Quantum codes from codes over Gaussian integers with respect to the Mannheim metric

0.13 MB·English

by  Murat Güzeltepe

Quantum codes from codes over
Gaussian integers with respect to the
Mannheim metric
Mehmet O¨zen, Murat Gu¨zeltepe
2 Department of Mathematics, SakaryaUniversity,TR54187 Sakarya, Turkey
1
0
2 Abstract

n
In this paper, some nonbinary quantum codes using classical codes
a
J over Gaussian integers are obtained. Also, some of our quantum codes
are better than or comparable with those known before, (for instance
3
1 [[8,2,5]]4+i).

]
T AMS Classification: 94B05, 94B60, 81P70
I Keywords: Nonbinary quantum codes, MDS codes, Mannheim metric, Hamming
.
s metric.
c
[

1 Introduction
2
v
4 AnimportantclassofquantumcodesareCalderbank-Shor-Steane(shortlyCSS)
5 codes. In fact, CSS codes are obtained from two classical codes such that one
7
of these codes contains the other code. Moreover, the bit flip and the phase
3
flip error correcting capacities of a CSS code depends on the classical code
.
3
that contains the other code and the dual code of the other classical code,
0
respectively [1, pp. 450-451]. The possibility of correcting decoherence errors
0
1 in entangled states was discovered by Shor [2] and Steane [3]. Binary quantum
: CSScodeshavebeenconstructedinseveralways(forinstance[3,10,11,12]). In
v
[10], good quantum codes of minimum distance three and four for such length
i
X n are obtained via Steane’s construction and the CSS construction. In [11],
r a large number of good quantum codes of minimum distance five and six by
a
Steane’sConstructionweregiven. In[12],somequantumerrorcorrectingcodes,
including an optimal quantum code [[27,13,5]] , were presented. Later, some
results were generalized to the case of nonbinary stabilizer codes [5, 6, 7, 8]. A
connection between classical codes and nonbinary quantum codes was given in
[5, 6, 7]. However,the theory explained in [5, 6, 7] is not nearly as complete as
inthe binarycase. The closesttheorytothe binarycaseofnonbinarystabilizer
codes was presented in [8].
On the other hand, the Mannheim metric was introduced by Huber in [9].
It is well known that the Euclideanmetric is the relevantmetric for maximum-
likelihood decoding. Although the Mannheim metric is a reasonable approx-
imation to it, it is not a priori, a natural choice. However, the codes being
1
proposed are very useful in coded modulation schemes based on quadrature
amplitude modulation (QAM)-type constellations for which neither Hamming
nor Lee metric is appropriate.
The rest of this paper is organized as follows. In section 2, classical codes
over Gaussian integer ring with respect to the Mannheim metric are given.
In Section 3, error bases are defined and quantum codes with respect to the
Mannheim distance are constructed.
2 Codes over Gaussian integers
Gaussian integers are a subset of complex numbers which have integers as real
and imaginary parts. Let p = a2 + b2 = ππ = N(π) 1 (mod4), where
≡
π =a+ib is a Gaussian integer, π =a ib denotes the conjugate of π and p is
−
anoddprimeinteger. Here,N(π)denotestheconjugateofπ. LetGdenotesthe
Gaussian integers and G the residue class of G modulo π, where the modulo
π
function µ:G G is defined according to
π
→
ςπ
µ(ς)=ς mod π =ς π. (1)
− ππ
(cid:20) (cid:21)
[] denotes rounding ofcomplex numbers. The rounding ofcomplex numbers to
·
Gaussian integers can be done by rounding the real and imaginary parts sepa-
ratelytotheclosestinteger. Hence,G becomesafinitefieldwithcharacteristic
π
p. Let α,β G and γ = β α (modπ). Then, the Mannheim weight of γ is
π
∈ −
defined as w (γ) = Re(γ) + Im(γ). Also, the Mannheim distance d be-
M m
| | | |
tween α and β is defined as d (α,β)=w (γ). Let C be code oflength n over
M M
G and let c = c , c , , c be a codeword. Then the Mannheim
π 0 1 n−1
···
n−1
weight of c is eq(cid:0)ual to (Re(c ) + Im(cid:1)(c )). Note that A Mannheim errorof
i i
| | | |
i=0
weightone takes on onePof the four values 1, i [9]. It is well known that the
± ±
Hamming weight of c is the number of the non-zero entries of c. We give an
example to compare a classical code with respect to these metrics.
Example 1 Let p=17. Then,
G = 0, 1, i, 2, 2i (1+i), (1 i), (2 i), (1+2i) .
4+i
{ ± ± ± ± ± ± − ± − ± }
Let the generator matrix of C over G be 1+i, 1 . Then, the set of
4+i
−
the codewords of C is
(cid:0) (cid:1)
(0,0), ( 1+i,1), (1 i, 1), ( 1 i,i),
− − − − −
(1+i, i), ( 1 2i,2), (1+2i, 2), (2 i,2i),
 − − − − − 
C = ( 2+i, 2i), ( 2,1+i), (2i,1 i), (2, 1+i), .
 −(2, 1 −i), (−i,2 i), (i, 2−+i), (1,−1+2i), 
− − − − −
( 1, 1 2i)
The minimum−Ma−nnh−eim distance of the code C is 3 and the minimumHam-
ming distance of the code C is 2. Let us assume that at the receiving end we
get the vector r = ( 1+i, 0 ). The minimum Mannheim distance between r
−
and the codewords of C is 1, namely, d (r,( 1+i, 1 )) = 1. Thus, we can
M
−
correct this error with respect to the Mannheim metric. But, we can not correct
2
this error with respect to the Hamming metric since d (r,( 1+i, 1 )) = 1
H
−
and d (r,( 0, 0 ))=1.
H
We now define a block code C of length n over G as a set of codewords
π
c= c , c , ,c withcoefficients c G . Letα ,α G be two
0 1 n−1 i π 1 2 π
··· ∈ ∈
differentelementsofordersp 1suchthatαp−1/4 =iandαp−1/4 = i. Hence,
(cid:0) (cid:1)− 1 2 −
xp−1/4 i and xp−1/4+i are factored as (x α ) x α5 x αp−4 and
− − 1 − 1 ··· − 1
(x α ) x α5 x αp−4 ,respectively. Also(cid:0),thepo(cid:1)lyno(cid:16)mialsxp−1(cid:17)/2+1
− 2 − 2 ··· − 2
and xp−1(cid:0) 1 are(cid:1)fact(cid:16)ored as (cid:17)
−
(x α ) x α5 x αp−4 (x α ) x α5 x αp−4 (2)
− 1 − 1 ··· − 1 − 2 − 2 ··· − 2
(cid:0) (cid:1) (cid:16) (cid:17) (cid:0) (cid:1) (cid:16) (cid:17)
and
xp−1/2+1 xp−1/2 1 , (3)
−
(cid:16) (cid:17)(cid:16) (cid:17)
respectively. A monic polynomialg(x) inG [x] is the generatorpolynomialfor
π
acycliccodeifandonlyif g(x) xn 1,whereG [x]isthesetofallpolynomials
π
| ±
with coefficients inG . Hence, Using (2), wealwayscanconstructtwo classical
π
codes C , C of length n=(p 1)/2 over G such that C C .
1 2 π 2 1
− ⊂
3 Nonbinary quantum CSS codes
Let p be an odd prime, let p = ππ 1 (mod4). A p-ary quantum code
≡
Q of length n and size K is a K dimensional subspace of a pn dimensional
− −
Hilbertspace. ThisHilbertspaceisidentifiedwiththen foldtensorproductof
p dimensional Hilbert space, that is, ( p)⊗n = p p− p, where denotes
− C C ⊗C ···C C
complex numbers. We denote by u the vectorsofa distinguishedorthonormal
| i
basis of p, where the labels u range over the elements of the finite field H .
π
C
For u = u , u , ,u ,v = v , v , ,v Gn, let
0 1 ··· n−1 0 1 ··· n−1 ∈ π
u v = u v be the usual inner product on Gn. For (u v),(u′ v′) G2n, set
· (cid:0)i i (cid:1) (cid:0) π | | (cid:1) ∈ π
(u v) (u′ v′) = Tr(vu′ v′u), where Tr : Gπk Gπ is the trace map. For
the| int∗ePger|k =1 then (u−v) (u′ v′)=(vu′ v′u→). Let w =(u v) (u′ v′)=
′ ′ |′ ∗ | − | − |
(u u v v ) = (w w ) (mod π), for i = 0,1,2,...,n 1, we define the
i − i| i − i i| i −
Mannheim weight of w and the Mannheim distance between (u v) and (u′ v′)
| |
as
Re(w ) + Im(w ) + + Re(w ) + Im(w )
0 0 n−1 n−1
wtM(w)= +| Re w|′ |+ Im |w′ ···+ | + Re w|′ | + Im |w′ 2
&" 0 0 ··· n−1 n−1 #, '
(cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:16) (cid:17)(cid:12)
dM((u|v),(u′(cid:12)(cid:12)|v′)) = w(cid:12)(cid:12)tM((cid:12)(cid:12)w), respe(cid:12)(cid:12)ctively. (cid:12)(cid:12)Let C ⊂ G2π(cid:12)(cid:12)n. (cid:12)(cid:12)Then the du(cid:12)(cid:12)al
code C⊥∗ of C is defined to be
C⊥∗ = (u v) G2n :(u v) (u′ v′)=0forall(u′ v′) C .
| ∈ π | ∗ | | ∈
Definition 1 T(cid:8)heunitaryoperatorsweredefinedin[8]asX u = ((cid:9)a+u) , Z u =
a b
| i | i | i
ξ(bu) u , where a,b are elements of the finite fields F , and ξ is a primitive pth
p
| i
root of unity.
3
Definition 2 WedefinetheunitaryoperatorsasX u = µ(a+u) , Z u =
a b
ξµ−1(bu) u , where a,b G , ξ is a primitive pthroot|oif un|ity, andthiefun|ctiion
π
| i ∈
µ:F G defines µ(g)=g [gπ/p]π.
p π
→ −
Also, we define the Hadamard gate as
1
H = (a ), a =ξ(s−1)(t−1) (mod p), 1 s,t p=π.π =N(π).
gate s,t s,t
N(π) ≤ ≤
For exampple, let π =2+i. Then,
1 1 1 1 1
1 ξ ξ2 ξ3 ξ4
1  
H = 1 ξ2 ξ4 ξ ξ3 .
gate
√5 1 ξ3 ξ ξ4 ξ2 
 
 1 ξ4 ξ3 ξ2 ξ 
 
 
Note that H H† = H† H = I , where H† denotes the conjugate
gate gate gate gate p gate
transpose of H and I denotes the identity matrix in p dimensions.
gate p
Theorem 1 (CSS Code Construction) Let C and C denote two classical lin-
1 2
ear codes over G with the parameters [n,k ,d ] and [n,k ,d ] such that
π 1 M1 π 2 M2 π
C C . Then, there exists an [[n,k k ,d ]] quantum code with minimum
2 ⊆ 1 1− 2 M π
distance d = min d ,d⊥ , where d⊥ denotes the minimum Mannheim
M M1 M2 M2
distance of the dual code C⊥ of the code C .
(cid:8) 2 (cid:9) 2
Proof. Let x be a codeword of C . Then, we define the quantum state
1
1
x+C = x+y ,
2
| i √C | i
2 yX∈C2
′
where + is bitwise addition modulo π. If x is an element of C such that
1
′ ′
x x C then, x+C = x +C , and thus the state x+C depends
2 2 2 2
− ∈ | i | i
only upon the coset of C1/C2.(cid:12)(cid:12) The nuEmber of cosets of C2 in C1 is equal to
C /C sothe dimensionofth(cid:12)e quantumcodeisN(π)k1−k2. Hence,wedefine
1 2
| | | |
the quantum code Q as the vector space spanned by t he state x+C
C1,C2 | 2i
for all x C . Therefore, the quantum code Q is an [[n,k k ,d ]] .
∈ 1 C1,C2 1− 2 M π
We now explain the minimum Mannheim distance d of the quantum code
M
Q equals = min d ,d . Suppose that a bit flip error occurs at only
C1,C2 { M1 M2}
one qubit in n qubit and a phase flip error occurs at only one qubit in n qubit.
If x+C was the original state then the corrupted state is
2
| i
1 ξµ−1((x+y)e2) (x+y+e ) mod π .
1
√C2 yX∈C2 b | i
b
To detect where bit flip error occurred it is convenient to introduce an ancilla
containing sufficient qubits to store the syndrome for the code C , and initially
1
intheallzerostate 0 . Weusereversiblecomputationtoapplytheparitycheck
| i
matrixH forthecodeC ,taking x+y+e 0 to x+y+e H (x+y+e ) =
1 1 1 1 1 1
| i| i | i| i
4 b b b
x+y+e H (e ) , since (x+y) C is annihilated by the parity check ma-
1 1 1 1
| i| i ∈
trix. The effect of this operation is to produce the state:
b b
1 ξµ−1((x+y)e2) x+y+e H (e ) .
1 1 1
√C2 yX∈C2 b | i| i
b b
Error detection for the bit flip error is completed by measuring the ancilla to
obtainthe resultH (e )anddiscardingthe ancilla. This showsthat the bitflip
1 1
error correcting capacity of the quantum code Q depends on the classical
C1,C2
code C . We now show the phase flip error correcting capacity of the quantum
1 b
code Q depends on the dual code C⊥ of the classical code C . The latest
C1,C2 2 2
state of the corrupted state, discarding the ancilla, is:
1 ξµ−1((x+y)e2) x+y .
√C2 yX∈C2 b | i
We apply the Hadamard gates to each qubit, taking the state to
1 ξµ−1((x+y)(e2+z)) z ,
C N(π)n b | i
2 Xz yX∈C2
p
′
wherethesumisoverallpossiblevaluesfornbitz. Settingz z+e (modπ),
2
≡
we obtain
1 µ−1 xz′ ′
ξ (cid:16) (cid:17) z +e . b
n 2
N(π) /C
2 z′X∈C2⊥ (cid:12) E
(cid:12)
Note that if z′ Cp⊥ then ξµ−1(yz′) =(cid:12) C ,band if z′ / C⊥ then
∈ 2 y∈C2 | 2| ∈ 2
y∈C2ξµ−1(yz′) =0. ThislookPsjustlikeabitfliperrordescribedbythe vector
e . To determine the error e , we introduce an ancilla qubit and reversibly
2 2
aPpply the parity check matrix H for C⊥ to obtain H e , and correct the error
2 2 2 2
e , obtaining the state
b2 b
b
b 1 ξxz′ z′ .
n
N(π) /C
| 2|z′X∈C2⊥ (cid:12) E
(cid:12)
p (cid:12)
The error correcting is completed by applying the inverse Hadamard gates,
H† , toeachqubit. Thistakesus backtothe initialstate withe =0. Hence,
gate 2
the proof is completed.
We use the Mannheim metric to determine the positions and the value of
b
the errors e ,e .
1 2
Let the minimum Mannheim distance of C and C⊥ be d . Then, the
1 2 m
number ofbtheberrors corrected by the quantum code QC1,C2 obtained from the
classical codes C , C is equal to
1 2
n n n
4 +42 + +4t ,
1 2 ··· t
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
where t= (dm 1)/2 and the symbol · gives the binomial coefficient.
⌊ − ⌋
(cid:18) · (cid:19)
5
Theorem 2 Let C = C C⊥ be code in G2n such that C C⊥∗, where C
2 1 π ⊂ 1
and C denote two classical codes, and C⊥ denotes the dual code of C . Then,
2 (cid:0) (cid:12) (cid:1) 1 1
there exists an [[n,K,dM]]π(cid:12) quantum code with the minimum distance
d =min wt (w): w C⊥∗ C ,
M M
∈ \
where K =dim(C⊥∗) dim(C)(cid:8). (cid:9)
−
The proof of Theorem 2 can be easily seen from the proof of Theorem 1.
Example 2 Let π = 4 +i. Let the generator polynomial of the code C be
1
g (x)=1+2i+( 1+i)x ix2+x3 and let the generator polynomial of the code
1
− −
C beg (x)=1 i+(2 i)x+( 1+i)x2 ix3 ix4+x5. Hence,usingthecodes
2 2
− − − − −
C and C we obtain a quantum code with parameters [[8,2,5]] with respect
1 2 4+i
to the Mannheim metric since the minimum distance of C and C⊥ are 5. Let
1 2
the quantum state ψ = 1 i, 2 i, 1+i, i, i, 1, 0, 0 . If
| i − − − − −
the operator IIIX X III acts on this state, then the corrupted state becomes
1 1 (cid:12) (cid:11)
1 i, 2 i, 1+i, (cid:12)1 i, 1 i, 1, 0, 0 . Thequantumcode[[8,2,5]]4+i
− − − − −
with respect to the Mannheim metric overcomes this error since the minimum
(cid:12) (cid:11)
(cid:12)Mannheim distance of the classical code C is equal to 5. Also, the number of
1
the bit flip errors corrected by this quantum code is 480.
On the other hand, let F be a finite field of characteristic 17 and let
17
C , C be the classical codes with respect to the Hamming metric such that
1 2
C C . Then a quantum code with parameters [[8,2,4]] can be obtained.
2 1 17
⊂
The code [[8,2,4]] is a maximum distance separable (shortly MDS) since this
17
code attains the quantum singleton bound, namely, 172 =178−2.4+2. Also, the
number of the bit flip errors corrected by this quantum code is 128. For the
length n = 8, a quantum code having the minimum distance greater than 4 is
not obtained with respect to the Hamming metric. So, it is obvious that the
quantumcodeobtainedhereisbetterthanthequantumcodeofthesamelength
with respect to the Hamming metric.
In Table I, some CSS codes compared with respect to the Hamming metric
and the Mannheim metric are given. The CSS codes constructed from classical
codes with respect to the Hamming metric can be found in the HM column of
Table I. The CSS codes constructed from classical codes with respect to the
MannheimmetriccanbefoundintheMMcolumnofTableI.Itisobviousthat,
some ofthe quantumcodes obtainedin this paper cancorrectmore errorsthan
the quantumMDS codes ofthe samelength giveninTable I. Using a computer
program, we compute the minimum Mannheim distance of the codes given in
Table I.
6
TableI:SomeCSScodescomparedwithrespecttotheHammingmetricandtheMannheim
metric.
p α1 α2 h1 g2 HM MM
5 i −i x3−ix2−x+i x3+ix2−x−i [[4,2,2]] [[4,2,2]]
2+i 2+i
13 2 -2 (1−i)−x+x2 (1−i)+x+x2 [[6,2,3]] [[6,2,4]]
3+2i 3+2i
13 2 -2 −i−2x+2ix2+x3 1−i+x+x2 [[6,1,3]] [[6,1,4]]
3+2i 3+2i
13 2 -2 x−2 x+2 [[6,4,2]] [[6,4,2]]
3+2i 3+2i
13 2 -2 −1+ix+x2 −i+(−1+i)x+x2 [[6,2,2]] [[6,2,2]]
3+2i 3+2i
−1+i+(2−i)x
−i+(−2i)x
17 1+i −2+i +(1−i)x2−ix3 [[8,1,4]] [[8,1,5]]
+x3+x4 4+i 4+i
+ix4+x5
(−2+i)+(1+i)x (2−i)+(1+i)x
17 1+i −2+i [[8,2,4]] [[8,2,5]]
+(2−i)x2+x3 −(2−i)x2+x3 4+i 4+i
17 1+i −2+i −1+(1+i)x+x2 −1−(1+i)x+x2 [[8,4,3]] [[8,4,≥3]]
4+i 4+i
17 1+i −2+i −(1+i)+x 1+i+x [[8,6,2]] [[8,6,≥2]]
4+i 4+i
19 −1+i −2+i −2+x 2+x [[14,12,2]] [[14,12,≥2]]
5+2i 5+2i
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