Download Construction X for quantum error

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Construction X for quantum error-correcting codes
Petr Lisoněk · Vijaykumar Singh
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Abstract Construction X is known from the theory of classical error control
codes. We present a variant of this construction that produces stabilizer quantum error control codes from arbitrary linear codes. Our construction does not
require the classical linear code that is used as an ingredient to satisfy the dual
containment condition. We prove lower bounds on the minimum distance of quantum codes obtained from our construction. We give examples of record breaking
quantum codes produced from our construction.
Keywords Quantum error-correcting code · Linear code
Mathematics Subject Classification (2000) 94B05 · 94B15
1 Quantum stabilizer codes
∗
Let Fq denote the finite field with
Pqn elements and Fq := Fq \ {0}. For a ∈ F4 let
a = a2 . For x, y ∈ Fn
let
hx,
yi
=
x
y
be
their
Hermitian
inner product. For
4
i=1 i i
n
n
⊥h
:= {u ∈ F4 : (∀x ∈ C) hu, xi = 0} be the Hermitian
a subspace C ⊂ F4 let C
dual of C. For two subspaces U, V ⊂ Fn
4 denote U + V := {u + v : u ∈ U, v ∈ V }.
We will write W = U ⊕ V if W = U + V and U ∩ V = {0} and we will say that
U ⊕ V is the direct sum of U and V .
For x ∈ Fn
4 let wt(x) denote the Hamming weight (or just weight) of x, that
is, the number of non-zero coordinates of x. By an [n, k]q linear code we mean
a k-dimensional subspace of Fn
q . By an [n, k, d]q linear code we mean an [n, k]q
linear code C such that the minimum weight of any non-zero vector in C is d. For
a linear code C we denote by wt(C) the minimum weight of any non-zero vector
Research reported in this paper was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Collaborative Research Group “Mathematics
of Quantum Information” of the Pacific Institute for the Mathematical Sciences (PIMS).
P. Lisoněk · V. Singh
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
E-mail: [email protected], [email protected]
2
Petr Lisoněk, Vijaykumar Singh
in C. As is well known, wt(C) also equals the minimum Hamming distance of any
two distinct codewords in C.
A quantum error-correcting code (or just quantum code) is a code that protects
quantum information from corruption by noise (decoherence) on the quantum
channel in a way that is similar to how classical error-correcting codes protect
information on the classical channel. We recommend [3] as an excellent reference
on binary quantum error-correcting codes. In accordance with literature we denote
by [[n, k, d]] the parameters of a binary quantum code that encodes k logical qubits
into n physical qubits and has minimum distance d. (Sometimes such parameters
are denoted [[n, k, d]]2 to emphasize that the quantum code is binary; however as
we only deal with binary quantum codes in this paper, we will omit the subscript 2
throughout.) The minimum distance of a quantum code C determines the error
detection/correction capabilities of C in a way that is similar to how the minimum
distance determines such capabilities of classical codes: For fixed n and k, the
higher d is, the more error control the code achieves.
An important class of quantum codes are stabilizer quantum codes. Binary
stabilizer quantum codes are equivalent to quaternary additive codes, that is,
additive subgroups of Fn
4 [3]. If we further restrict our attention to linear subspaces
of Fn
4 , then the following theorem expresses the parameters of the quantum code
that can be constructed from a classical linear, dual containing code.
Theorem 1 [3, Theorem 2] [1, Theorem 17.24] Given a linear [n, k, d]4 code C
such that C ⊥h ⊆ C, we can construct an [[n, 2k − n, d]] quantum code.
2 A variant of Construction X
Construction X is well known from the theory of classical error control codes, see
for example [5, §18.7] or [1, Section 14.1]. In general this construction does not
produce dual containing codes, which however is a requirement for an application
of Theorem 1 to construct quantum codes. In this section we give a variant of
Construction X that produces quantum codes from arbitrary quaternary linear
codes.
Theorem 2 For an [n, k]4 linear code C denote e := n − k − dim(C ∩ C ⊥h ).
Then there exists an [[n + e, 2k − n + e, d]] quantum code with d ≥ min{wt(C),
wt(C + C ⊥h ) + 1}.
We will now work towards the proof
2. For x ∈ Fn
4 let ||x|| = hx, xi
Pnof Theorem
3
be the norm of x. Note that ||x|| = i=1 xi can only attain values 0 or 1. More
specifically, ||x|| equals the parity of wt(x). Also
||x + y|| = ||x|| + ||y|| + Tr(hx, yi)
where Tr(a) := a + a is the trace from F4 to F2 . A subset S ⊂ Fn
4 is called
orthonormal if hx, yi = 0 for any two distinct x, y ∈ S and hx, xi = 1 for any
x ∈ S.
Proposition 1 Let D be a subspace of Fn
4 and assume that M is a basis for
D ∩ D⊥h . Then there exists an orthonormal set B such that M ∪ B is a basis for
D.
Construction X for quantum error-correcting codes
3
Proof First note that if S is a subspace of Fn
4 such that there exist x, y ∈ S
with hx, yi =
6 0, then there exists z ∈ S with ||z|| = 1. Indeed for c ∈ F∗4 we
have ||cx + y|| = ||x|| + ||y|| + Tr(c hx, yi), and if c assumes all values in F∗4 ,
then Tr(c hx, yi) attains both values in F2 . Hence there exists c ∈ F∗4 such that
||cx + y|| = 1.
Now let W be a subspace of Fn
4 such that
D = (D ∩ D⊥h ) ⊕ W.
(1)
Let ` := dim(W ). For each 0 ≤ i ≤ ` we will construct an orthonormal set Si that
is a basis for an i-dimensional subspace Ti of W such that
W = Ti ⊕ (Ti⊥h ∩ W ).
(2)
The process is iterative. Take S0 := ∅ and suppose that for some 0 ≤ i < ` the
set Si is an orthonormal basis for Ti such that dim(Ti ) = i and (2) holds. Let x be
a non-zero vector in Ti⊥h ∩ W . Then there exists y ∈ Ti⊥h ∩ W such that hx, yi =
6 0.
Indeed if such y did not exist, then by (2) and (1) we would have that x ∈ D⊥h ,
which however would be a contradiction to (1) being a direct sum since we know
that x ∈ W . By the first paragraph of this proof there exists z ∈ Ti⊥h ∩ W with
||z|| = 1. Set Si+1 := Si ∪ {z}. Clearly Si+1 is an orthonormal set. Let Ti+1 be
the subspace spanned by Si+1 . By (2) we have z 6∈ Ti , hence dim(Ti+1 ) = i + 1.
We want to show that
⊥h
∩ W ).
(3)
W = Ti+1 ⊕ (Ti+1
⊥h
⊥h
First we show that Ti+1 ∩ Ti+1
∩ W = {0}. Let v ∈ Ti+1 ∩ Ti+1
∩ W . Since
⊥h
v ∈ Ti+1 , we have v = u + cz where u ∈ Ti and c ∈ F4 . Since v ∈ Ti+1 , we have
for each w ∈ Ti and each d ∈ F4
0 = hu + cz, w + dzi = hu, wi + d hu, zi + c hz, wi + cd||z|| = hu, wi + cd.
We must have c = 0 or else hu, wi + cd would not stay constant as d runs through
F4 . Thus hu, wi = 0 for all w ∈ Ti , hence u ∈ Ti⊥h . Since u ∈ Ti and Ti ∩Ti⊥h = {0}
⊥h
by (2) it follows that u = 0. Hence v = 0 and Ti+1 ∩ Ti+1
∩ W = {0}.
⊥h
Next we show that W = Ti+1 + (Ti+1 ∩ W ). Let w ∈ W . By assumption
W = Ti + (Ti⊥h ∩ W ). Hence there exist vectors x ∈ Ti and y ∈ Ti⊥h ∩ W such
that w = x + y. Let x0 := x + hy, zi z and y 0 := y − hy, zi z. Clearly x0 ∈ Ti+1 and
for any u + dz ∈ Ti+1 (where u ∈ Ti and d ∈ F4 ) we have
hy − hy, zi z, u + dzi = hy, ui + d hy, zi − hy, zi hz, ui − d hy, zi ||z||
= d hy, zi − d hy, zi = 0,
⊥h
⊥h
thus y 0 ∈ Ti+1
∩ W . Hence W = Ti+1 + (Ti+1
∩ W ). This completes the proof
that (2) implies (3) assuming that the vector z is chosen as described above. Such
choice of z is always possible and moreover it can be done efficiently as it only
involves basic linear algebra. After completing the `-th step of this process, we
have T` = W and S` is an orthonormal basis for T` . This completes the proof of
Proposition 1.
u
t
4
Petr Lisoněk, Vijaykumar Singh
2.1 Proof of Theorem 2
Assume the notation as in Theorem 2. We will now prove this theorem.
Note
e = dim(C ⊥h ) − dim(C ∩ C ⊥h ) = dim(C + C ⊥h ) − dim(C).
Denote s := dim(C ∩ C ⊥h ). Let G be a block matrix


Ms×n
0s×e
G =  A(n−e−2s)×n 0(n−e−2s)×e 
Be×n
Ie×e
where the subscripts denote the sizes of the blocks, and 0, I are the zero matrix
and the identity matrix respectively. For a matrix P let r(P ) denote the set of
rows of P . The matrix G is constructed such that r(M ) is a basis for C ∩ C ⊥h ,
r(M ) ∪ r(A) is a basis for C, r(M ) ∪ r(B) is a basis for C ⊥h , and r(B) is an
orthonormal set. The existence of such B follows from Proposition 1. Note that
r(M ) ∪ r(A) ∪ r(B) is a basis for C + C ⊥h .
Let E be the linear code for which G is a generator matrix. Let S denote the
union of the first s rows of G and the last e rows of G, i.e., S is the set of rows of
the matrix
Ms×n 0s×e
.
Be×n Ie×e
By construction, each vector in S is orthogonal to each row of G, thus each vector
in S belongs to E ⊥h . Also, the vectors in S are linearly independent. Since
dim(E ⊥h ) = n + e − (n − s) = e + s = |S|
it follows that S is a basis for E ⊥h . Since S is a subset of E by construction, it
follows that E ⊥h ⊆ E.
Let x be a non-zero vector in E and in accordance with the vertical block
2
e
structure of G write x = (x1 | x2 ) where x1 ∈ Fn
4 and x ∈ F4 . So x is a linear combination of rows of G. If none of the last e rows of G enters this linear combination
with a non-zero coefficient, then x1 ∈ C \ {0}, thus wt(x) = wt(x1 ) ≥ wt(C). If
some of the last e rows of G enter this linear combination with a non-zero coefficient, then x1 ∈ C + C ⊥h \ {0} and wt(x) = wt(x1 ) + wt(x2 ) ≥ wt(C + C ⊥h ) + 1.
Thus E is an [n + e, k + e, d]4 code with d ≥ min{wt(C), wt(C + C ⊥h ) + 1} and
E ⊥h ⊆ E. An application of Theorem 1 to the code E now finishes the proof of
Theorem 2.
3 Finding ingredient codes
In Theorem 2 we have e = 0 if and only if C ⊥h ⊆ C, in which case we simply
obtain Theorem 1. On the other hand, if e is large then wt(C + C ⊥h ) + 1 will be a
pessimistic lower bound on the minimum weight of E. Thus it seems reasonable to
focus on codes for which e is positive but small. Thus characterizing such codes is
an interesting problem as it leads to applications of our construction. This problem
appears to be similar to one of central problems in quantum coding theory, namely
characterizing dual containing (or, equivalently, self-orthogonal) linear codes, and
Construction X for quantum error-correcting codes
5
one may hope that similar techniques will be applicable to tackling both problems
for a given class of codes. This is indeed the case for the class of cyclic codes, as
we will show in this section.
For the sake of brevity we will abbreviate “linear quaternary cyclic code” to
“cyclic code.” Recall that if C is a cyclic code of block length n with n odd, then C
is given by a principal ideal hg(x)i in F4 [x]/(xn − 1) where g is called the generator
polynomial for C. Let m be the order of 4 modulo n, then F4m is the splitting field
of xn − 1 containing a primitive nth root of unity, let us call it β. Then a defining
set of C is the set {k : g(β k ) = 0, 0 ≤ k < n}; note that this depends on the
choice of β. We can make amcanonical choice for β by fixing a primitive element α
of F4m and letting β := α(4 −1)/n . Thus we now need a canonical choice for α; in
Section 4 we let α be defined by the PrimitiveElement function in Magma.
For n and m as above and a ∈ {0, . . . , n − 1} the set {a4j mod n : 0 ≤ j < m}
is called a cyclotomic coset modulo n. It is well known that a defining set of a cyclic
code of length n is the union of some cyclotomic cosets modulo n. Let Zn denote
the set of integers modulo n. Clearly defining sets should be considered as subsets
of Zn . For S ⊂ Zn denote S := Zn \ S and −2S := {(−2s) mod n : s ∈ S}.
Proposition 2 If C is a quaternary linear cyclic code with defining set Z, then
dim(C ⊥h ) − dim(C ∩ C ⊥h ) = |Z ∩ −2Z|.
Q
Proof Let n be the block length of C. Let k∈Z (x − β k ) be the generator polyQ
nomial for C, then the generator polynomial for C ⊥h is k∈−2Z (x − β k ) and the
Q
generator polynomial for C ∩ C ⊥h is k∈Z∪−2Z (x − β k ). Then
dim(C ⊥h ) − dim(C ∩ C ⊥h ) = n − | − 2Z| − (n − |Z ∪ −2Z|)
= |Z ∪ −2Z| − | − 2Z| = |Z ∩ −2Z|.
u
t
The fact that any defining set is the union of some cyclotomic cosets modulo n
allows us to design a simple backtracking algorithm to enumerate all cyclic codes
with a given block length and a given upper bound on the value dim(C ⊥h ) −
dim(C ∩ C ⊥h ) = |Z ∩ −2Z| which is denoted e in Theorem 2. We start with Z = ∅
and we add one cyclotomic coset to Z at a time. We note that if Z 0 ⊇ Z, then
|Z 0 ∩ −2Z 0 | ≥ |Z ∩ −2Z|, which gives an easy backtracking rule for enforcing an
upper bound on |Z ∩ −2Z|.
3.1 A special lower bound for certain cyclic codes
In cases when the defining set of a cyclic code has a certain special structure we
can prove the following refinement of Theorem 2. Note that if n is divisible by 3,
then {0}, { n3 } and { 2n
3 } are singleton cyclotomic cosets modulo n and moreover
each of them is fixed under the map S 7→ −2S that operates on subsets S of Zn .
Theorem 3 Assume that n is divisible by 3 and let C be an [n, k]4 cyclic code
with defining set Z such that Z ∩ −2Z ⊆ {0, n3 , 2n
3 }. Denote e := |Z ∩ −2Z|. Then
there exists an [[n + e, 2k − n + e, d]] quantum code with d ≥ min{wt(C), wt(Cu ) +
1, wt(C + C ⊥h ) + 2} where the minimum is taken over the cyclic codes Cu with
defining set Z \ {u} for each u ∈ Z ∩ −2Z.
6
Petr Lisoněk, Vijaykumar Singh
Proof Denote n = 3` and let ω denote a primitive cube root of unity in F4 . For
t ∈ {0, 1, 2} define the polynomials
bt (x) :=
x3` − 1
x−β
tn
3
`−1
=
X 3i+2
x3` − 1
t 3i+1
2t 3i
.
=
x
+
ω
x
+
ω
x
x − ωt
i=0
Pn−1
i
As is common in the theory of cyclic codes, with a polynomial
i=0 ai x ∈
n
n
F4 [x]/(x − 1) we associate the vector (a0 , a1 , . . . , an−1 ) ∈ F4 and with a slight
abuse of notation we use the same symbol for both objects. Note that {b0 , b1 , b2 }
is an orthonormal set since n is odd.
We now follow the structure of the proof of Theorem 2, where for the rows of
⊥h
B we take the set U = {bt : tn
\ C for each
3 ∈ Z ∩ −2Z}. Note that bt ∈ C
bt ∈ U and the vectors in U are linearly independent, as required in the first part
of the proof of Theorem 2.
For the second part of the proof (the claim about the minimum distance of the
quantum code), let E be the quaternary linear code as introduced in the proof of
Theorem 2. Let x be a non-zero vector in E and consider x as a linear combination
of rows of G. This time we split the proof into three cases: (i) No rows of B enter
the linear combination with a non-zero coefficient, (ii) exactly one row of B enters
the linear combination with a non-zero coefficient, (iii) at least two rows of B
enter the linear combination with a non-zero coefficient. In cases (i) and (iii) the
argument is the same as in the proof of Theorem 2, with the obvious adjustment
of the term +1 to +2. In case (ii) let bt denote the row of B that enters the linear
combination with a non-zero coefficient. Then x belongs to the span of C and bt ,
which is easily seen to be precisely the cyclic code with the defining set Z \ { tn
3 }:
Q
note that bt is divisible by k∈Z\{ tn } (x − β k ).
u
t
3
4 New quantum codes
Just as a case study, in order to demonstrate that our results do produce quantum
codes with better parameters than hitherto known codes, we applied the backtracking algorithm outlined in the previous section to enumerate cyclic codes C
with e ≤ 3 and modest block length. Then all codes appearing in the lower bounds
of Theorems 2 and 3 are cyclic. We computed the minimum weight of these codes
using the built-in function in Magma [2]. In Table 1 we report some examples of
new quantum codes whose minimum distance is higher than the minimum distance of the best known quantum code with the same length and dimension as
recorded in [4]. We denote
S by Cy(n ; a1 , . . . , at ) the cyclic code with block length
n and the defining set ti=1 Cai , where Ca denotes the cyclotomic coset modulo n
containing a.
It should be noted that secondary constructions such as [3, Theorem 6] applied
to codes listed in Table 1 produce many more record breaking codes; for example
our [[53, 17, 10]] code alone leads to at least 28 new record breaking codes improving
the lower bounds of [4].
In the course of our computations we have also come across dual containing
cyclic codes that produce quantum codes improving the lower bounds of [4] merely
by applying Theorem 1; these codes are listed in Table 2.
Construction X for quantum error-correcting codes
quantum code
[[52, 26, 7]]
[[53, 17, 10]]
[[54, 8, 12]]
[[54, 24, 8]]
[[65, 31, 9]]
[[86, 8, 19]]
[[86, 12, 17]]
[[86, 24, 15]]
[[86, 56, 8]]
[[92, 30, 14]]
[[92, 48, 10]]
C
Cy(51 ;
Cy(51 ;
Cy(51 ;
Cy(51 ;
Cy(63 ;
Cy(85 ;
Cy(85 ;
Cy(85 ;
Cy(85 ;
Cy(91 ;
Cy(91 ;
7
0, 1, 6, 35)
0, 1, 2, 6, 17, 22)
0, 1, 3, 9, 17, 22, 34, 35)
0, 1, 3, 17, 34, 35)
0, 2, 3, 11, 15, 31, 42)
0, 10, 13, 15, 18, 21, 29, 34, 37, 41, 57)
0, 2, 6, 7, 9, 10, 14, 18, 30, 41)
0, 7, 9, 15, 17, 18, 21, 37, 57)
0, 7, 30, 34, 57)
0, 2, 3, 9, 14, 34)
0, 1, 14, 19, 39)
Table 1 New quantum codes from Theorems 2 and 3.
quantum code
[[85, 9, 19]]
[[85, 13, 17]]
[[93, 3, 21]]
C
Cy(85 ; 3, 7, 9, 10, 17, 19, 21, 30, 37, 57)
Cy(85 ; 3, 10, 13, 19, 21, 29, 30, 37, 57)
Cy(93 ; 1, 5, 9, 13, 17, 23, 33, 34, 45)
Table 2 New quantum codes from Theorem 1 (C ⊥h ⊂ C).
5 Conclusion
We have presented a construction of stabilizer quantum codes from linear quaternary codes that do not need to satisfy the dual-containment (equivalently, selforthogonal) requirement. We have given a general lower bound on the distance of
the resulting quantum code, as well as a more specialized lower bound applicable
to certain cyclic codes. We have shown that these lower bounds produce record
breaking codes. We noted that it is an interesting problem to characterize classes
of codes that are “nearly missing” the dual-containment condition, as such codes
can be used as ingredients to our construction. We provided such characterization
for cyclic codes.
Also, one should note that there is a considerable freedom in choosing the
orthonormal set B whose existence is proved in Proposition 1, and in general
different choices of set B will lead to inequivalent codes. A randomized method
for constructing the set B can be extracted from our proof of Proposition 1; this
can be then used in searches for good codes.
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