
00
∣∣∣∣∣r︷ ︸︸ ︷BD1
D2
n−k−r−s︷ ︸︸ ︷C1
I0
k+s︷ ︸︸ ︷C2
E2
0
).
In order for the first r rows of HX to be -orthogonal to the last s rows of HZ , wemust have D2 = 0, which implies s = 0. Also, by using the last n − k − r rows, C1
can be eliminated all together. Therefore, we can always put H into the following formwhich we call the standard form:
rn− k − r
( r︷︸︸︷I0
n−k−r︷ ︸︸ ︷A1
0
k︷ ︸︸ ︷A2
0
∣∣∣r︷︸︸︷BD
n−k−r︷︸︸︷0I
k︷︸︸︷CE
). (3.1)
We now define two matrices, X and Z, which will enable us to find a basis for Qwith easy encoding procedure.
8
Definition 3.6 Given a stabilizer matrix H in standard form as above, the XH-matrixand ZH-matrix are defined to be
XH = (
r︷︸︸︷0
n−k−r︷︸︸︷Et
k︷︸︸︷I |
r︷︸︸︷Ct
n−k−r︷︸︸︷0
k︷︸︸︷0 )
ZH = (
r︷︸︸︷0
n−k−r︷︸︸︷0
k︷︸︸︷0 |
r︷︸︸︷At2
n−k−r︷︸︸︷0
k︷︸︸︷I ).
We denote the i-th row of XH and ZH by XHi and ZHi respectively. 4
Proposition 3.7 Given a stabilizer matrix H.(a) The rows in ZH are linearly independent and -orthogonal with the rows in H.(b) The rows of XH are linearly independent and -orthogonal with the rows in H.Moreover, XHj is -orthogonal with all the ZHi except ZHj .
Proof. Straight forward verification. ♦
Because of the previous result, we can define a basis for the quantum code Q asso-ciated to H to be the states with stabilizer
〈H1, . . . ,Hn−k, (−1)x1ZH1 , . . . , (−1)xkZHk 〉
where (x1, . . . , xk) ∈ 0, 1k.
3.3 Quantum circuit for encoding
In order to encode any quantum message, we first need the following proposition.
Proposition 3.8 Given a stabilizer matrix H = (HX | HZ) in standard form (3.1).For any i-th row of H, there is a vector U i = (uix | uiz) such that U i is -orthogonal toall ZH and is -orthogonal to all the rows in H except the i-th row.
Proof. We take
U =( 0 0 0
0 I 0
∣∣∣ I 0 00 0 0
)and let U i be the i-th row of U . The claim will then be easily verified. ♦
To encode the |0〉⊗k state, we start with the state |0〉⊗n and measure each of theobservables H1, . . . ,Hn−k, Z
H1 , . . . , Z
Hk in turn. We then will have a state stabilized by
〈±H1, . . . ,±Hn−k,±ZH1 , . . . ,±ZHk 〉 where the plus or minus signs depend on the mea-surement outcomes. Then by applying the set of operators XH1 , . . . , XHk and thosefound by proposition 3.8, we can change the state to an arbitrary encodes basis state|x1, . . . , xk〉encoded.
9
Figure 3.1: Quantum circuit for measuring a single qubit operator M with eigenvalues ±1.
We now consider the network that performs the measurement. We first note that todo a measurement on a operator M with eigenvalues ±1, the quantum circuit in figure3.1 may be used:
Then, the following example should suffice to demonstrate how encoding procedureis done in general.
Example 3.9 We consider the five-qubit code which has stabilizer matrix
H =
1 0 0 0 10 1 0 0 10 0 1 0 10 0 0 1 1
∣∣∣∣∣∣∣∣∣1 1 0 1 10 0 1 1 01 1 0 0 01 0 1 1 1
.
Hence, we have
X = (00001 | 10010) , Z = (00000 | 11111).
Therefore, to measure the observablesH1, . . . ,Hn−k, ZH1 , . . . , Z
Hk , the circuit in figure
3.2 may be used.After the measurement, we may then apply X and those operators found in propo-
sition 3.8 to convert the quantum state into any encoded state as desired.4
10
Figure 3.2: Quantum circuit for measuring the generators and Z of the 5-qubit code.
11
Chapter 4
Calderbank-Shor-Steaneconstruction
4.1 CSS construction
Definition 4.1 Let C be a classical [n, k, d] code containing its dual C⊥ and let H bethe parity-check matrix of C. Then the CSS code corresponding to C is the quantumcode associated to the stabilizer matrix
H =
(H 00 H
).
4
By construction, it is trivial that H is indeed a stabilizer matrix.
Theorem 4.2 Let C be a classical [n, k, d] code containing its dual C⊥. Then the cor-responding CSS code is a [[n, 2k − n, d]] quantum code.
Proof. The theorem follows easily from theorem 3.4 and the construction of H. ♦
4.2 Enlargement of CSS quantum codes
Definition 4.3 The i-th generalized distance di of a linear code C is the minimum sizeof the support of a i-th dimensional subcode of C. 4
Note that d1 is the minimum distance of the code and d2 is the minimum weight ofthe bitwise OR of two different nonzero codewords, i.e., d2 = min∑i(ai∨bi) | a, b ∈ C.We also have the following inequality for d1 and d2.
Proposition 4.4 For any linear code C with parameters [n, k, d],⌈3d1
2
⌉≤ d2.
12
Proof. Take any two distinct element u, v ∈ C such that the corresponding d2 isachieved. Assume wt(u) = d1 = d and the general case can then be proved by induction.By rearranging terms, we can assume the first d entries of u are 1’s. If d2 < d3d
2e, either
wt(v) < d or wt(u+ v) < d, a contradiction as v, u+ v ∈ C. ♦
To construct better quantum codes, we need the enlarged CSS construction whichwas first invented by Steane [17]. He proved that the quantum code constructed hasparameter [[n, k+k′−n,mind, d3d′
2e]]. Cohen, Encheva and Litsyn later improved the
minimal distance to mind, d′2 [5]. We will give an alternative proof to the theorembased on the symplectic geometry formalism introduced earlier.
Theorem 4.5 Given a classical binary error correcting code C = [n, k, d] which con-tains its dual, C⊥ ⊆ C, and which can be enlarged to C ′ = [n, k′ > k + 1, d′], a nonde-generate quantum code of parameters [[n, k + k′ − n,mind, d′2]] can be constructed.
Proof. Let H ′ be the parity-check matrix of C ′ and H ′ and B together check C.Consider
H =
AB BH ′ 00 H ′
,where A is a (k′ − k) × (k′ − k) matrix such that AB, (A + I)B and B generate thesame set. One possible choice is
A =
0100 . . . 00010 . . . 00001 . . . 0
. . .0000 . . . 11100 . . . 0
.
By construction, u, v rows in H ′ implies
u, v ∈ C ′⊥ ⇒ u ∈ C⊥ ⊂ C ′ ⇒ u · v = 0.
Similar calculations can show H is indeed a stabilizer matrix.We now let S be the vector space generated by the rows of H. If (α | β) ∈ 0, 12N
is such that(α | β) (u | v) = α · v ⊕ u · β = 0,∀(u | v) ∈ S.
Then we consider two cases: (1) either α or β is 0; (2) they are both nonzero.
(1) W.l.o.g., take β to be zero, then α ∈ C, so wt(α) ≥ d.
(2) If they are both nonzero, we have α, β ∈ C ′. If α = β, then by the requirement ofA, α = β is in C and so wt(α | β) ≥ d. If α 6= β, wt(α | β) ≥ d′2 by definition.
Now, as all the vectors with weights under mind, d′2 are outside of S⊥ , the corre-sponding quantum code is nondegenerate. ♦
13
Chapter 5
Algebraic function fields and codes
This chapters serves as a reference on definitions and theorems needed for later devel-opment. All proofs of statements can be found in [19].
5.1 Review on algebraic function fields
Definition 5.1 An algebraic function field F/K of one variable over K is an extensionfield F ⊇ K such that F is a finite algebraic extension of K(x) for some element x ∈ Fwhich is transcendental over K. 4
Definition 5.2 A valuation ring of the function field F/K is a ring O ⊇ F such that:(1) K ⊂ O ⊂ F , and (2) for any z ∈ F , z ∈ O or z−1 ∈ O. 4
Proposition 5.3 Let O be a valuation ring of the function field F/K. Then (a) O isa local ring, i.e., O has a unique maximal ideal P .(b) P is a principal ideal.(c) If P = tO then any 0 6= z ∈ F has a unique representation of the form z = tnu forsome n ∈ Z, u ∈ O∗.
Proof. See proposition I.1.5 and theorem I.1.6 in [19]. ♦
Definition 5.4 (a) A place P of the function field F/K is the maximal ideal of somevaluation ring O of F/K. Any element t ∈ P such that P = tO is called a prime elementof P .
(b) PF := P | P is a place of F/K.(c) To any place P ∈ PF we associate a function vP : F → Z ∪ ∞ by defining
vP (0) :=∞ and vP (z) := n when z = tnu with t a prime element of P . 4
Definition 5.5 Let P be a place of F/K and OP be its valuation ring. For x ∈ OP
we define x(P ) ∈ OP to be the residue class of x modulo P , for x ∈ F\OP we definex(P ) := ∞. We note that this symbol ∞ is different from that used in the previousdefinition but confusion should not arise. 4
14
Proposition 5.6 In a function field F/K, for any element 0 6= x ∈ F , the set P ∈PF | vP (x) 6= 0 is finite.
Proof. See corollary I.3.4 in [19]. ♦
Definition 5.7 The free abelian group which is generated by the places of F/K isdenoted by DF , the divisor group of F/K. The element of DF are called divisors ofF/K. In other words, a divisor is a formal sum
D =∑P∈PF
nPP with nP ∈ Z, almost all nP = 0.
4
Definition 5.8 (x) =∑P∈PF vP (x)P. 4
Remark 5.9 (x) is in DF by proposition 5.6. 4
Definition 5.10 For a divisor A ∈ DF we set
L(A) := x ∈ F | (x) ≥ −A ∪ 0.
The dimension of the vector space L(A) over K is denoted by dim(A). 4
Definition 5.11 An adele of F/K is a mapping
α :
PF → FP 7→ αP ,
such that α ∈ OP for almost all P ∈ PF .
AF := α | α is an adele of F/K
is called the adele space of F/K. It is also a vector space over K. 4
Definition 5.12 A Weil differential of F/K is a K-linear map ω : AF → K vanishingon AF (A) + F for some divisor A ∈ DF . We call
Ω := ω | ω is a Weil differential of F/K
the module of Weil differentials of F/K. For A ∈ DF let
Ω(A) := ω | ω vanishes on AF (A) + F.
Also, for ω 6= 0, let
M(ω) := A ∈ DF | ω vanishes on AF (A) + F.
4
15
Theorem 5.13 Let 0 6= ω ∈ Ω. Then there is a uniquely determined divisor W ∈M(ω)such that A ≤ W for any A ∈M(ω).
Proof. Theorem I.5.10 in [19]. ♦
Definition 5.14 The divisor (ω) of a Weil differential ω 6= 0 is the uniquely determineddivisor of F/K satisfying(1) ω vanishes on AF ((ω)) + F .(2) If ω vanishes on AF (A) + F , then A ≤ (ω).
A divisor W is called a canonical divisor of F/K if W = (ω) for some ω ∈ Ω. 4
Theorem 5.15 Let A ∈ DF and W = (ω) a canonical divisor of F/K. Then L(W−A)and Ω(A) are isomorphic as K-vector space. In particular,
dim(W − A) = dimA− degA+ g − 1.
Proof. See theorem I.5.14 in [19]. ♦
Definition 5.16 The genus g of F/K is defined by
g := maxdegA− dimA+ 1 | A ∈ DF.
4
Theorem 5.17 The genus g of F/K is a non-negative integer.
Proof. See remark I.4.16 in [19]. ♦
Theorem 5.18 (Riemann-Roch Theorem) Let W be a canonical divisor of F/K. Then,for any A ∈ DF ,
dimA = degA+ 1− g + dim(W − A).
Proof. See theorem I.5.15 in [19]. ♦
Corollary 5.19 For a canonical divisor W , we have
degW = 2g − 2 and dimW = g.
Proof. To show dimW = g, take A = 0. To show degW = 2g− 2, take A = W . ♦
Theorem 5.20 If A is a divisor of F/K of degree ≥ 2g − 1, then
dimA = degA+ 1− g.
Proof. See theorem I.5.17 in [19]. ♦
16
Definition 5.21 Let P ∈ PF .(a) For x ∈ F , let ιP (x) ∈ AF be the adele whose P -component is x, and all other
components are 0.(b) For a Weil differential ω ∈ Ω, define its local component ωP : F → K by
ωP (x) := ω(ιP (x)).
4Proposition 5.22 Let P ∈ PF be a place of degree one and ω ∈ Ω. Let (ω) =∑ni=1 aPiPi be such that aPi ≥ −1 for Pi = P .
(a) ωP (1) = 0⇔ aP ≥ 0.(b) For x ∈ F with vP (x) ≥ 0,
ωP (x) = x(P )ωP (1).
Proof. See proofs of theorems II.2.7 and II.2.8 in [19]. ♦Theorem 5.23 Let ω ∈ Ω and α = (αP ) ∈ AF . Then ωP (αP ) 6= 0 for at most finitelymany places P , and
ω(α) =∑P∈PF
ωP (αP ).
Proof. See proposition I.7.2 in [19]. ♦
5.2 Algebraic geometric codes
For this section, we will always have the following notations:
F/GFq is an algebraic function fields of genus g;P1, . . . , Pn are pairwise distinct places of F/GFq of degree 1;D = P1 + . . .+ Pn;G is a divisor of F/GFq such that suppG ∩ suppD = ∅.
Definition 5.24 The algebraic geometric codes CL(D,G) and CΩ(D,G) associated withthe divisors D and G are defined by
CL(D,G) := (x(P1), . . . , x(Pn)) | x ∈ L(G) ⊆ GF nq (5.1)
CΩ(D,G) := (ωP1(1), . . . , ωPn(1)) | ω ∈ ΩF (G−D) ⊆ GF nq . (5.2)
4Theorem 5.25 CΩ(D,G) = CL(D,G)⊥.
Proof. See theorem II.2.8 in [19]. ♦Theorem 5.26 Let 2g − 2 < degG < n. Then CΩ(D,G) is an [n, k, d] code withparameters
k = n+ g − 1− degG , d ≥ degG− (2g − 2).
Proof. See theorem II.2.7 in [19]. ♦
17
Chapter 6
Quantum algebraic geometric codes
The following presentation is based on the paper Asymptotically Good Quantum Codesby A. Ashiknmin, S. Litsyn and M. A. Tsfasman [2].
Definition 6.1 Let θ ∈ (GF ∗q )n. For a code C ⊆ GF nq , we define
C⊥θ = x ∈ GF (q)n |n∑i=1
θixiyi = 0,∀y ∈ C.
4
Theorem 6.2 If F is an algebraic function field over GFq of genus g with at least n′ ≥3g places of degree 1. Then for any n ≤ n′−g, and for any a = 2g−1, . . . ,minn, n′/2+g/2− 1, there exists an [n, k = n+ g− 1− a, d ≥ a− 2g+ 2]-code C such that C ⊇ C⊥θfor some θ ∈ (GF ∗q )n.
Moreover, if q = 2m,m ∈ N, there exists such a code with C ⊇ C⊥.
Proof. Let G be a positive divisor of degree a, i.e., G ≥ (0), and P ′ = P1, . . . , Pn′is the set of places in F of degree 1 such that SuppG ∩ P ′ = ∅. Set
D′ = P1 + . . . ,+Pn′ ∈ DF .
Now let E be an effective divisor of degree n′ + g − 2 − 2a, which implies thata ≤ n′/2 + g/2 − 1. Define A to be D′ − 2G − E, then for a canonical divisor W , wehave deg(W + A) = g be corollary 5.19. By theorem 5.15,
dimΩ(−A) = dim(W + A).
By Riemann-Roch Theorem (theorem 5.18),
dim(W + A) = g + 1− g + dim(W − A) ≥ 1.
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Therefore, ∃ nonzero ω0 ∈ Ω(−A).
Consider the map
Ω(−A) → CΩ(D′, D′ − A) (6.1)
ω 7→ (ωP1(1), . . . , ωPn′ (1)). (6.2)
The kernel is Ω(2G+ E). Let ω0Pij(1) = 0 for Pi1 , . . . , Pim ≡ P0. Then
ω0 ∈ Ω(2G+ E − (D′ −m∑j=1
Pij)) = Ω(−A+m∑j=1
Pij).
As ω0 6= 0, by theorem 5.20, we must have
deg(−A+m∑j=1
Pij) ≤ 2g − 2,
i.e.,−(2− g) +m ≤ 2g − 2
which implies m ≤ g.
Let P = P1, . . . , Pn ≡ P ′\P0. Then n ≥ n′ − g. Put D = P1+, . . . ,+Pn ∈ DF , wehave ω0 ∈ Ω(2G+ E −D). Define θ = (ω0P1(1), . . . , ω0Pn(1)).
We consider
C = CL(D,G)⊥θ = x ∈ GF nq |
∑i
θixiyi = 0,∀y ∈ CL(D,G).
For x, y ∈ L(G), we have
0 = ω0(xy) (6.3)
=∑P∈PF
ω0P (xy) (6.4)
=n∑i=1
ω0Pi(xy) (6.5)
=n∑i=1
x(Pi)y(Pi)ω0Pi(1) (6.6)
=n∑i=1
x(Pi)y(Pi)θi, (6.7)
where (6.3) and (6.5) follow from the fact that ωo ∈ Ω(2G + E −D) and x, y ∈ L(G);(6.4) follows from theorem 5.23; (6.6) follows from proposition 5.22.
Therefore, C ⊇ CL(D,G). Also, x ∈ C⊥θ ⇒ x ∈ CL(D,G)⊥ ⇒ x ∈ C. Hence,C ⊇ C⊥θ .
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If q = 2m, any element in GFq is a square. Let θi = η2i . Consider the multiplication
map
mη : GF nq → GF n
q (6.8)
x 7→ (η1x1, . . . , ηnxn). (6.9)
Define C ′ = mη(C) which will then contain its dual, i.e., C ′ ⊇ C ′⊥.Now for n ≥ a ≥ 2g − 1, by theorem 5.25 and 5.26, CL(D,G)⊥ = CΩ(D,G) is a
[n, k = n + g − 1 − a, d ≥ a − 2g + 2] linear code. The multiplication map mθ−1 fromCΩ(D,G) to C is a isomorphism of vector spaces. Hence, C is a linear code with thesame parameters.
Recalling the restrictions on a, i.e., 2g − 1 ≤ a ≤ n′/2 + g/2 − 1, we have n′ ≥ 3g.The theorem is then just a restatement of the previous discussion. ♦
Now, let us define R = k/n and δ = d/n and do some analysis on them. First of all,by the previous theorem, we have for n ≤ n′−g and 2g−1 ≤ a ≤ minn, n′/2+g/2−1,
R = 1− a+ 1 + g
n, (6.10)
δ =a− 2g + 2
n. (6.11)
From the equations, we see that R rises with the incease of n and with the decrease ofa, while δ rises with the increase of a and with the decrease of n. Also, it is easily seenthat R+ δ = 1− g/n. Hence, for fixed δ, we should choose the maximum n available tomaximize R. Therefore, we will always choose n = n′ − g from now on.
Applying theorem 6.2 to a families of algebraic function fields F ’s over GFq, q asquare, such that
lim supg→∞
N(F )
g(F )=√q − 1,
where N(F ) is the number of places of degree one in F and g(F ) is the genus of F (forexistence of such families, see [21, 6]), we get the following corollary:
Corollary 6.3 Let q be an even power of a prime. Then for any
α ∈(
2√q − 2
,1
2+
1√q − 2
)there exists families of codes with asymptotic parameters
R = 1− α +1
√q − 2
, (6.12)
δ ≥ α− 2√q − 2
, (6.13)
with the property C ⊇ C⊥θ for some θ ∈ (GF ∗q )n.It q is an even power of 2, there exists such codes with the property C ⊇ C⊥.
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Proof. We have taken n = n′ − g, so
R =n− a+ g − 1
n(6.14)
= 1− α +g
n′ − g(6.15)
= 1− α +1
√q − 2
, (6.16)
where α will lie in the prescribed range. The rest of the proof can be done by similarcalculations. ♦
To use the generalized CSS construction, we need a triple C ′ ⊃ C ⊇ C⊥. Therefore,we take two divisors G′ < G, then CL(D,G′) ⊂ CL(D,G) and we have the oppositeinclusion for duals. Taking SuppG′ ∪ P = ∅, ω0 can then remain the same. Therefore,we have proved the following:
Corollary 6.4 Let q = 22m. Then for any pair of real numbers (α, α′) such that
2
2m − 2≤ α′ < α ≤ 1
2+
1
2m − 2
there exists families of triples of 22m-ary codes C ′ ⊃ C ⊇ C⊥ with asymptotic parameters
R′ = 1− α′ + 1
2m − 2, δ′ ≥ α′ − 2
2m − 2
R = 1− α +1
2m − 2, δ ≥ α− 2
2m − 2.
♦
By using all the machineries developed, we can now construct an asymptoticallygood quantum code Q and we will state how it can be done in a step by step manner.
Algorithm 6.5 1. Start with a family of algebraic function fields over GF22m withthe property that
limg→∞
N(F )
g(F )= 2m − 1.
Each algebraic function field will then give us a triple of 22m-ary codes C ′ ⊃ C ⊇C⊥.
2. Let C ′ and C be an [n′, k′, d′] code and an [n, k, d] codes respectively. Binaryexpand C and C⊥ with respect to a self-orthogonal basis to get a triple of binarycodes D′ ⊃ D ⊇ D⊥ with parameters
nD′ = nD = 2mn,
kD′ = 2mk′ , kD = 2mk,
dD′ ≥ d′ , dD = d.
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3. By the generalized CSS construction, each triple give us a quantum stabilizer code[[2mn, 2m(k + k′ − n),≥ mind, d′2]]. The corresponding asymptotic parametersare
RQ = R +R′ − 1 (6.17)
δQ ≥ 1
2mminδ, 3δ′
2 (6.18)
where R,R′, δ, δ′ are the parameters of algebraic geometric GF22m-ary codes.4
Let us now try to find a dependence between RQ and δQ. Letting γ = 12m−2
. Fromthe previous section, we see that for q = 22m and for any pair of real numbers (α, α′)such that 2γ ≤ α′ < α ≤ 1
2+ γ,
RQ = 1− (α + α′) + 2γ (6.19)
δQ ≥ 1
2mminα− 2γ ,
3
2(α′ − 2γ). (6.20)
For any fixed δQ, in order to maximize RQ, we want α − 2γ = 32(α′ − 2γ), i.e.,
α′ = 23(α + γ). Hence,
RQ = 1− 5
3α +
4
3γ (6.21)
δQ ≥ 1
2m(α− 2γ). (6.22)
Therefore, for any m ≥ 3 and δ ≤ 12m
(12− 1
2m−2), we get
RQ = 1− 2
2m − 2− 10
3mδQ. (6.23)
Immediately, we see that the asymptotic parameters of QAG codes are separatedfrom zero and so they are asymptotically good as claimed.
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Figure 6.1: Nonconstructive bound of equation 2.4 and the bound of equation 6.23.
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Chapter 7
Examples
7.1 Rational function field
Definition 7.1 F = K(x) where x is transcendental over K is called the rationalfunction field over K. 4
Remark 7.2 F = GFq(x) has genus g = 0 and q + 1 places of degree one which arePα | α ∈ K∪P∞. Moreover, for 0 ≤ degG ≤ n−2, k = 1+degG and d = n−degG.4
Let K = GFq and F = K(x). Following the previous chapter, we let G = a ·P∞, a ∈N and D =
∑α∈K,α 6=0 Pα with degD = n = q−1. Define E = (n−2−2a)·P∞ and impose
the condition that a ≤ n/2 − 1 so that E is positive. We now set A = D − 2G − E =∑α∈K,α 6=0 Pα−(n−2)·P∞. If we choose as a basis for L(G) the functions 1, x, . . . , xa then
the geometric Goppa code CL(D,G) is given by matrix ζ ij with ζ a primitive elementof K. Classically, this code is called Reed-Solomon code. In the quantum case, we foundan element θ ∈ CΩ(D, 2G + E) = CL(D, 2G + E)⊥. By theorem 6.2, wt(θ) = n. Wenow consider the image of the multiplication map mθ(CL(D,G)) and its dual is equalto C ′ := CL(D,G)⊥θ . We note that the dual code of a code can be easily found byGaussian elimination on the generator matrix of the code. Again, by theorem 6.2, weget the following parameters for C ′:
n = q − 1 , k = q − 2− a , d ≥ a+ 2
where 1 ≤ a ≤ q−12− 1.
Now, we set q = 2m,m ∈ N and we find η such that η2i = θi. Then we set C = mη(C
′)and we get C ⊇ C⊥. We formulate the above discussion into the following proposition.
Proposition 7.3 Let F = GFq(x) where x is transcendental over GFq and q = 2m, wehave for any 1 ≤ a′ < a ≤ b(2m − 1)/2 − 1c, there exists an [[m(2m − 1),m(2m − 3 −(a+ a′)), d ≥ mina+ 2, 3
2(a′ + 2)]] quantum code Q. ♦
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We note that since the construction is similar to that of the classical Reed-Solomoncodes, we may identify the codes found in proposition 7.3 as the Quantum Reed-SolomonCodes. We also remark that proposition 7.3 is an improvement of theorem 6 in [10].
We now want to consider another class of QAG codes which is the quantum analogof the classical BCH codes. Firstly, let n|q− 1 and let ζ ∈ GFq be a primitive n-th rootof unity. We consider the rational function field F = GFq(x) and let Pi := x− ζ i−1 andD := P1 + · · · + Pn. We also define G := a · P0 + b · P∞ with 0 ≤ a, b ≤ n − 2, andE := (n− 2− 2(a+ b)) ·P∞ with (a+ b) ≤ n/2− 1. We note that the generator matrixof CL(D, a · P0 + b · P∞) is given by
1 ζ−a ζ−2a · · · (ζn−1)−a
1 ζ−a+1 ζ−2a+2 · · · (ζn−1)−a+1
......
......
1 ζ−a+(a+b) ζ−2a+2(a+b) · · · (ζn−1)−a+(a+b)
. (7.1)
Following previous exposition, we find η such that η2i = θi where θ ∈ CΩ(D, 2G+E)
and consider C := mη((mθ(CL(D,G))⊥) which is a [n, n−(a+b)−1, d ≥ (a+b)+2]-codecontaining its dual.
Then by binary expanding and using the Steane’s construction, we get the followingproposition.
Proposition 7.4 Let F = GFq(x) where x is transcendental over GFq, q = 2m andn|q−1, we have for any 0 ≤ c′ < c ≤ n/2−1, there exists an [[mn,m(n−(c+c′)−2), d ≥minc+ 2, 3
2(c′ + 2)]] quantum code Q. ♦
We note that weakly self-dual BCH codes can also be used to construct quantumcodes through Steane’s construction, but their parameters are no better than thosefound in the previous proposition. Some more specific discussion on Quantum codesbased on BCH codes can be found in [17, 9, 20].
7.2 Elliptic function field
Definition 7.5 An algebraic function field F/K (where K is the full constant field ofF ) is said to be an elliptic function field if the following conditions hold:(1) the genus of F/K is 1;(2) there exists a divisor A ∈ DF with degA = 1. 4
We consider an elliptic curve E defined by the equation:
y2 + y = u(x)
where u(x) ∈ GF2[x] is a polynomial of degree 3.
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We now consider the points on E over GFq with q = 2m. We follow [18] in thissection and define a subset S of E(GFq) to be
S := α ∈ GFq | (α, β, 1) ∈ E(GFq), α 6= 0.
Let s = #S and d ∈ N be such that 2s− d ≥ 1.We now define G := a · P∞ and D :=
∑α∈S(Pα + Pα) where Pα := (α, β, 1) and
Pα := (α, β + 1, 1).Set l := ba/2c and l∗ := b(a− 3)/2c and define a function f on E by
f(x, z) :=∏α∈S
(x+ αz).
Defining
ωi := xizs−1−iydx
f, 0 ≤ i ≤ s− l − 2
ω∗j := xjzs−jdx
f, 0 ≤ j ≤ a− l∗ − 2.
They then serve as a basis for Ω(D −G) and their residue can be calculated to be
ResPα(xldx
f) =
αl
f ′(α), ResPα(xly
dx
f) =
αly(α)
f ′(α)
wheref ′(α) =
∏α′∈S,α′ 6=α
(α + α′).
The image of the following map is then a [n, n− d,≥ d]2m-code with n = 2s.
Res : Ω(D −G)→ CΩ(D,G) ⊂ GF nq (7.2)
ω 7→ (ResP1(ω), . . . , ResPs(ω), ResP1(ω), . . . , ResPs(ω)). (7.3)
To find quantum codes, following the previous definition, we define
D′ :=∑α∈S
(Pα + Pα) (7.4)
G := a · P∞ (7.5)
E := (2s− 1− 2a) · P∞. (7.6)
We then find an element θ ∈ CΩ(D′, 2G+E), it can have at most one component = 0.W.l.o.g., let the first component be zero, we then re-define D :=
∑α∈S(Pα+Pα)−P1 and
find η ∈ GF n−1q such that η2
i−1 = θi, 2 ≤ i ≤ n. We now consider mη((mθ(CΩ(D,G))⊥)which contains its dual by theorem 6.2. The parameters of the quantum code will thenbe
[[n = m(2s− 1) , k = m(2s− (a+ a′)− 1) , d ≥ mina, 3a′
2]]
where 1 ≤ a′ ≤ a ≤ 2s− 1.
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7.3 Hermitian function field
Definition 7.6 The Hermitian function field H over GFq2 is defined by the curve
xq+1 + yq+1 + zq+1 = 0.
H := GFq2(x, y) with yq + y = xq+1.
4
Remark 7.7 The genus of H is g = q(q − 1)/2. 4
We shall show that H has 1 + q3 points in P2(GFq2). In fact, if x, y or z = 0, thenw.l.o.g., assume z = 0 and y = 1 and we need to solve the equation xq+1 = 1 over GFq2
which has q + 1 solutions. Therefore, H has 3(q + 1) points with xyz = 0. Otherwise,take z = 1 and y ∈ GF ∗q2 such that yq+1 6= 1. For each y, there are r+ 1 solutions for x.Hence, H has 3(q + 1) + (q − 2)(q + 1)2 = 1 + q3 points.
Choose α, β ∈ GFq such that αq + α = βq+1 = −1, and setting
u =β
y − βx, v = xu− α
we can transformxq+1 + yq+1 + zq+1 = 0
to the equationvq + v = uq+1.
We now take G = a · Q where Q = (0, 1, 0) with q2 − q < a < q3, and D =∑P∈H,P 6=Q P . The classical Hermitian code CL(D,G) is thus a [q3, a− g+ 1,≥ q3−a]q2-
code. Also, we note that elements viuj with i ≥ 0, 0 ≤ j ≤ q − 1 and iq + j(q + 1) ≤ aform a basis of the space L(a · Q) and so the generator matrix can be found readily.Again by theorem 6.2, we have the following proposition.
Proposition 7.8 Let H be a Hermitian function field over GFq2 and q = 2m, we havefor any q2 − q ≤ a′ < a ≤ q3, there exists an [[2mq3, 2m(q3 + q2 − q − 2− (a+ a′)), d ≥mina− q2 + q + 2, 3
2(a′ − q2 + q + 2)]] quantum code Q. ♦
Similar to the case of BCH code, we may consider Hermitian code that is naturallyself-dual. This situation is described by the next theorem.
Theorem 7.9 (a) For r ≤ s, we have Cr ⊆ Cs where Cr := CL(D, r ·Q).
(b) C⊥r = Cq3+q2−q−2−r
(c) If q2 − q − 2 ≤ r ≤ q3 + q2 − q − 2. Then k = r + 1 − q(q − 1)/2 and d ≥ q3 − rwhere k = dimCr and d is the minimum distance of Cr.
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Proof. See Propositions VII.4.2 and VII.4.3 in [19]. ♦
To construct quantum codes throught the Steane’s method, we want classical codesC ′ and C such that C ′ ⊃ C ⊇ C⊥. By theorem 7.9, we thus have the following corollary.
Corollary 7.10 For r, s ∈ Z such that 12(q3 + q2− q− 2) ≤ r < s ≤ q3 + q2− q− 2, we
have Cs ⊃ Cr ⊃ C⊥r where
dimCr = r + 1− q(q − 1)
2, d ≥ q3 − r
dimCs = s+ 1− q(q − 1)
2, d ≥ q3 − s.
♦
We now let q = 2m,m ∈ Z. By binary expanding the finite field GFq2 , we havebinary codes D′ and D such that
nD′ = nD = 2mq
kD′ = 2m(s+ 1− q(q − 1)
2) , kD = m(r + 1− q(q − 1)
2)
dD′ = q3 − s , dD = q3 − r,for 1
2(q3 + q2 − q − 2) ≤ r < s ≤ q3 + q2 − q − 2. Hence, by theorem 4.5, we have the
following corollary.
Corollary 7.11 Associated to the Hermitian function field H over GFq2, we have afamily of quantum codes QH with parameters
n = 2mq3 (7.7)
k = 2m(s+ r + 2− q2 + q))− q3 (7.8)
d ≥ minq3 − r, 3
2(q3 − s) (7.9)
where 12(q3 + q2 − q − 2) ≤ r < s ≤ q3. ♦
We now set a := q3 + q2− q− 2− r and a′ := q3 + q2− q− 2− s, we get the followingfamily of codes:
[[2mq3 , 2m(q3 + q2 − q − 2− (a+ a′) , ≥ mina− q2 + q + 2,3
2(a′ − q2 + q + 2]]
where q2 − q − 2 ≤ a′ < a ≤ 12(q3 + q2 − q − 2).
Comparing thre result with proposition 7.8, we see that construction by Ashikhmin,Litsyn and Tsfasman is more general, but the weakly self-duality of Hermitian code doesmanage to improve the lower bound by 2.
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Chapter 8
Conclusion
So we have succeed in finding some asymptotically good codes, but it seems that weshould be able to do much better than what we have done. For instance, the parametersof QAG codes are still far from attaining the quantum lower bound in equation 2.4
R(δ) ≥ 1− δ log2 3−H(δ).
So as it is guaranteed that there are good codes that are even better than our QAGcodes, why are we wasting time on QAG codes? The answer lies on the fact that thelower bound is nonconstructive. In other words, those imaginary codes appearing inthe lower bound cannot be constructed in polynomial time in n while the QAG codescan be. In fact, recalling our three steps in algorithm 6.5, step (1) is polynomiallyconstructible, see [21], step (2) is polynomially constructible because binary expansionis, step (3) is also polynomially constructible (c.f. section 3.3). Therefore, our QAGcodes are polynomially constructible.
On the other hand, if we know improvement is possible, what should we go aboutin achieving that? First of all, if manipulation of spins (or other quantum quantities)in higher dimension is possible, we should try to generalise our QAG codes to higherdimension and it may improve their parameters. For instance, the Gilbert-Varshamovbound is achieved when q ≥ 49 in the classical case. Secondly, in constructing QAGcodes, we were basically converting classical algebraic-geometric codes to quantum onesthrough the Steane’s construction. One may wonder if the theory of algebraic functionfields could be applied directly to the quantum case without recuring to our classicalcounter part.
Some research has been done in the first direction [1, 3, 13], but it is still unclearabout how to combine the techniques devised on higher dimensional quantum codeswith the algebraic-geometric constructions. For the second direction, it may requireanother quantum leap in ideas of QECC constructions similar to the invention of CSSconstruction and the Steane’s improvement on it.
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