量子編碼的數學性質與建構

國

立

交

通

大

學

應 用 數 學 系

碩 士 論 文

量 子 編 碼 的 數 學 性 質 與 建 構

Mathematical Properties and

Construction of Quantum Codes

研 究 生 ：蔡睿翊

指導教授 ：翁志文 教授

中 華 民 國 一 百 零 三 年 七 月

量 子 編 碼 的 數 學 性 質 與 建 構

Mathematical Properties and Construction of Quantum

Codes

研 究 生 : 蔡睿翊　 Student : Jui-Yi, Tsai

指導教授 : 翁志文

Advisor : Chih-Wen Weng

國

立

交

通

大

學

應 用 數 學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

in Partial Fulfillment of Requirements

for the Degree of Master

in Applied Mathematics

July 2014

Hsinchu, Taiwan, Republic of China

中 華 民 國 一 百 零 三 年 七 月

量 子 糾 錯 碼 的 數 學 性 質 與 建 構

研究生：蔡睿翊

指導教授：翁志文 教授

國立交通大學

應用數學系

摘 要

Mathematical Properties and

Construction of Quantum Codes

Student：Jui-Yi, Tsai Advisor：Dr. Chih-Wen Weng

Department of Applied Mathematics

National Chiao Tung University

Hsinchu 300, Taiwan, R.O.C.

Abstract

This thesis introduces about the quantum error correcting codes in the viewpoint of classical error correcting codes. We also introduce some construction and charac-terization of quantum error correcting codes. Thereafter, we give a construction of quantum error correcting codes associated with graphs, which generalizes a previous result that excludes the binary case so that it is valid for all cases.

誌

謝

首先，非常感謝指導教授翁志文老師，對我在學習及研究的方

法與態度上有很多指引與指正，這對我在研究上有相當大的幫助。

也很感謝家人在這兩年間對我的關懷與督促，讓我即使在外地不致

偏離該走的路，遇上困難時也不致感到灰心喪志。

感謝系上師長 (翁志文老師、康明軒老師、楊一帆老師等) 的教

導，讓我有機會接觸到更加高深的代數與組合學的知識；感謝系辦

職員的協助，讓我們有非常好的學習環境；感謝人社系段馨君老師

的鼓勵及分享，使我能找到自己可以努力下去的方向；感謝系上各

位同學 (特別是黃于哲、黃建順兩位同研究室的同學的分享與建議、

組合數學組的許博喻、林凡軒、楊凱帆、余冠儒等同學們在課業方

面及 L

_{TEX 等方面的協助) 及學長姊 (特別是林易萱、李光祥、鄭硯}

仁、蔡志奇這四位學長，他們分別在我的課業、研究、助教工作及

生活各方面上幫了我很多忙)、學弟妹們還有友聲合唱團團員們的陪

伴與扶持，使我這兩年間的學校生活增添了許多色彩；也感謝真耶

穌教會新竹教會及關東橋教會的傳道、長執、同靈們及大清交團契、

新竹高級班給予我的勉勵及代禱，使我在這兩年間得到心靈上的成

長，也使我有堅持下去的動力。

最後，最感謝的，就是主耶穌。若非祢一路帶領，我想就沒有

今天的我了。謹此表示對在這兩年間遇到的所有師長，朋友的感謝。

願一切榮耀歸於天上的真神！

睿翊 謹誌于新竹交大

2014 年 7 月 24 日

Contents

Abstract (in Chinese) i

Abstract (in English) ii

Acknowledgment iii

1 Introduction 1

2 Preliminaries 2

3 Comparison between Classical and Quantum Error Correcting Codes 4

3.1 Basic Definitions and Structures of Classical Codes and Quantum Codes . . . 4 3.2 Error Detection and Correction of Classical Codes and Quantum Codes 5 3.3 Bounds in Classical Codes and Quantum codes . . . 10 3.4 Some Simple Lemmas . . . 10

4 The Characterization of Quantum Codes Using Logic Functions 12

Chapter 1

Introduction

Quantum communication like quantum coding theory and quantum cryptogra-phy has been developed in the recent decades. This is a great improvement in com-munication theory. Its concepts are based on quantum mechanics, but we will not mention anything about quantum mechanics in this thesis. We will give the strict mathematical definition of quantum error correcting codes based on [3]. The results from [6] and [7] will be introduced. Thereafter, we give a construction of quantum error correcting codes associated with graphs, which generalizes a previous result in [7] that excludes the binary case so that it is valid for all cases.

The thesis is organized as follows: Chapter 2 introduces the notations which will be used in this thesis. To compare the similarities and differences between classical error correcting codes and quantum error correcting codes, chapter 3 recalls the definitions and propositions of classical error correcting codes; those of quantum error correcting codes will also be introduced in parallel. Chapter 4 introduces a method to characterize quantum error correcting codes by logic functions fromFn

p to

Fp introduced in [6]. Chapter 5 introduces about the construction of quantum error

correcting codes using the graph-theoretical method introduced in [7], in which only non-binary quantum codes are discussed. Thus we will generalize the result in [7].

Chapter 2

Preliminaries

In this section, we will introduce the notations which will be used in this thesis. Throughout the thesis, let Fp = {0, 1, · · · , p − 1} be the finite field of p elements.

Sometimes we also treat an element i ∈ Fp as its corresponding integer. Let ω be

the p-th primitive root of unity. Let A = (ai,j) be an m× n matrix and B an s × t

matrix. Then the Kronecker tensor product A_{⊗ B of A and B is defined by the} following ms_{× nt matrix} A⊗ B =      a11B a12B · · · a1nB a21B a22B · · · a2nB ... ... . .. ... am1B am2B · · · amnB     .

Note that (A⊗ B)(C ⊗ D) = AC ⊗ BD for matrices A, B, C, D of suitable sizes. The n-th Kronecker tensor power of a vector space V over a field F is defined by

V⊗n := span{v1 ⊗ v2⊗ · · · ⊗ vn|vi ∈ V }.

The Hermitian inner product of complex vectors u = (u1,· · · , un)T, v = (v1,· · · , vn)T

is defined by ⟨u, v⟩ := n ∑ i=1 uivi,

where _{· stands for complex conjugation. In the the last two sections, we will} fre-quently use the block notations of matrices. Therefore we should introduce the

following notation of a matrix M and a column vector u: M =    ∆1 ∆2 ∆1 M [∆1|∆1] M [∆1|∆2] ∆2 M [∆2|∆1] M [∆2|∆2]   , u = ( u[∆1] u[∆2] ) ,

where M [∆1|∆2] means the sub-matrix of M with rows indexed by elements in ∆1

and the columns by elements in ∆2, and u[∆1] means the sub-column vector with

Chapter 3

Comparison between Classical and

Quantum Error Correcting Codes

In this chapter, we are going to recall the definitions and propositions of classical error correcting codes and introduce those of quantum error correcting codes to compare the similarities and differences between them.

3.1

Basic Definitions and Structures of Classical

Codes and Quantum Codes

The classical (linear) codes are vector spaces over finite fields; whereas the quan-tum codes are vector spaces over the complex number field C.

Definition 3.1. A classical (linear) [n, k]p-code (or a classical (n, K)p-code,

where K = pk _{=|C|) is a k-dimensional subspace C of F}n p.

Definition 3.2. A quantum [[n, k]]p-code (or a quantum ((n, K))p-code, where

k = logpK) is a K-dimensional subspace Q of (Cp)⊗n.

Remark 3.3. Error-correcting coding theory depends on what basis of a vector

space is chosen.

(i) Classical coding theory:

{ei = (0,· · · , 0, 1, 0, · · · , 0)T ∈ Fnp | 1 ≤ i ≤ n},

(ii) Quantum coding theory:

{ei1+1⊗ · · · ⊗ ein+1 | ij ∈ {0, 1, . . . , p − 1}}.

It is a convention in Quantum coding theory (Dirac notation) to write |i⟩ for ei+1

and |i1i2· · · in⟩ for ei1+1⊗ ei2+1· · · ⊗ ein+1.

3.2

Error Detection and Correction of Classical

Codes and Quantum Codes

An error can occur when passing a message, so we have to correct it to the right one. Here we introduce the error detection and correction of classical codes and quantum codes in an algebraic point of view.

An error in classical codes is simply a vector over a finite field, and the codewords is interrupted by an error under addition.

Definition 3.4. An error is a nonzero vector e independent to C.

The quantum codewords may be interrupted by the errors under matrix multi-plication. Here is the definition of the errors in quantum codes.

Definition 3.5. (i) For a, b _{∈ F}p, define two linear operators X(a) and Z(b) on

Cp _by

X(a)|x⟩ = |x + a⟩, Z(b)|x⟩ = ωb·x|x⟩,

where x_{∈ F}p. Then X(a) is called a bit error; Z(b) is called a phase error.

(ii) For a = (a1,· · · , an)T, b = (b1,· · · , bn)T ∈ Fnp,

X(a) = X(a1)⊗ · · · ⊗ X(an), Z(b) = Z(b1)⊗ · · · ⊗ Z(bn).

(iii) _En := {ωtX(a)Z(b)|0 ≤ t ≤ p − 1, a, b ∈ Fnp} is called the quantum error

An element of E ∈ En is called an error of quantum code. Note that E|u⟩ =

ωt+b·u|u+a⟩ for u ∈ Fn

p and E = ωtX(a)Z(b). In fact, from the previous definition,

we have the following proposition.

Proposition 3.6. The following (i)-(ii) holds.

(i) With respect to the basis _{{|0⟩, |1⟩, · · · , |p − 1⟩}, the operator X(1) is a cyclic} matrix, and the operator Z(1) is a diagonal matrix as follows.

X(1) =        0 1 1 0 1 . .. . .. 0 0 1 0        , Z(1) =        1 0 ω ω2 . .. 0 ωp−1        .

(ii) X(a) = X(1)a_{, Z(b) = Z(1)}b_{, X(a)}T _{= X(−a) = X(a)}−1_{, Z(b)}T _{= Z(−b) =}

Z(b)−1, and Z(b)X(a) = ω−b·aX(a)Z(b) for all a, b∈ Fp.

Moreover, _En is a group of order p2n+1.

Here we compare the Hamming weight with the quantum weight.

Definition 3.7. (i) The Hamming weight wtH(e) of an element e∈ Fnp is the

number of nonzero entries in e. The Hamming distance of u, v ∈ Fn p is

defined by d(u, v) := wtH(u−v). Note that the Hamming distance is a metric

(i.e. its value is always nonnegative, and it satisfies the symmetry and the triangle inequality).

(ii) The quantum weight wtQ(E) of an element E = ωtX(a)Z(b) ∈ En is the

number of nonzero pairs (ai, bi) in the two vectors a, b∈ Fnp.

For classical codes, the Hamming distance is used to describe the ability of error detection and correction. As for quantum codes, we use the Hermitian inner products of quantum codewords in a quantum code Q. The orthogonality is usually used to describe the ability of error detection and correction.

Definition 3.8. For two quantum codewords u, v∈ Q. Then

(i) If u = γv for some nonzero γ ∈ C, then we say u, v are totally

indistin-guishable.

(ii) If _{⟨u, v⟩ = 0, then we say u, v are totally distinguishable in Q.}

Here is the comparison between the definitions of error detection of classical codes and of quantum codes.

Definition 3.9. C can detect an error e if d(u, v + e) > 0 for distinct u, v∈ C. Definition 3.10. For a quantum ((n, K))p-code Q with K ≥ 2 and an error E ∈ En,

Q can detect an error E if u, v are totally distinguishable implies that u, Ev are

totally distinguishable.

Note that 3.10 is equivalent to ⟨u, Ev⟩ = λE⟨u, v⟩, where λE ∈ C depends only on E but is independent of u, v.

Here is the comparison between the definitions of error correction of classical codes and of quantum codes.

Definition 3.11. C can correct an error e if d(v, v + e) < d(w, v + e) for all

distinct v, w∈ C.

Definition 3.12. For a quantum ((n, K))p-code Q, where K ≥ 2. Q can correct

errors of weight at most t if, for any totally distinguishable u, v∈ Q and any errors

E1, E2 ∈ En with wtQ(E1), wtQ(E2) ≤ t, E1u, E2v are totally distinguishable; in

other words, _⟨E1u, E2v⟩ = 0.

Here is the comparison between the definitions of minimum distance of classical codes and of quantum codes.

Definition 3.13. The minimum distance of C with|C| ≥ 2 is at least d if C can

detect errors of Hamming weight at most d_{− 1; in other words, 0 < wt}H(e) < d

Note that the previous definition is equivalent to the common one; that is,

d = min{d(u, v)|u, v ∈ C are distinct.}.

This can be verified by the triangle inequality of Hamming distance.

Definition 3.14. A quantum ((n, K))p-code Q with K ≥ 2 has minimum

dis-tance at least d if Q can detect errors of quantum weight at most d− 1; in other

words, _{⟨u, v⟩ = 0 implies ⟨u, Ev⟩ = 0 for any error E ∈ E}n with wtQ(E)≤ d − 1.

Note that the definitions of the minimum distance of classical codes and quantum codes are similar because the minimum distance d is given by the detection of classical errors of Hamming weight d − 1 in classical case and the detection of quantum errors of quantum weight d− 1 in the quantum case.

Here is a special property often used in quantum codes.

Definition 3.15. A quantum code Q is a d-pure code if for any u, v∈ Q and any

errors E ∈ En with 0 < wtQ(E) < d, u, Ev are totally distinguishable; in other

words, ⟨u, Ev⟩ = 0.

This property can help us to distinguish a codeword from another one interrupted by an error of quantum weight less than d.

Remark 3.16. (i) A (quantum or classical) code has minimum distance

ex-actly d if it has minimum distance at least d, but does not have minimum distance at least d + 1.

(ii) From definition 3.14, a quantum ((n, K))p-code Q (or a quantum [[n, k]]p-code)

is an ((n, K,_{≥ d))}p-quantum code (or an [[n, k,≥ d]]p-quantum code) for some

d ≥ 1 means a pk_{-dimensional (or K-dimensional) quantum code in (}Cp₎⊗n

with minimum distance at least d; if Q has minimum distance exactly d, then

Q is a quantum ((n, K, d))p-code (or a quantum [[n, k, d]]p-code).

Proposition 3.17. A d-pure quantum ((n, K))p-code Q with K ≥ 2 has minimum

distance ≥ d.

Proof. Suppose E ∈ En with wtQ(E) ≤ d − 1. Let c1, c2 ∈ Q be codewords with

⟨c1, c2⟩ = 0. Now if E = ωkI, then wtQ(e) = 0. Thus ⟨c1, Ec2⟩ = ωk⟨c1, c2⟩ = 0. If

wtQ(E)̸= 0, then 1 ≤ wtQ(E)≤ d − 1, and so ⟨c1, Ec2⟩ = 0 by definition 3.14.

Note that, by definition 3.15, for K = 1 (i.e. k = 0), Q is a d-pure quan-tum ((n, 1, d))p-code (or a d-pure quantum [[n, 0, d]]p-code) since any two vectors in

{Ec|E ∈ En, 0 ≤ wtQ(E) ≤ d − 1} are orthogonal, where c is the non-zero vector

that spans Q.

Here is the comparison between the abilities of error correction of classical codes and of quantum codes.

Theorem 3.18. An [n, k, d]p-code C can correct errors of weight at most

⌊_d−1

2

⌋

. Proof. Let e be an error of weight at most⌊d−1

2

⌋

. From the triangle inequality, we have d(w, v + e)≥ d(w, v) − d(v, v + e) ≥ d − ⌊ d− 1 2 ⌋ ≥ ⌈ d− 1 2 ⌉ > d(v, v + e).

(The inequality d(w, v) _{≥ d comes from the definition of minimum distance; the} other one wtH(e) = d(v, v + e) ≤

⌊_d−1

2

⌋

is from the hypothesis.)

Theorem 3.19. If Q is an quantum ((n, K, d))p-code with K ≥ 2, then Q can

correct errors of weight at most ⌊d−1

2

⌋

.

Proof. Let E1, E2 ∈ Enwith wtQ(E1), wtQ(E2)≤

⌊_d−1

2

⌋

. It is clear that wtQ(E1E2)≤

wtQ(E1) + wtQ(E2)≤ d − 1. Since Q has minimum distance d, Q can detect E1E2;

that is, for any totally distinguishable c1, c2 ∈ Q, we have ⟨c1, E1E2c2⟩ = 0,

3.3 Bounds in Classical Codes and Quantum codes

In classical and quantum coding theory, the parameters (n, K, d) or (n, k, d) de-termine the efficiency of communication (k/n) and the ability of error correction(d), but there are some restrictions called the Hamming bound and the Singleton

bound, causing that we can not obtain both high efficiency of communication and

good ability of error correction. For the proofs, please see [3] in detail.

Theorem 3.20. (classical Hamming bound) If C is an (n, K, d)p code, then

pn≥ K · ⌊d−1 2 ⌋ ∑ i=0 (p− 1)i ( n i ) . (3.1)

Theorem 3.21. (classical Singleton bound) If C is an (n, K, d)p code, then

K ≤ pn−d+1. (3.2)

Theorem 3.22. (quantum Hamming bound) If Q is a d-pure quantum ((n, K, d))p

-code, then pn≥ K · ⌊d−1 2 ⌋ ∑ i=0 (p2− 1)i ( n i ) . (3.3)

Theorem 3.23. (quantum Singleton bound) If Q is a quantum ((n, K, d))p-code,

then

K ≤ pn−2d+2. (3.4)

3.4

Some Simple Lemmas

In this section, we provide some simple lemmas, which will be used many times in the thesis.

Lemma 3.24. Let p be a prime and ω the p-th root of unity. Then∑_u∈Fn pω

u·v _{= 0}

for all non-zero vectors v∈ Fn p.

Proof. For u = (u1,· · · , un)∈ Fnp and v = (v1,· · · , vn)∈ Fnp \ {0}, we have ∑ u∈Fn p ωu·v = ∑ (u1,··· ,un)∈Fnp ωu1v1+···+unvn ₌ ∑ (u1,··· ,un)∈Fnp ωu1v1· · · ωunvn =  ∑ u1∈Fp ωu1v1   · · ·  ∑ un∈Fp ωunvn  

For v̸= 0, there is some vi ̸= 0, which is certainly an inverse of some other element

inFp. Thus viFp =Fp, and so

∑

ui∈Fpω

uivi _{= 0 (because 1 + ω +}· · ·+ωp−1 _{= 0).} Lemma 3.25. Let p be a prime, ω a p-th primitive root of unity and Vp = (ωij)i,j∈Fp

a p_{× p Vandermonde matrix of the form} Vp =      1 1 · · · 1 1 ω · · · ωp−1 .. . ... . .. ... 1 ωp−1 · · · ω(p−1)2     .

Then Vp is invertible with Vp−1 = (1/p)Vp, and so the Kronecker tensor product

Vp⊗ Vp⊗ · · · ⊗ Vp of m Vp’s is invertible.

Proof. (i) By lemma 3.24, we have (VpVp)i,j = p−1 ∑ k=0 ωikω−kj = p−1 ∑ k=0 ωk(i−j) = { 0, if i_{̸= j} p, if i = j. Thus Vp is invertible with Vp−1 = (1/p)Vp.

(ii) By the multiplication rule of the Kronecker tensor product, we have (Vp ⊗ Vp⊗ · · · ⊗ Vp)(Vp−1⊗ Vp−1⊗ · · · ⊗ Vp−1)

=(VpVp−1)⊗ (VpVp−1)⊗ · · · ⊗ (VpVp−1)

=Ip⊗ Ip⊗ · · · ⊗ Ip = Ipm,

Chapter 4

The Characterization of Quantum

Codes Using Logic Functions

In this chapter, we introduce a characterization of quantum codes using logic functions introduced in [5] and [6] (for logic functions, see [1] and [2]).

Throughout this section, let Q be a K-dimensional quantum code of (_Cp₎⊗n_with

an orthonormal basis_{vi =

∑ u_∈Fn

pvi(u)|u⟩|1 ≤ i ≤ K}, where vi :F

n

p → C, 1 ≤ i ≤

K are functions.

Theorem 4.1. Assume K ≥ 2. Then Q is a quantum ((n, K, ≥ d))p-code if and

only if for any subset E⊆ {1, 2, · · · , n} with |E| = d − 1, d ≥ 2, Ec _{={1, 2, · · · , n} \}

E,|Ec| = n − d + 1 and w, w′ ∈ Fd_p−1, 1≤ i, j ≤ K, we have ∑ u[Ec_]=u′[Ec_] vi(u)vj(u′) = { 0, if i̸= j η(w, w′), if i = j, (4.1) where the sum is indeed over u, u′ ∈ Fn

p with u[Ec] = u′[Ec] and u[E] = w, u′[E] =

w′, and η(w, w′)∈ C is a constant independent of i (depends on w, w′).

Proof. ” _{⇒ ” : Let E = X(a)Z(b) ∈ E}n be an error of quantum weight at most

d− 1, where a satisfies a[E] = w − w′, a[Ec_{] = 0, and b ∈ F}n

p is a vector of d− 1

variables satisfying b[Ec_{] = 0. Then by definition 3.10,}

λEδij =⟨vi, Evj⟩ = ∑ x∈Fd−1 p ωb[E]·x′ ∑ u[Ec_]=u′_[Ec_] vi(u)vj(u′), (4.2)

where δij = 0 if i̸= j; δij = 1 otherwise, λE depends only on E and is independent

of i, j, and the second sum is indeed over u, u′ ∈ Fn

p with u[Ec] = u′[Ec] and

u[E] = x, u′[E] = x− a[E]. In matrix form, (4.2) becomes Ωy =

{

0, if i ̸= j

λE(1, 1,· · · , 1)T, if i = j,, where Ω is an pd−1_{× p}d−1 _{matrix indexed by F}d−1

p with b[E], x′-entry ωb[E]·x

′

, and

y = ∑

u[Ec_]=u′[Ec_]

vi(u)vj(u′) (being indexed depending on x′)

is a (d− 1)-dimensional column vector over Fp . Since the matrix Ω is invertible by

lemma 3.25, we find the column vector

y =

{

0, if i̸= j

λEΩ−1(1, 1,· · · , 1)T_{, if i = j,}.

Hence the result follows by considering the x = w entry of vector y.

”⇐ ” : To show that Q has minimum distance at least d, let E = ωt_X(a)Z(b)∈

En be an error of quantum weight at most d− 1. Without loss of generality, we can

assume t = 0. Choose E such that|E| = d−1 and a, b ∈ Fn

p satisfy (a[Ec], b[Ec]) =

(0, 0). Pick two totally distinguishable codewords v =∑K_i=1αivi, w =

∑K j=1βjvj ∈ Q. Note that Ew = K ∑ j=1 βjevj = K ∑ j=1 βj ∑ u′∈Fn p vj(u′)e|u′⟩ = K ∑ j=1 βj ∑ u′∈Fn p vj(u′)ωb·u ′ |u′_{+ a⟩.}

Thus for u[E] = x, u′[E] = x− a[E], we have

⟨v, Ew⟩ = K ∑ i,j=1 αiβj ∑ u=u′_+a ωb·u′vi(u)vj(u′) = K ∑ i,j=1 αiβj ∑ x∈Fd−1 p ωb[E]·(x−a[E]) ∑ u[Ec_]=u′[Ec_] vi(u)vj(u′) = K ∑ i=1 αiβi ∑ x∈Fd−1 p ωb[E]·(x−a[E]) ∑ u[Ec_]=u′_[Ec_] vi(u)vi(u′) = ∑ x∈Fd−1p

(The last equality is obtained by the condition: when i̸= j,∑_u[Ec_]=u′[Ec_]vi(u)vj(u′) =

0; when i = j, ∑_u[Ec_]=u′[Ec_]vi(u)vi(u′) = η(x, x− a[E]), which is independent to i),

completing the proof.

Theorem 4.2. Let K ≥ 1. Then Q is a pure quantum ((n, K, ≥ d))p-code if and

only if for any subset E⊆ {1, 2, · · · , n} with |E| = d − 1, d ≥ 2, Ec ₌{1, 2, · · · , n} \

E,|Ec| = n − d + 1 and w, w′ ∈ Fd_p−1, we have ∑ u[Ec_]=u′_[Ec_] vi(u)vj(u′) = { 0, if w_{̸= w}′ δi,jp1−d, if w = w′, (4.3)

where the sum is indeed over u, u′ ∈ Fn

p with u[Ec] = u′[Ec], and u[E] = w, u′[E] =

w′ _{∈ F}d−1 p .

Proof. ”⇒ ” : Let a set E of cardinality d − 1 and w, w′ ∈ Fd−1

p be given. Choose

E = X(a)Z(b)∈ En be an error of quantum weight at most d− 1, where a satisfies

a[E] = w_{− w}′, a[Ec_{] = 0, and b ∈ F}n

p is any vector satisfying b[Ec] = 0. Since Q

is a pure ((n, K, d))p-quantum code, for all 1≤ i, j ≤ K,

δwtQ(E),0 =⟨vi, evj⟩ = ∑ x′=x−a[E] ωb[E]·x′ ∑ u[Ec_]=u′[Ec_] vi(u)vj(u′). (4.4) Note that ∑ x∈Fd−1 p ∑ u[Ec_]=u′_[Ec_] vi(u)vj(u′) = ∑ u_∈Fn p

vi(u)vj(u) = δi,j.

In matrix form, (4.4) becomes Ωy =

{

(δi,j, 0, 0,· · · , 0)T, if w = w′(a[E] = 0);

(0, 0, 0,· · · , 0)T_{, if w}̸= w′_(a[E]̸= 0),

where Ω and y are as described in the proof of theorem 4.1. Note that the first column of Ω−1 is p1−d_{(1, 1,· · · , 1)}T_{. Thus}

y =

{

δi,jp1−d(1, 1,· · · , 1)T, if w = w′(a[E] = 0);

(0, 0, 0,· · · , 0)T, if w̸= w′(a[E]̸= 0), proving the necessary condition.

”⇐ ” : To show that Q is d-pure, let E = ωt_X(a)Z(b)∈ E

n with 1 ≤ wtQ(E) ≤

d− 1. Without loss of generality, we can assume t = 0. Choose E = {i|1 ≤ i ≤ n, (ai, bi) ̸= (0, 0)} so that |E| = d − 1, (a[E], b[E]) ̸= (0, 0) and (a[Ec], b[E]c) =

(0, 0) for a, b∈ Fn_p. Pick two codewords v =∑K_i=1αivi, w =

∑K

j=1βjvj ∈ Q. Then

for u[E] = x, u′[E] = x− a[E], we have

⟨v, Ew⟩ = K ∑ i,j=1 αiβj ∑ u=u′+a ωb·u′vi(u)vj(u′) = K ∑ i,j=1 αiβj ∑ x_∈Fd−1 p ωb[E]·(x−a[E]) ∑ u[Ec_]=u′_[Ec_] vi(u)vj(u′) = {∑K i,j=1δi,jp 1−d_α iβj ∑ x∈Fd−1p ω b[E]·x _{= 0, if a[E] = 0} 0, if a[E]̸= 0 (The last equality is obtained by the second condition: when a[E]_{̸= 0,}

∑ u[Ec_]=u′[Ec_]

vi(u)vj(u′) = 0;

when a[E] = 0, b[E]̸= 0 and ∑ u[Ec_]=u′_[Ec_]

vi(u)vj(u′) = δi,jp1−d),

completing the proof.

In fact, the functions vi :Fnp → C can be obtained simply by vi(u) = ωfi(u), where

1 ≤ i ≤ K and u ∈ Fn

p and fi are functions from Fnp to Fp so that the functions

vi(u) = ωfi(u) satisfy the conditions (4.1) and (4.3) in the previous theorems (see

[6]). Hence these functions can be used to construct (pure) quantum codes. Such function is called a logic function (for p = 2, a Boolean function).

Chapter 5

Graph-theoretical Method

In this chapter, we will discuss about a method to construct quantum error correcting codes introduced by Schlingemann and Werner in [7], in which only the case for the odd primes is discussed, therefore we are going to improve the method by applying vi(u) = ωu

T

iBw+wTAw for u_i ∈ Fk

p (1≤ i ≤ K) to theorem 4.2 so that it

is valid for all primes.

Throughout this section, we assume the following hypotheses: Let p be a prime and ω the p-th primitive root of unity. Let X, Y be sets with cardinality |X| = k and |Y | = n. Let d ≥ 2 and (n, k, d) satisfy the quantum Singleton bound n ≥

k + 2(d− 1). Let A be an n × n matrix with rows and columns indexed by Y , B an k× n matrix with rows indexed by X and columns indexed by Y . Define a linear

function f : (_Cp₎⊗k _{→ (C}p₎⊗n _by

f (|u⟩) = ∑

w∈Fn p

ωuTBw+wTAw|w⟩ (5.1) for u = (x1, x2,· · · , xk)T ∈ Fkp. Let Q = f ((Cp)⊗k), the image of f . Here we will

use the function (5.1)to reprove the result in [7] in three steps.

In step 1, we shall give a definition and prove two lemmas, which give a necessary condition and a sufficient condition of quantum pairs, respectively:

with |E| = d − 1, u ∈ Fk

p and e∈ Fdp−1, the following implication holds:

uT

B[X|Ec]− eT(A + AT)[E|Ec] = 0⇒ u = 0 and B[X|E]e = 0, (5.2) where Ec_{= Y \ E.}

First we prove the necessary condition:

Lemma 5.2. If (A, B) is an [[n, k, d]]p-quantum pair, then the sub-matrix B[X|Ec]

of B has rank k over Fp, and the intersection of row space of B[X|Ec] and (A +

AT)[E|Ec] is the zero space for any (d− 1)-subset E of Y .

Proof. Taking e = 0 in (5.2), we find that B[X|Ec_{] has rank k. Suppose u}T_B[X|Ec_{] =}

eT_{(A + A}T_)[E|Ec_{] is a vector in the intersection of row spaces of B[X|E}c_{] and}

(A + AT)[E|Ec]. Then u = 0 by (5.2). Hence the vector uTB[X|Ec] = 0. Now we prove the sufficient condition:

Lemma 5.3. If the sub-matrix B[X|Ec_{] of B has rank k over} F

p, the sub-matrix

(A + AT)[E|Ec] of A + AT has rank d− 1 over Fp and the intersection of row spaces

of B[X|Ec_{] and (A + A}T_)[E|Ec_{] is the zero space for any (d−1)-subset E of Y , then}

(A, B) is an [[n, k, d]]p-quantum pair.

Proof. Suppose that uT_B[X|Ec_{]− e}T_{(A + A}T_)[E|Ec_{] = 0. Then u}T_B[X|Ec_{] =}

eT_{(A + A}T_)[E|Ec_{] = 0 since it is in the intersection of row spaces of B[X|E}c_{] and}

(A + AT)[E|Ec]. Since rank(B[X|Ec]) = k, the row vectors of B[X|Ec] are linearly independent, thus u = 0. And rank((A + AT_)[E|Ec_{]) = d− 1 implies that e = 0 by}

similar argument.

We shall call such a pair (A, B) in lemma 5.3 a pure [[n, k, d]]p-quantum pair.

In step 2, we shall prove

Theorem 5.4. For any v, v′ ∈ Cp⊗k, if (A, B) is an [[n, k, d]]p-quantum pair, then

⟨f(v′_{)|f(v)⟩ = p}n_⟨v′_{|v⟩. In other words, f preserves orthogonality. In particular,}

Proof. Let v =∑ u∈Fk p v(u)|u]⟩ ∈ Cp⊗k, v′ = ∑ u′_∈Fk p v′(u′)|u′⟩ ∈ Cp⊗k,

where v(u), v′(u′)∈ C. Then w =f (v) = ∑ u∈Fk p ∑ w∈Fn p v(u)ωuTBw+wTAw|w⟩, w′ =f (v′) = ∑ u′_∈Fk p ∑ w∈Fn p v′(u′)ωu′TBw′+w′TAw′|w′⟩. Now, we can compute the Hermitian inner product:

⟨w′_{, w⟩ =}∑ u,u′ [ v′(u′)v(u) ( ∑ w=w′ ωt )] , where t = (u− u′)TBw.

Since (A, B) is an [[n, k, d]]p-quantum pair, rankB[X|Ec] = k. Hence, by lemma

3.24, we have ∑ w ω(u−u′)TBw= { pn_{, if u = u}′_, 0, otherwise. This follows that

⟨w′_{, w⟩ = p}n ∑

u=u′

v′(u′)v(u) = pn⟨v′, v⟩.

Hence f preserves the orthogonality, and so f maps the basis of (Cp₎⊗k _{to that of}

(Cp₎⊗n_{. Hence f is 1-1, which means the image of f has dimension p}k_.

In step 3, we shall prove

Theorem 5.5. If (A, B) is an [[n, k, d]]p-quantum pair, then Q = f ((Cp)⊗k) has

Proof. Let w =f (v) = ∑ u∈Fk p ∑ w∈Fn p v(u)ωuTBw+wTAw|w⟩ ∈ Q, w′ =f (v′) = ∑ u′_∈Fk p ∑ w′∈Fn p v′(u′)ωu′TBw′+w′TAw′|w′⟩ ∈ Q.

be totally distinguishable (hence v′, v are totally distinguishable). For any E = X(l)Z(s) ∈ En with wtQ(E) ≤ d − 1, we have E|w[E], w[Ec]⟩ = ωs[E]·w[E]|w[E] +

l[E], w[Ec_{]⟩ for all (d − 1)-subset E of Y and s, l ∈ F}n

p with s[Ec] = l[Ec] = 0. It

follows that Ew = ∑ u∈Fk p ∑ w∈Fn p v(u)ωuTBw+wTAwe|w⟩ = ∑ u∈Fk p ∑ w[E],w[Ec_]

v(u)ωs[E]·w[E]+uTB(w,w[Ec])+(w[E],w[Ec])TA(w[E],w[Ec])|w[E] + l[E], w[Ec]⟩, and so ⟨w′_{, Ew⟩ =} ∑ u,u′∈Fk p ∑ w′_=w+l ωrv′(u′)v(u),

where

r =s[E]· w[E] + uTB[X|E]w[E] − uTB[X|E]l[E] + uTB[X|Ec]w[Ec]

+ w[E]TA[E|E]w[E] − l[E]TA[E|E]w[E] − w[E]TA[E|E]l[E] − l[E]TA[E|E]l[E]

+ w[Ec]TA[Ec|E]w[E] − w[Ec]TA[Ec|E]l[E]

+ w[E]TA[E|Ec]w[Ec]− l[E]TA[E|Ec]w[Ec]

+ w[Ec]TA[Ec|Ec]w[Ec]− u′TB[X|E]w′[E]− u′TB[X|Ec]w′[Ec]

− w′_[E]T

A[E|E]w′[E]− w′[E]TA[E|Ec]w′[Ec]

− w′_[Ec_]T_A[Ec_|E]w′_{[E]− w}′_[Ec_]T_A[Ec_|Ec_]w′_[Ec_]

=s[E]· w[E] + (u − u′)TB[X|E]w[E] + (u − u′)TB[X|Ec]w[Ec]

− uT

B[X|E]l[E] − l[E]TA[E|E]l[E] − l[E]TA[E|E]w[E] − w[E]TA[E|E]l[E] − l[E]T_A[E|Ec_]w[Ec_{]− w[E}c_]T_A[Ec_|E]l[E]

=s[E]· w[E] + (u − u′)TB[X|E]w[E] + (u − u′)TB[X|Ec]w[Ec]

− uT

B[X|E]l[E] − l[E]TA[E|E]l[E]

− l[E]T_{(A + A}T_{)[E|E]w[E] − l[E]}T_{(A + A}T_)[E|Ec_]w[Ec_]

and the previous lemmas, we can simplify the inner product ⟨w′, Ew⟩ as follows: ⟨w′_{, Ew⟩ =} ∑ u,u′∈Fk p ∑ w=w′ ωrv′(u′)v(u) = ω−l[E]TA[E|E]l[E]∑

u,u′ v′(u′)v(u) · [ ∑ w[E]

ωs[E]·w[E]+(u−u′)TB[X|E]w[E]−l[E]T(A+AT)[E|E]w[E]−uTB[X|E]l[E]

]

·

[ ∑ w[Ec_]

ω(u−u′)TB[X|Ec]w[Ec]−l[E]T(A+AT)[E|Ec]w[Ec]

] = pn−d+1ω−l[E]TA[E|E]l[E]∑ u,u′ v′(u′)v(u)· [ ∑ w[E]

ω(s[E]−l[E]T(A+AT)[E|E])w[E]

]

= pn−d+1ω−l[E]TA[E|E]l[E]

[ ∑ w[E]

ω(s[E]−l[E]T(A+AT)[E|E])w[E]

]

⟨v′_{, v⟩ = 0.}

The result of theorems 5.4 and 5.5 shows that if (A, B) is an [[n, k, d]]p-quantum

pair, then Q is a quantum [[n, k, d]]p-code. Below we quote Theorem 4.1 and 4.2 to

prove a stronger result, which was given in [6].

Theorem 5.6. If (A, B) is a pure [[n, k, d]]p-quantum pair, then Q is a pure quantum

[[n, k,≥ d]]p-code.

Proof. Order the vectors in _Fk

p as u1, u2, . . . , upk, and define functions v_i : Fn_p → C

by vi(w) = 1 √ pnω uT iBw+w T_Aw for w_{∈ F}n

p. Then by Theorem 5.4, Q has the following orthonormal basis

  vi = ∑ w∈Fn p vi(w)|w⟩|1 ≤ i ≤ pk   . By theorem 4.2, it suffices to show that

1 pn ∑ w[Ec_]=w′_[Ec_] ωuTiBw+wTAwωuj T_Bw′_+w′T_Aw′ = { 0, w′[E]̸= w[E]

where the sum is over w, w′ ∈ Fn

p such that with w[Ec] = w′[Ec], and the two

prefixed parts w[E] and w′[E] of E. This follows by the following computation

vi(w)vj(w′)

=ωujTBw′+w′TAw′−(uiTBw+wTAw)

=ωuTjB(w′[E],w[Ec])−uTiB(w[E],w[Ec])+(w′[E],w[Ec])TA(w′[E],w[Ec])−(w[E],w[Ec])TA(w[E],w[Ec])

=ωuTjB[X|E]w′[E]−uTiB[X|E]w[E]+w′[E]TA[E|E]w′[E]−w[E]TA[E|E]w[E]

· ω{(uj−ui)TB[X|Ec]+(w′[E]−w[E])T(A+AT)[E|Ec]}w[Ec]_.

Now we take the part related to Ec_{, and we have}

∑ w[Ec_]

ω{(uj−ui)TB[X|Ec]+(w′[E]−w[E])T(A+AT)[E|Ec]}w[Ec]_.

Since (A, B) is a pure [[n, k, d]]p-quantum pair, then (5.2) implies that ui = uj,

w′[E] = w[E] and the summation becomes pn−d+1_{. Otherwise, the summation}

becomes 0 by lemma 3.24 again, completing the proof.

Actually, the previous result also shows that, by defining fi : Fnp → Fp by

fi(x) = uTi Bx + xTAx for ui ∈ Fkp(1 ≤ i ≤ pk), we can obtain an [[n, k, d]]p

-quantum code Q = span{∑_x∈Fn p ω

fi(x)|x⟩|1 ≤ i ≤ pk} (see [6]). Now, let R be the

(k + n)× (k + n) matrix with rows and columns indexed by (X ∪ Y ) of the form

R = ( 0 B BT _{A + A}T ) . (5.3)

Then lemma 5.3 is equivalent to that the sub-matrix

R[X∪ E|Ec] = (

B[X|Ec_]

(A + AT_)[E|Ec_]

)

of R has rank k + d−1 for any E ⊆ Y with |E| = d−1. Hence we have the following corollaries from the previous theorem. The corollaries are used for MDS quantum codes (i.e. the codes reaching Singleton bound).

Corollary 5.7. Suppose n = k + 2(d− 1), and the square (k + d − 1) × (k + d − 1)

sub-matrix R[X_{∪ E|E}c_{] is invertible for any E ⊆ Y with |E| = d − 1. Then Q is a}

Proof. By Lemma 5.3 and Theorem 5.6, Q is a pure quantum [[n, k, t]]p-code for

some t ≥ d. By using quantum Singleton bound and the assumption, we have

n− 2d + 2 = k ≤ n − 2t + 2, so t = d.

Here we give examples for the applications of theorems 5.4 and 5.5 and corollary 5.7.

Example 5.8. Let X = _{x0}, Y = {y0, y1, y2, y3, y4} and E a 2-subset of Y .

Con-sider the graph G1 with vertex set V (G1) = X∪ Y as below:

.. x0 _. y0 . y1 . y2 . y3 . y4 Figure 1. G₁

Then its adjacency matrix is

R =                 x0 y0 y1 y2 y3 y4 x0 0 1 1 1 1 1 y0 1 0 1 0 0 1 y1 1 1 0 1 0 0 y2 1 0 1 0 1 0 y3 1 0 0 1 0 1 y4 1 1 0 0 1 0                 ,

which is of the form (5.3) with

A =       0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0      , B = ( 1 1 1 1 1)

Then we can check that f (|a⟩) = ∑ w∈F5 p ωa(1,1,1,1,1)·w+wTAw|w⟩ = ∑ w∈F5 p

ωa∑5i=1ui+∏i(mod 5)uiui+1|w⟩,

where a = 0, 1, 2,· · · , p − 1, form a basis of Q = f((Cp₎⊗1_{), and dim Q = p by the}

orthogonality. Also, we check that for any subset E of Y with|E| = 2 and |Ec| = 3

the sub-matrix R[X∪ E|Ec_{] is invertible. According the the edge relation between}

X∪ E and Ec, there are only two situations of R[X∪ E|Ec_]:

 1 1 10 0 1 1 0 0   ,  1 1 11 0 1 1 1 0   ,

both of which have determinant 1. Hence we can construct a pure quantum [[5, 1, 3]]p

-code Q = span_{∑_x∈F5

pω

(i,i,i,i,i)·x+xT_Ax

|x⟩|0 ≤ i ≤ 4, i ∈ Fp} explicitly.

Here is an example from [1]. We interpret it with the graph-theoretical method.

Example 5.9. Let X = ϕ, Y = {y0, y1, y2, y3, y4, y5} and G′1 be the graph with

vertex set V (G′₁) = Y as below.

.. y5 _. y0 . y1 . y2 . y3 . y4 Figure 2. G′₁

Then consider its adjacency matrix R = A + AT =                 y0 y1 y2 y3 y4 y5 y0 0 1 0 0 1 1 y1 1 0 1 0 0 1 y2 0 1 0 1 0 1 y3 0 0 1 0 1 1 y4 1 0 0 1 0 1 y5 1 1 1 1 1 0                 , with A =         0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0         .

For any E _{⊆ Y with |E| = 3, we have rank(R[E|E}c_{]) = 3 since the row vectors of}

R[E|Ec_{] are never linearly dependent for any choice of E. Hence we can construct}

a pure quantum [[6, 0, 4]]2-code

Q = span{∑ x_∈F6 2 (−1)f (x)|x⟩}, where f (x) = xTAx = x0x1+ x1x2+ x2x3+ x3x4+ x4x0+ x5(x0 + x1+ x2+ x3+ x4).

Example 5.10. [4] For (n, k, d) = (6, 2, 3), we can not construct binary(p = 2)

quan-tum code this way, because these parameters violates Hamming bound when p = 2. However, we can use this method to construct non-binary quantum codes with these parameters as follows: Let (n, k, d) = (6, 2, 3), p = 3 and consider the graph G2 with

vertex set V (G2) = X ∪ Y , where X = {x0, x1}, Y = {y0, y1, y2, y3, y4, y5} and the

.. x0 . x1 . y0 . y1 . y2 . y3 . y4 . y5 Figure 3. G2

Then its weighted adjacency matrix is

R = ( 0 B BT A + AT ) =                       x0 x1 y0 y1 y2 y3 y4 y5 x0 0 0 1 1 1 1 1 1 x1 0 0 1 −1 1 −1 1 −1 y0 1 1 0 −1 1 0 1 1 y1 1 −1 −1 0 1 −1 0 0 y2 1 1 1 1 0 1 −1 0 y3 1 −1 0 −1 1 0 −1 −1 y4 1 1 1 0 −1 −1 0 −1 y5 1 −1 1 0 0 −1 −1 0                       ,

which is in the form (5.3) with

A =         0 −1 1 0 1 1 0 0 1 −1 0 0 0 0 0 1 −1 0 0 0 0 0 −1 −1 0 0 0 0 0 −1 0 0 0 0 0 0         and B = ( 1 1 1 1 1 1 1 −1 1 −1 1 −1 ) .

For any E ={yi, yj}(0 ≤ i ̸= j ≤ 5), it is clear that rank(B[X|Ec]) = 2. The column

space of (A + AT_)[E|Ec_{] is spanned by{(1, 0)}T_{, (0, 1)}T_{}. So rank((A+A}T_)[E|Ec_{]) =}

2. In addition, the intersection of their row spaces contains only the zero vector, we can conclude that (A, B) is a pure [[6, 2, 3]]p-quantum pair for p = 3. Hence

Q = span{∑_x∈F6 3ω uT iBx+x T_Ax |x⟩|1 ≤ i ≤ 9, ui ∈ F23} is a pure quantum [[6, 2, 3]]3

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