Non-Additive Quantum Codes from Goethals and Preparata Codes

0.17 MB·English

by  Markus Grassl

Non-Additive Quantum Codes from
Goethals and Preparata Codes
Markus Grassl Martin Ro¨tteler
Institute for Quantum Optics and Quantum Information NEC Laboratories America, Inc.
Austrian Academy of Sciences 4 Independence Way, Suite 200
Technikerstraße 21a, 6020 Innsbruck, Austria Princeton, NJ 08540, USA
Email: [email protected] Email: [email protected]
8
0 Abstract—Weextendthestabilizerformalismtoaclassofnon- codes have the same parameters as the additive CSS codes
0
additive quantum codes which are constructed from non-linear that can be obtained using the same underlying binary codes.
2
classicalcodes.Asanexample,wepresentinfinitefamiliesofnon-
In this paper, we extend the approach of combining unitary

n additive codes which are derived from Goethals and Preparata
images of stabilizer states to arbitrary stabilizer codes as
a codes.
startingpoint.Forastabilizercode[[n,k,d]],thesearchgraph
J
I. INTRODUCTION has only 2n−k vertices, and a clique of size K yields a
4
1 Recently,severalnewnon-additivequantumerror-correcting quantumcodeofdimensionK×2k.Whatismore,wepresent
codes(QECCs)havebeenconstructedthathavehigherdimen- some infinite families of non-additive quantum codes which

] sionthanadditiveQECCswiththesamelengthandminimum are constructed using non-linear binary codes, avoiding the
h
p distance[5],[16],[17].Thefirstexampleofsuchacodeisthe NP-hard problem of finding a maximal clique in the search
- code ((5,6,2)) of [11] which has been found via numerical graph. Finally, we show how to obtain encoding circuits for
t
n optimization. Afterwards, the code has been identified as the the resulting non-additive codes using encoding circuits for
a span of a particular state and its image under five unitary the underlying stabilizer codes.
u
transformations (see also [7]). The recently discovered codes
q ((9,12,3)) and ((10,24,3)) (see [16], [17]) start with a so- II. ABRIEFREVIEWOFTHESTABILIZERFORMALISM
[
called graph state which corresponds to a stabilizer state, i.e., We start with a brief review of the stabilizer formalism for

1 a stabilizer code with parameters [[n,0,d]] (see [8], [13]). A quantumerror-correctingcodesandtheconnectiontoadditive
v
basis of the quantum code is obtained by this initial state codes over GF(4) (see, e.g., [4], [6]). A stabilizer code
4
4 together with its image under some tensor products of Pauli encoding k qubits into n qubits having minimum distance d,
1 matrices and identity (Pauli operators). The distance between denoted by C = [[n,k,d]], is a subspace of dimension 2k of
2 any pair of these states can be defined as the minimal weight thecomplexHilbertspace(C2)⊗n ofdimension2n.Thecode
1. of a Pauli operator transforming one state into the other (see is the joint eigenspace of a set of n−k commuting operators
0 below). The problem of finding a code of high dimension S ,...,S which are tensor products of the Pauli matrices
1 n−k
8 can be stated as finding a maximal clique in a search graph (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
0 0 1 0 −i 1 0
whose vertices are all images of the initial state. There is σ = , σ = , σ = ,
: x 1 0 y i 0 z 0 −1
v an edge between two states if their distance is at least the
i prescribed minimum distance. Using the formalism of graph or identity. The operators S generate an Abelian group S
X i
states, Cross et al. show in [5] that it is sufficient to consider with 2n−k elements, called the stabilizer of the code. It is
r
a Z-only operators in order to define the search graph, but this a subgroup of the n-qubit Pauli group Pn which itself is
has the disadvantage that the distance between two of these generated by the tensor product of n Pauli matrices and
statesisnotnecessarilyequaltothenumberofZ operatorsin identity. We further require that S does not contain any non-
the tensor product. Moreover, constructing a code ((n,K,d)) trivialmultipleofidentity.ThenormalizerofS inP ,denoted
n
requires to find a clique of size K in a search graph with byN,actsonthecodeC =[[n,k,d]].Itispossibletoidentify
2n vertices. Fixing one basis state, the graph can be slightly 2k logical operators X ,...,X and Z ,...,Z such that
1 k 1 k
simplified. However, for the code ((10,24,3)) the simplified these operators commute with any element in the stabilizer S,
graph still has 678 vertices and 149.178 edges. and such that together with S they generate the normalizer
A different approach for constructing non-additive QECCs N of the code. The operators X mutually commute, and so
i
based on Boolean functions and projection operators has do the operators Z . The operator X anti-commutes with the
j i
been presented in [1]. Evaluating the Boolean function at the operator Z if i=j and otherwise commutes with it.
j
projectionoperators,thecodeisgivenasthesumoftheimage It has been shown that the n-qubit Pauli group corresponds
of products of the projections. Finally, we mention the non- toasymplecticgeometryandthatonecanreducetheproblem
additive QECCs obtained by the method given in [12]. Those of constructing stabilizer codes to finding additive codes over
GF(4) that are self-orthogonal with respect to a symplectic As the elements of the normalizer N commute with all
inner product [3], [4]. Up to a scalar multiple, the elements elements of the stabilizer S, two elements t and t define
1 2
of P can be expressed as σaσb where (a,b) ∈ F2 is a the same character if t−1t ∈N. Hence the set of characters
1 x z 2 2 1
binary vector. Choosing the basis {1,ω} of GF(4), where corresponds to the cosets of N in P . If T is a set of coset
n
ω is a primitive element of GF(4) with ω2+ω+1=0, we representatives, we can write the decomposition of the full
get the following correspondence between the Pauli matrices, space as
(cid:77)
elements of GF(4), and binary vectors of length two: (C2)⊗n = tC. (2)
operator GF(4) F2 t∈T
2
I 0 (00) Note that measuring the eigenvalues of the generators Si
σ 1 (10) projects onto one of these space tC, corresponding to the
x
σ ω2 (11) character χt given by the sequence of eigenvalues. In terms
y
σ ω (01) of the classical codes, the eigenvalues correspond to an error-
z
syndrome which is obtained by computing the symplectic
This mapping extends naturally to tensor products of n Pauli
inner product of the received vector with the n−k vectors
matrices being mapped to vectors of length n over GF(4) or
correspondingtothegeneratorsofthestabilizer,i.e.,abasisof
binary vectors of length 2n. We rearrange the latter in such a thecodeC.ForallvectorsofthedualcodeC∗ corresponding
way that the first n coordinates correspond to the exponents
tothenormalizerN,theinnerproductiszero.Sothedifferent
of the operators σx and write the vector as (a|b), i.e., spaces tC correspond to cosets C∗+t of the code C∗.
g =σa1σb1 ⊗...⊗σanσbn =ˆ (a|b)=(gX|gZ). (1) As for a fixed code C two spaces t1C and t2C are either
x z x z
identical or orthogonal, we can define the distance of them as
Two operators corresponding to the binary vectors (a|b) and follows:
(c|d) commute if and only if the symplectic inner product
dist(t C,t C):=min{wgt(p):p∈P |pt C =t C}. (3)
a · d − b · c = 0. In terms of the binary representation, 1 2 n 1 2
the stabilizer corresponds to a binary code C which is self- Here wgt(p) is the number of tensor factors in the n-qubit
orthogonal with respect to this symplectic inner product, and Pauli operator p that are different from identity. Clearly,
the normalizer corresponds to the symplectic dual code C∗. dist(t C,t C) = dist(t−1t C,C). The distance (3) can also
1 2 2 1
ThestabilizertogetherwiththelogicaloperatorsZi generatea be expressed in terms of the associated vectors over GF(4).
self-dual code. In terms of the correspondence to vectors over Lemma 1: The distance of the spaces t C and t C equals
1 2
GF(4),thestabilizerandnormalizercorrespondtoanadditive theminimumweightinthecosetC∗+t −t ,whereweuset
1 2 i
code over GF(4) and its dual with respect to an symplectic todenotebothann-qubitPaulioperatorandthecorresponding
inner product, respectively, which we will also denote by C vector over GF(4).
and C∗. The minimum distance d of the quantum code is Proof: Direct computation shows
given as the minimum weight in the set C∗ \ C which is dist(t C,t C)=dist(C∗+t ,C∗+t )
1 2 1 2
lower bounded by the minimum distance d∗ of C∗. If d=d∗,
=dist(C∗+(t −t ),C∗)
the code is said to be pure, and for d ≥ d∗, the code is said 1 2
to be pure up to d∗. =min{wgt(c+t1−t2):c∈C∗}
=min{wgt(v):v ∈C∗+t −t }.
1 2
III. THEUNIONOFSTABILIZERCODES
With this preparation, we are ready to present the general
Note that we have defined a stabilizer code C as the joint
construction of the union of stabilizer codes (see also [7]).
eigenspace of the commuting operators S generating the
i The quantum code will be defined as the span of some of the
stabilizer S. The term stabilizer suggests that the code is
summands in (2).
the joint +1 eigenspace of the operators. However, for each
Definition 2 (union stabilizer code): Let C = [[n,k,d ]]
0 0
of the generators S we may choose either the eigenspace
i be a stabilizer code and let T = {t ,...,t } be a subset
0 1 K
with eigenvalue +1 or the eigenspace with eigenvalue −1.
of the coset representatives of the normalizer N of the code
This gives rise to a decomposition of the space (C2)⊗n into 0
C in P . Then the union stabilizer code is defined as
2n−k mutuallyorthogonalspaceswhichcanbelabeledbythe 0 n
(cid:77)
eigenvalues of the n−k generators Si, or equivalently, by the C = tC0.
characters χ of the stabilizer group S. Moreover, the n-qubit t∈T0
Pauli group Pn acts transitively on these spaces. WithoutlossofgeneralityweassumethatT0 containsidentity.
From now on we fix the code C as the the joint +1 With the union stabilizer code C we associate the (in general
eigenspace corresponding to the trivial character. Let t ∈ Pn non-additive) union normalizer code given by
be an arbitrary n-qubit Pauli operator. Then we can define a (cid:91)
C∗ = C∗+t={c+t :c∈C∗,i=1,...,K},
character of S on the generators Si as follows 0 i 0
(cid:40) t∈T0
χt(Si):= +1 if t and Si commute, where C0∗ denotes the additive code associated with the
−1 if t and Si anti-commute. normalizer N0 of the stabilizer code C0. We will refer to
 
SX SZ i=i(cid:48), we get
1 1
 . . 
 .. ..  (cid:104)c |t†Et |c (cid:105)=±(cid:104)c |E|c (cid:105). (5)
  j i i j(cid:48) j j(cid:48)
 
 SnX−k SnZ−k  As the code C0 is pure up to the minimum distance of C0∗ ⊂
 ZX ZZ  C∗, equation (5) vanishes as well.
 .1 .1  Remark 4: Wenotethatsimilartostabilizercodes,thetrue
 . . 
 . .  minimum distance of a union stabilizer code might be higher.
 
 ZX ZZ  The true minimum distance is given by
 k k 
  min{dist(c+ti,c(cid:48)+ti(cid:48)):ti,ti(cid:48) ∈T0,
 X Z 
 X..1 X..1  c,c(cid:48) ∈C0∗|c+ti−(c(cid:48)+ti(cid:48))∈/ C(cid:101)0}
 . .  =min{wgt(v):v ∈(C∗−C∗)\C(cid:101)0},
X Z
X X
k k where (C∗−C∗)={a−b:a,b∈C∗} denotes the set of all
 
 tX1.. tZ1..  dthifefeardednictievseocflovseucrtoerosfinCC∗.0∗, and C(cid:101)0 is the symplectic dual of
. .
 tX tZ  IansotardbeilrizteorccoondsetruCct uannidonusqeuaantsuemarcchodgersa,pwhewmhaoysestavretrtwicieths
K K 0
are the mutually orthogonal translates {tC : t ∈ T } of the
0 0
Fig.1. Arrangementsofthevectorsassociatedwithaunionstabilizercode. stabilizer code. Two vertices are connected by an edge if and
only if the distance between them is at least d, where d is
the desired minimum distance. For simplicity, we also require
both, the vectors t and the corresponding unitary operators,
i that the code C is pure up to d ≥ d. The distance between
0 0
as translations.
two translates can be computed using Lemma 1. This allows
A union stabilizer code can be defined in terms of binary
to use stabilizer codes C of arbitrary dimension, and hence
0
vectors as shown in Fig. 1. The first n−k rows correspond to
allows to go beyond the case of stabilizer states (or graph
the binary vectors (cf. (1)) associated with the generators S
i states) as, e.g., in [5], [17]. We note that the construction of
of the stabilizer S of the code C . They generate the classical
0 non-additive quantum codes of [2] is also based on taking the
code C . The next k rows correspond to the logical operators
0 union of orthogonal images of a stabilizer code.
Z , followed by the k logical operators X . The last K rows
j i
correspond to the K translations ti defining the cosets of the IV. UNIONSTABILIZERCODESFROMBINARYCODES
classical code C∗ and the unitary images of the stabilizer
0 A. CSS-like codes
code C , respectively. We use curly brackets to stress the fact
0
Given two linear binary codes C = [n,k ,d ] and C =
that the set of operators T need not be closed under group 1 1 1 2
0 [n,k ,d ] with C⊥ ⊂ C , the so-called CSS construction
operation. In general, the quantum code is not invariant under 2 2 2 1
(see, e.g., [14]) yields a quantum error-correcting code C =
these generalized logical X-operators. On the other hand, if
[[n,k +k −n,d]] with d ≥ min(d ,d ). Starting with this
T is closed under group operation, the resulting code will be 1 2 1 2
0
CSS code, we consider unions of cosets of the binary codes
a stabilizer code where a basis of T defines an additional set
0
C , i.e.,
of logical X-operators. i (cid:91)
Theorem 3: LetC beaunionstabilizercodeasinDefinition C(cid:101)i = Ci+t(i)
2. The dimension of C is |T0|2k = K2k, and the minimum t(i)∈Ti
distance is lower bounded by the minimum distance d of the such that the minimum distance of the codes C(cid:101)i is at least
union normalizer code C∗. d(cid:101)≤ d. Using the translations {(t(1)|t(2)) : t(1) ∈ T1,t(2) ∈
Proof: As T0 is a subset of the coset representatives T } we obtain a CSS-like union stabilizer code of dimension
2
of the normalizer N0, the spaces tiC0, each of which has |T1|·|T2|·2k1+k2−n whose minimum distance is at least d(cid:101).
doifmtheensuionnion2kc,oadreeims Kutu2akl.lyFioxritnhgogaonnoarlt.hHoneonrcmeatlhebadsiimse{n|csio(cid:105)n: If G1 = (cid:0)GH122(cid:1) and G2 = (cid:0)GH211(cid:1) are generator matrices of
j the codes C , where H is a generator matrix of the dual code
i i
j = 1,...,2k} of the stabilizer code C0, the set {ti|cj(cid:105) : i = C⊥, the corresponding vectors are as shown in Fig. 2.
1,...,K, j =1,...,2k}isanorthonormalbasisoftheunion i
stabilizercode.LetE ∈P beann-qubitPaulierrorofweight B. Enlargement construction
n
0 < wgt(E) < d. For basis states |ci,j(cid:105) = ti|cj(cid:105) ∈ tiC0 and Steane has presented a construction that allows to increase
|ci(cid:48),j(cid:48)(cid:105)=ti(cid:48)|cj(cid:48)(cid:105)∈ti(cid:48)C0 we consider the inner product the dimension of a CSS code [14]. For this, he starts with the
(cid:104)c |E|c (cid:105)=(cid:104)c |t†Et |c (cid:105). (4) CSSconstructionappliedtoabinarycodeC =[n,k,d]which
i,j i(cid:48),j(cid:48) j i i(cid:48) j(cid:48) contains its dual, yielding a CSS code C = [[n,2k−n,d]].
0
Fori(cid:54)=i(cid:48),wehavedist(t C ,t C )=min{wgt(c+t −t ): Using a code C(cid:48) = [n,k(cid:48) > k+1,d(cid:48)] which contains C, he
i 0 i(cid:48) 0 i i(cid:48)
c ∈ C∗} ≥ d, and hence (4) vanishes for wgt(E) < d. For obtainsaquantumcode[[n,k+k(cid:48)−n,min(d,(cid:100)3d(cid:48)/2(cid:101))]].The
0
 
H 0      
2 X Z 00 I0 00 I0
 
 0 H1   Z   00 0I   00 0I 
   −Q→1  −Q→c  
 G 0 
 12  X 0I 00 0I 00
  (cid:110) (cid:111) (cid:110) (cid:111)  
 0 G21  T0 T0Q1  c1...0 0...0 
 tt(1(1....11)) tt(K(1....222))  Fenigc.od3i.ngcTirracnusitfsorQm1ataionndoQfct.he union stabilizer code gcivKen0by th0e0inverse
. .
. .
 tt(K(...111)) tt((1...22))  fthroeSmteenRalnaeeregdhe-maMseucnlotlencrsotcnrousdctretuescd[t1iao5n]f.aomTfhil[ey1c4oo]fdteaosdtdahrieteivocehbatqaiuninaenodtfuacmpopdcleoysdinegs
K1 K2 RM(r,m)⊂RM(r,m)⊥ =RM(m−r−1,m)
Fig. 2. Arrangements of the vectors associated with a CSS-like union ⊂RM(m−r,m).
stabilizercode.
Inparticular,forr =2andm≥5thisyieldsadditiveQECCs
C =[[2m,2m−(cid:0)m(cid:1)−2m−2,6]], while using only the CSS
resulting code can also be considered as a union stabilizer co1nstruction, one o2btains C =[[2m,2m−2(cid:0)m(cid:1)−2m−2,8]].
code. If D is a generator matrix of the complement of C 0 2
in C(cid:48) and A is a fixed point free, invertible linear map, the AstheGoethalscodeG(m)istheunionofKG =2(m2)−2m+2
translations can be defined as cosets of R(m), we can construct a CSS-like union stabilizer
code based on C . The minimum distance of the resulting
T ={(vD|vAD):v ∈Fk(cid:48)−k}. 0
0 2 non-additive code is 8 and its dimension is K2 dim(C ) =
G 0
The key observation [14] for proving the lower bound on 22m−6m+2.
the minimum distance is that the weight of an operator g = Replacing the Goethals code by the Preparata code P(m),
(gX|gZ) can be expressed in terms of the Hamming weight we have KP = 2(m2)−m+1 cosets of R(m). This results in a
of the binary vectors and their sum: CSS-like union stabilizer code with minimum distance 6 and
wgt((gX|gZ))= 1(wgt(gX)+wgt(gZ)+wgt(gX +gZ)). dimension KP2 dim(C0)=22m−4m.
Both the union stabilizer codes based on Goethals codes
2
andthosebasedonPreparatacodesaresuperiortotheadditive
As T is closed under addition and the properties of A ensure
0
that0(cid:54)=tX (cid:54)=tZ (cid:54)=0foranynon-zeroelement(tX|tZ)∈T , codesderivedfromReed-Mullercodes.Theparametersofthe
0
the weight of all three binary vectors is lower bounded by d(cid:48). first codes in these families are as follows:
enlarged RM Goethals Preparata
V. QUANTUMCODESFROMREED-MULLER,GOETHALS,
ANDPREPARATACODES [[64,35,6]] ((64,230,8)) ((64,240,6))
UsingtheCSS-likeconstructionoftheprevioussection,we [[256,210,6]] ((256,2210,8)) ((256,2224,6))
now construct some families of non-additive quantum codes.
[[1024,957,6]] ((1024,2966,8)) ((1024,2984,6))
For this, we use the Goethals codes G(m) and the Preparata
codes which are nonlinear binary codes of length n=2m for However, applying the enlargement construction to extended
m ≥ 4, m even. Some of the properties of these codes are primitive BCH codes results in stabilizer codes with parame-
summarized as follows (see, e.g., [10]): ters [[2m,2m −5m−2,8]] and [[2m,2m −3m−2,6]] (see
• Both the Goethals code G(m) and the Preparata code [14]).
P(m) are unions of cosets of the Reed-Muller code
VI. ENCODINGCIRCUITS
R(m) := RM(m−3,m). Furthermore, they are nested
In [9] we have shown how to compute a quantum circuit
subcodes of RM(m−2,m), i.e.,
consisting of Cliffordgates only that transformsany stabilizer
RM(m−3,m)⊂G(m)⊂P(m)⊂RM(m−2,m). S given by the binary (n−k)×2n matrix (X|Z) into the
stabilizer of a trivial code given by (0|I0), where I is an
• The parameters of the codes are
identity matrix of size n−k. The corresponding trivial code
RM(m−3,m)=R(m)=[2m,2m−(cid:0)m(cid:1)−m−1,8] corresponds to the mapping |φ(cid:105) (cid:55)→ |0...0(cid:105)|φ(cid:105). We denote
2
G(m)=(2m,22m−3m+1,8) the resulting quantum circuit that corresponds to the inverse
P(m)=(2m,22m−2m,6) encoding circuit of (X|Z) by Q1. Note that we can apply Q1
to all the operators defining the code as illustrated in Fig. 3.
RM(m−2,m)=[2m,2m−m−1,4].
Further note, that for the trivial stabilizer code, the “encoded”
 X Y Y X H • • • (cid:103) (cid:103) |0(cid:105)
 Z Z • H •(cid:103) (cid:103) (cid:103) |0(cid:105)
|ψ (cid:105) Z P • H • • 
i
 Z••• P ••• HH • (cid:103) • • (cid:103) |i(cid:105),i=0,...,5
Fig.4. Inverseencodingcircuitforthenon-additivecode((5,6,2)).ThefirstsetofgatesincludingtheHadamardtransformationsimplementstheinverse
encodingcircuitQ1 forthestabilizercode[[5,0,3]],followedby5CNOTand2ToffoligatesimplementingtheclassicalcircuitQc.
X-andZ-operatorsareweight-onePaulioperatorsσx andσz, ACKNOWLEDGMENTS
respectively, acting on the last k qubits. As the transformed
We acknowledge fruitful discussions with Vaneet Aggarwal
translations TQ1 define cosets of the normalizer, we may
0 and Robert Calderbank. Markus Grassl would like to thank
choosethemsuchthattheyaretensorproductsofoperatorsσ
x NEC Labs., Princeton for the hospitality during his visit. This
andidentityactingonthefirstn−kqubitsonly.Thenwehave
work was partially supported by the FWF (project P17838).
the trivial union code spanned by a set of K2k basis states
of the form |c (cid:105)|j(cid:105), where {|j(cid:105) : j = 0,...,2k −1} is the REFERENCES
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