A group theoretic framework is introduced that simplifies the description of
known quantum error-correcting codes and greatly facilitates the construction
of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1
error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors,
and 1 to 29 qubits correcting 5 errors.Comment: RevTex, 4 pages, no figures, submitted to Phys. Rev. Letters. We have
  changed the statement of Theorem 2 to correct it -- we now get worse rates
  than we previously claimed for our quantum codes. Minor changes have been
  made to the rest of the pape

Calderbank, A. R.

Rains, E. M

Shor, P. W.

Sloane, N. J. A.

English

arXiv

A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits correcting 5 errors

Rains, E. M.

Caltech Authors - Main

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Quantum Error Correction and Orthogonal Geometry
A. R. Calderbank,1 E. M. Rains,2 P. W. Shor,1 and N. J. A. Sloane1
1AT&T Research, Murray Hill, New Jersey, 07974
2Institute for Defense Analyses, Princeton, New Jersey, 08540
(July 2, 1996)
A group theoretic framework is introduced that simplifies
the description of known quantum error-correcting codes and
greatly facilitates the construction of new examples. Codes
are given which map 3 qubits to 8 qubits correcting 1 error,
4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2
errors, and 1 to 29 qubits correcting 5 errors.
PACS: 03.65.Bz
A quantum error-correcting code is a way of encoding
quantum states into qubits (two-state quantum systems)
so that error or decoherence in a small number of indi-
vidual qubits has little or no effect on the encoded data.
The existence of quantum error-correcting codes was dis-
covered only recently [1]. These codes have the poten-
tial to be important for quantum computing and quan-
tum communication; for example, they could be used
as repeaters for quantum cryptography. Although the
subject of quantum error-correcting codes is relatively
new, a large number of papers have already appeared.
Many of these describe specific examples of codes [1–11].
However, the theoretical aspects of these papers have
been concentrated on properties and rates of the codes
[9,12–16], rather than on combinatorial recipes for con-
structing them. This letter introduces a unifying frame-
work which explains the codes discovered to date and
greatly facilitates the construction of new examples.
The basis for this unifying framework is group theo-
retic. It rests on the structure of certain subgroups of
O(2n) and U(2n) called Clifford groups [17]. We first
construct a subgroup E of O(2n). This group E is an ex-
traspecial 2-group, and provides a bridge between quan-
tum error-correcting codes in Hilbert space and binary
orthogonal geometry. We obtain the Clifford groups by
taking the normalizer L of E in O(2n) or the normal-
izer L′ of the group E′ generated by E and iI in U(2n).
Since the natural setting for quantum mechanics is com-
plex space, it might appear more appropriate to focus
on the complex group L′. However, we shall begin by
discussing the real Clifford group L, since its structure
is easier to understand, and L is all that is required for
the construction of the known quantum error-correcting
codes.
The Group Theoretic Framework. The extraspecial 2-
group E is realized as an irreducible group of 21+2n or-
thogonal 2n× 2n matrices. The center Ξ(E) is {±I} and
the extraspecial property is that E¯ = E/Ξ(E) is elemen-
tary abelian (hence a binary vector space). Let V denote
the vector space Zn2 (where Z2 = {0, 1}) and label the
standard basis of R2
n
by |v〉, v ∈ V . Every element e of
E can be written uniquely in the form
e = X(a)Z(b)(−I)λ (1)
where λ ∈ Z2, X(a) : |v〉 → |v + a〉, Z(b) : |v〉 →
(−1)b·v|v〉, for a, b ∈ V .
Consider the jth qubit in a quantum channel which
transmits n qubits. Let vj be the vector with a 1 in
the jth bit and 0’s in the remaining bits. Then X(vj)
is the transformation which applies the Pauli matrix
σx =
(
0 1
1 0
)
to the jth qubit and does nothing to the
remaining n−1 qubits. The transformation Z(vj) applies
the Pauli matrix σz =
(
1 0
0 −1
)
to the jth bit and does
nothing to the other n − 1 qubits. In the language of
quantum error correction, X(vj) is a bit error and Z(vj)
is a phase error in the jth qubit. The element X(a)Z(b)
corresponds to bit errors in the qubits for which aj = 1
and phase errors in the qubits for which bj = 1.
Observe that E is the group of tensor products ±w1⊗
. . . ⊗ wn where each wj is one of σx, σz or σy = σxσz .
For the purposes of quantum error correction, we need
consider only errors of the types σx, σz, and σy, since
if we can correct these errors we can correct arbitrary
errors [4,9,12].
Define a quadratic form Q on E¯ = E/Ξ(E) by
Q(e¯) =
n∑
j=0
ajbj , (2)
where e = ±X(a)Z(b) is any element of E whose image
in E¯ is e¯. Then e2 = (−I)Q(e¯) and Q(e¯) = 0 or 1 ac-
cording as X(a) and Z(b) commute or anticommute. If
e = w1 ⊗ . . .⊗wn then Q(e¯) is the parity of the number
of components wj that are equal to σxσz.
The normalizer L of E in the real orthogonal group
O(R2
n
) (the subgroup of elements g such that g−1Eg =
E) acts on E by conjugation, fixing the center Ξ(E)
(g ∈ L acts on E as the permutation e→ g−1eg). Hence
there is a well-defined action of L on the binary vec-
tor space E¯ that preserves the quadratic form Q. The
quotient L/E is the orthogonal group O+(2n, 2), a fi-
nite classical group [18]. The group L appears in recent
connections between classical Kerdock error-correcting
codes, orthogonal geometry, and extremal Euclidean line
sets [17]. This group also appears [9] as the group of
1
Bell-state-preserving bilateral local transformations that
two experimenters (A and B) can jointly perform on n
pairs of particles (each pair being in a Bell state). Hence
there is a one-to-one correspondence between Bell states
and elements of E¯ (cf. Eqs. (39) and (67) of [9]). The
quadratic form Q(e¯) is 0 or 1 according as the Bell states
are symmetric or antisymmetric under interchange of A
and B. Observe that symmetry/antisymmetry of Bell
states is the physical invariant conserved by this presen-
tation of L.
The following are group elements that generate L,
shown together with their induced action on the binary
vector space E¯.
(1) H = 2−n/2[(−1)u·v]u,v∈V , which interchanges X(b)
and Z(b).
(2) Every matrix A in the general linear group GL(V )
determines a permutation matrix |v〉 → |vA〉 in O(R2
n
).
The action on E¯ induced by conjugation is X(a) →
X(aA), Z(b) → Z(bA−T). For example the quantum
xor taking |q1q2〉 → |q1(q1 ⊕ q2)〉 is represented by
(a1a2|b1b2)→ (a1a2|b1b2)


1 1 0 0
0 1 0 0
0 0 1 0
0 0 1 1

 . (3)
The back action of the xor on the phases is evident in
its effect on b1 and b2. Any orthogonal matrix in O(R
2n)
that normalizes both X(V ) and Z(V ) is of this type for
some A [17, Lemma 3.14].
(3) Hˆ2 =
1√
2
(
1 1
1 −1
)
⊗ I2n−1 , which applies a pi/2
rotation to the first qubit and leaves the other qubits
unchanged. This acts on E¯ by interchanging a1 and b1.
(4) Diagonal matrices dM = diag[(−1)
QM (v)], where
QM is a binary quadratic form on V for which the as-
sociated bilinear form QM (u + v) − QM (u) − QM (v) is
uMvT. Note that M is symmetric with zero diagonal.
The induced action on E¯ is given by
(a|b)→ (a|b)
(
I M
0 I
)
. (4)
These matrices are precisely the elements of L that in-
duce the identity on the subgroup Z(V ). In terms of
their effect on qubits, these are the transformations in L
that change the phases of the qubits while fixing their
values.
Remark. The group L′ is the normalizer of the group
E′ generated by E and iI in the unitary group U(2n).
Now we cannot define Q(e¯) = e2 because (ie)2 6= e2.
However, we still have the nonsingular alternating binary
form
(
X¯(a)Z¯(b), X¯(a′)Z¯(b′)
)
= a · b′ + a′ · b. (5)
The group L′ is generated by L and by diagonal trans-
formations dP = diag[i
TP (v)] where TP is a Z4-valued
quadratic form [17, Section 4]. The induced action on
E¯′ = E′/Ξ(E′) is described by
(a|b)→ (a|b)
(
I P
0 I
)
(6)
where P is symmetric and may have a nonzero diagonal.
For example, applying the transformation
(
1 0
0 i
)
to
each qubit corresponds to P = I. The quotient L′/E′ is
the symplectic group Sp(2n, 2) [18].
Quantum Error-Correcting Codes. A subspace S¯ of
E¯ is said to be totally singular if Q(s¯) = 0 for all s¯ ∈ S¯.
It follows that for s¯ = (a|b), s¯′ = (a′|b′) in S¯ the inner
product (s¯, s¯′) = Q(s¯+ s¯′)−Q(s¯)−Q(s¯′) satisfies
(s¯, s¯′) = a · b′ + a′ · b = 0. (7)
The group L acts transitively on totally singular sub-
spaces of a given dimension. Hence every k-dimensional
totally singular subspace is contained in the same number
of maximal (n-dimensional) totally singular subspaces. If
M¯ is a maximal totally singular subspace, then the group
M¯ has 2n distinct linear characters, and the correspond-
ing eigenspaces determine a coordinate frame F(M¯) (an
orthonormal basis of R2
n
). For example, the image of
Z(V ) in E¯ determines the coordinate frame |v〉, v ∈ V ,
and the image of X(V ) determines the coordinate frame
2−n/2
∑
v(−1)
u·v|v〉, u ∈ V . If S¯ ⊆ M¯ is a k-dimensional
totally singular subspace, then the group S¯ has 2k dis-
tinct linear characters. The 2n vectors in F(M¯) are parti-
tioned into 2k sets of size 2n−k with each set correspond-
ing to a different eigenspace. We view each eigenspace
as a quantum error-correcting code which maps n − k
qubits into n qubits. The 2n−k vectors from F(M¯) in
that eigenspace constitute the codewords.
More generally, we may use complex space, and so we
define a quantum error-correcting code encoding k qubits
into n qubits to be any 2k-dimensional subspaceC of C2
n
.
This code will protect against errors in a certain error set
E , which we take to be a subset of the extraspecial group
E. As remarked earlier, there is no loss of generality in
restricting to error sets in E.
For such a code C to protect against all errors in an
error set E , it is necessary and sufficient [9,15] that for
any two vectors |c1〉 and |c2〉 in C with 〈c1|c2〉 = 0, and
any two transformations e1 and e2 from E , we have
〈c1|e
−1
1 e2|c2〉 = 0, (8)
〈c1|e
−1
1 e2|c1〉 = 〈c2|e
−1
1 e2|c2〉. (9)
Note that since we are assuming that E is contained in
E, we can replace e−11 e2 by e1e2 in the above equations,
since e1 = ±e
−1
1 . An interesting special case occurs
when both sides of Eq. (9) are always equal to 0 for
e1 6= e2 ∈ E . This implies that there is a measurement
which will uniquely determine the error without affecting
the encoded subspace. After this measurement, the error
can subsequently be corrected by a unitary operation.
2
If both sides of Eq. (9) are not always 0, then there can
be two errors e1 and e2 between which it is impossible to
distinguish. However, these two errors are guaranteed to
have identical effects on vectors within the subspace C,
and so it is not necessary to distinguish between these
errors in order to correct the error.
We now can show the connection between orthogonal
geometry and quantum error correcting codes.
Theorem 1. Suppose that S¯ is a k-dimensional to-
tally singular subspace of E¯. Let S¯⊥ be the (2n − k)-
dimensional subspace orthogonal to S¯ with respect to
the inner product (7). Further suppose that for any two
vectors e1 and e2 in an error set E ⊆ E, either e¯1e¯2 ∈ S¯
or e¯1e¯2 /∈ S¯
⊥. Then the eigenspace C corresponding to
any character of the group S¯ is an error-correcting code
which will correct any error e ∈ E .
Proof. We first show that if e ∈ E, then e permutes
the 2k spaces Ci which are generated by the 2
k differ-
ent linear characters of S¯. Consider an element s¯ ∈ S¯
with eigenvalue λs. We write s for the associated repre-
sentation. Then for any |c〉 ∈ C we have s|c〉 = λs|c〉,
and se|c〉 = (−1)(s¯,e¯)es|c〉 = (−1)(s¯,e¯)λse|c〉, where (s¯, e¯)
is the inner product (7). Since (−1)(s¯,e¯)λs is indepen-
dent of c, this shows that the action of e permutes the
eigenspaces generated by the characters of S.
We will divide the proof into two cases, according as
e¯1e¯2 ∈ S¯ or e¯1e¯2 6∈ S¯
⊥.
Case 1. Suppose e¯1e¯2 ∈ S¯. It follows that for |c1〉 and
|c2〉 ∈ C with 〈c1|c2〉 = 0,
〈c1|e1e2|c2〉 = λe1e2〈c1|c2〉 = 0, (10)
and for all |c〉 ∈ C,
〈c|e1e2|c〉 = λe1e2〈c|c〉 = λe1e2 , (11)
establishing Eqs. (8) and (9).
Case 2. Suppose e¯1e¯2 /∈ S¯
⊥. It follows that for some
s¯ ∈ S¯, se1e2 = −e1e2s. Thus, for |c〉 ∈ C, se1e2|c〉 =
−e1e2s|c〉 = −λse1e2|c〉, so e1e2|c〉 /∈ C. Thus e1e2|c〉 is
in a different eigenspace, so
〈c1|e1e2|c2〉 = 0 (12)
for all c1, c2 ∈ C (including c1 = c2). Again Eqs. (8) and
(9) hold. 2
The quadratic form Q plays no role in this proof and
it is only necessary that S¯ satisfy (S¯, S¯) = 0 with respect
to the alternating form. This means the complex group
L′ can also be used for code construction.
The group L acts transitively on the totally singular
subspaces S¯ of dimension k, so there is some element
g ∈ L taking any particular k-dimensional totally singu-
lar subspace to the subspace corresponding to a quantum
code generated as in Theorem 1. This implies that some g
takes the canonical 2n−k-dimensional Hilbert space gen-
erated by the first n− k qubits to the encoded subspace.
Since L can be generated by xor’s and pi/2 rotations,
these quantum gates are therefore sufficient for encoding
any of these quantum codes.
Example 1. We first describe the code mapping 1
qubit into 5 qubits presented in Ref. [9], which contains
two codewords,
|c0〉 = |00000〉
+ |11000〉+ |01100〉+ |00110〉+ |00011〉+ |10001〉
− |10100〉 − |01010〉 − |00101〉 − |10010〉 − |01001〉
− |11110〉 − |01111〉 − |10111〉 − |11011〉 − |11101〉,
|c1〉 = |11111〉
+ |00111〉+ |10011〉+ |11001〉+ |11100〉+ |01110〉
− |01011〉 − |10101〉 − |11010〉 − |01101〉 − |10110〉
− |00001〉 − |10000〉 − |01000〉 − |00100〉 − |00010〉.
Essentially the same code is given in Ref. [5], but we use
the above presentation since it is fixed under cyclic per-
mutations. It is easily verified that X(11000)Z(00101)
fixes |c0〉 and |c1〉. Thus we may take the vector
(11000|00101) ∈ E¯ to be in the subspace S¯. Using the
fact that the code is closed under cyclic permutations, we
find that S¯ is a 4-dimensional totally singular subspace
generated by the vectors
11000 00101
01100 10010
00110 01001
00011 10100
(13)
[The fifth cyclic shift, (10001|01010), is also in this sub-
space.] The dual S¯⊥ is generated by S¯ and the additional
vectors (11111|00000) and (00000|11111). It is straight-
forward to verify that the minimal weight vectors in S¯⊥
have weight three [one such vector is (00111|00101)] and
thus the code can correct one error.
Example 2. Suppose we have a classical linear [n, k, d]
binary error correcting code C (i.e., it is over Zn2 , it is k-
dimensional, and it has minimal distance d, so that it
corrects t = [(d − 1)/2] errors). Suppose furthermore
that C⊥ ⊂ C. We can define a subspace S¯ to consist of
all vectors (v1|v2) ∈ E¯, where v1, v2 ∈ C
⊥. The dual S¯⊥
consists of all vectors (v1|v2) with v1, v2 ∈ C, showing
that the corresponding quantum error-correcting code
corrects t errors. The subspace S¯ is 2(n−k)-dimensional,
so the quantum code maps n − 2k qubits into n qubits.
This is the method described in Refs. [3,4].
Example 3. Consider the subspace S¯ obtained by
modifying the classical [8,4,4] Hamming code as follows:
01110100 00111010
00111010 00011101
00011101 01001110
11111111 00000000
00000000 11111111.
(14)
3
It is straightforward to verify that these vectors generate
a 5-dimensional totally singular subspace S¯ which is in-
variant under cyclic permutations of the last 7 bits, and
that S¯⊥ has minimal weight 3. This gives a quantum
code mapping 3 qubits into 8 qubits which can correct
one error. The same code was discovered by Gottesman
[11], also via group-theoretic techniques.
Example 4. By duplicating the 5-qubit code (13) and
adding two vectors, we can obtain a code which maps 4
qubits into 10 qubits and corrects one error:
01100 11110 10010 01100
00110 01111 01001 00110
00011 10111 10100 00011
10001 11011 01010 10001
11111 11111 00000 00000
00000 00000 11111 11111.
(15)
Example 5. The following construction is a gener-
alization of the 5-qubit code (13) inspired by classical
quadratic residue codes. It works for any prime p of
the form 8j + 5. We have not found good theoretical
bounds on the minimal distance, but for small primes
these codes are excellent. To construct the first vector
(a|b), put aj = 1 when j is a nonzero quadratic residue
mod p (that is, j = k2 mod p for some k) and put bj = 1
when j is a quadratic nonresidue. To obtain p−1 vectors
that generate the subspace S¯, take p − 2 cyclic shifts of
the first vector. For p = 13, the first basis vector is
0101100001101|0010011110010
and the remaining vectors are obtained by cyclic shifts.
The minimal weight of S¯⊥ was calculated by computer
to be 5. This gives a code mapping one qubit into 13
qubits which corrects 2 errors. For p = 29, ths subspace
S¯⊥ has minimal distance 11, so this construction gives a
code mapping 1 qubit to 29 which corrects 5 errors.
We next give a Gilbert–Varshamov lower bound for
the asymptotic rate of this class of codes. It matches
the Gilbert–Varshamov lower bounds known for general
quantum error-correcting codes [12].
Theorem 2. There exist quantum error-correcting
codes with asymptotic rate
R = 1− 2δ log2 3−H2(2δ) (16)
where δ is the fraction of qubits that are subject to de-
coherence and H2(δ) = −δ log2 δ − (1 − δ) log2(1 − δ) is
the binary entropy function.
Proof. Let Nk denote the number of k-dimensional
totally singular subspaces. We count pairs (e, S¯) where
e ∈ E is in E2 (i.e., e = e1e2 with e1, e2 in the error set E)
and S¯ is a k-dimensional totally singular subspace with
e¯ ∈ S¯⊥\S¯. Transitivity of L on singular points (e¯ 6= 0,
Q(e¯) = 0), and on nonsingular points (Q(e¯) 6= 0) implies
that each e ∈ E satisfies e¯ ∈ S¯⊥\S¯ for µNk subspaces
S¯, where the fraction µ ≈ 2−k. If |E2| < 2k then there
exists a k-dimensional totally singular subspace S¯ that
satisfies e¯ /∈ S¯⊥\S¯ for all e ∈ E2. Hence the achievable
rate R satisfies
1−R = log2 |E
2|/n
= log2
[
32δn
(
n
2δn
)]
/n (17)
= 2δ log2 3 +H2(2δ). 2
We would like to thank David DiVincenzo for discus-
sions about the group L as presented in [9].
[1] P. W. Shor, Phys. Rev. A. 52, 2493 (1995).
[2] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,
J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76,
722 (1996).
[3] A. R. Calderbank and P. W. Shor, Phys. Rev. A (to be
published). Also LANL e-print quant-ph/9512032.
[4] A. Steane, Proc. R. Soc. London A (to be published).
Also LANL e-print quant-ph/9601029.
[5] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek,
Phys. Rev. Lett. (to be published). Also LANL e-print
quant-ph/9602019.
[6] B. B. Plenio, V. Vedral, and P. L. Knight, LANL e-print
quant-ph/9603022 (unpublished).
[7] L. Vaidman, L. Goldenberg, and S. Wiesner, LANL e-
print quant-ph/9603031 (unpublished).
[8] P. W. Shor and J. A. Smolin, LANL e-print quant-
ph/9604006 (unpublished).
[9] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and
W. K. Wootters, LANL e-print quant-ph/9604024 (un-
published).
[10] S. L. Braunstein, LANL e-print quant-ph/9604036 (un-
published).
[11] D. Gottesman, LANL e-print quant-ph/9604038 (unpub-
lished).
[12] A. Ekert and C. Macchiavello, LANL e-print quant-
ph/9602022 (unpublished).
[13] B. Schumacher, LANL e-print quant-ph/9604023 (un-
published).
[14] B. Schumacher and M. A. Nielsen, LANL e-print quant-
ph/9604022 (unpublished).
[15] E. Knill and R. Laflamme, LANL e-print quant-
ph/9604034 (unpublished).
[16] S. Lloyd, LANL e-print quant-ph/9604015 (unpub-
lished).
[17] A. R. Calderbank, P. J. Cameron, W. M. Kantor, and
J. J. Seidel, “Z4 Kerdock codes, orthogonal spreads, and
extremal Euclidean line-sets,” Proc. London Math. Soc.
(to be published).
[18] P. B. Kleidman and M. W. Liebeck, The Subgroup Struc-
ture of the Finite Classical Groups (Cambridge Univer-
sity Press, Cambridge, England, 1990).
4


Quantum Error Correction and Orthogonal Geometry

VOLUME 78, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 20 JANUARY 1997
rror-
h map
ing 2Quantum Error Correction and Orthogonal Geometry
A. R. Calderbank,1 E. M. Rains,2 P. W. Shor,1 and N. J. A. Sloane1
1AT&T Labs—Research, Murray Hill, New Jersey 07974
2Institute for Defense Analyses, Princeton, New Jersey 08540
(Received 9 May 1996; revised manuscript received 3 July 1996)
A group theoretic framework is introduced that simplifies the description of known quantum e
correcting codes and greatly facilitates the construction of new examples. Codes are given whic
3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correct
errors, and 1 to 29 qubits correcting 5 errors. [S0031-9007(96)02177-1]
PACS numbers: 03.65.Bzn
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A quantum error-correcting code is a way of encodi
quantum states into qubits (two-state quantum systems
that error or decoherence in a small number of indiv
ual qubits has little or no effect on the encoded data. T
existence of quantum error-correcting codes was disc
ered only recently [1]. Although the subject is relative
new, a large number of papers on quantum error cor
tion have already appeared. Many of these describe
cific examples of codes [1–9]. However, the theoreti
aspects of these papers have been concentrated on pr
ties and rates of the codes [7,10–12], rather than on rec
for constructing them. This Letter introduces a unifyin
framework which explains all the codes discovered to d
and greatly facilitates the construction of new example
The basis for this unifying framework is group theoret
It rests on the structure of certain finite subgroupsE , L
in Os2nd and E0 , L0 in Us2nd [13]. Since the natura
setting for quantum mechanics is complex space, it mi
appear more appropriate to focus on the complex gro
E0 andL0. However, we shall begin by discussing the re
groupsE andL, since their structure is easier to understa
and they are sufficient for the construction of the kno
quantum error-correcting codes. We will first constru
the subgroupE of Os2nd. This groupE provides a bridge
between quantum error-correcting codes in Hilbert sp
and binary orthogonal geometry. We then construct
larger subgroupL , Os2nd as the normalizer ofE.
The groupE is the group of tensor products6w1 ≠
. . . ≠ wn where eachwj is either the identity or one o
the Pauli matrices x, sy or sz applied to thejth qubit.
Mathematically, the groupE is realized as an irreducibl
group of2112n orthogonal2n 3 2n matrices. The cente
of E, JsEd, is h6Ij and the groupE has theextraspecial
property that Ē ­ EyJsEd is elementary Abelian (henc
a binary vector space). LetV denote the vector spaceZn2
(where Z2 ­ h0, 1j) and label the standard basis ofR2
n
by jyl, y [ V . Every elemente of E can be written
uniquely in the form
e ­ XsadZsbd s2Idl, (1)
where l [ Z2, Xsad : jyl ! jy 1 al, Zsbd : jyj !
s21db?y jyl, for a, b [ V .0031-9007y97y78(3)y405(4)$10.00g
so
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Consider thejth qubit in a quantum channel which is
transmittingn qubits. Letyj be the vector with a 1 in
the jth bit and 0’s in the remaining bits. ThenXsyjd is
the transformation which applies the Pauli matrixsx ­
s 0 11 0 d to thejth qubit and does nothing to the remainin
n 2 1 qubits. The transformationZsyjd applies the Pauli
matrix sz ­ s
1 0
0 21 d to the jth bit and does nothing
to the othern 2 1 qubits. In the language of quantum
error correction,Xsyjd is a bit error andZsyjd is a phase
error in thejth qubit. The elementXsadZsbd corresponds
to bit errors in the qubits for whichaj ­ 1 and phase
errors in the qubits for whichbj ­ 1. It has been shown
that for the purposes of quantum error correction, w
need consider only errors of the typessx ­ Xsyjd, sz ­
Zsyjd, and sy ­ XsyjdZsyjd, since if we can correct
errors of these types int qubits, we can correct arbitrary
errors int qubits [4,7,10].
Our construction yields quantum codes as simultaneo
eigenspaces of the matrices in an Abelian subgroup ofE.
For this construction, we need a criterion for whether tw
elements ofE commute. Define a quadratic formQ on
Ē ­ EyJsEd by
Qsēd ­
nX
j­0
ajbjsmod2d , (2)
wheree ­ 6XsadZsbd is any element ofE whose image
in Ē is ē. Then e2 ­ s2IdQsēd and Qsēd ­ 0 or 1
accordingly asXsad and Zsbd commute or anticommute.
If e ­ w1 ≠ . . . ≠ wn, then Qsēd is the parity of the
number of componentswj that are equal tosxsz. Define
a binary inner product on̄E by sē, ē0d ­ Qsē 1 ē0d 1
Qsēd 1 Qsē0d. For ē ­ sajbd, ē0 ­ sa0jb0d,
sē, ē0d ­ a ? b0 1 a0 ? b smod2d. (3)
Now consider two elements ofE: e ­ 6XsadZsbd and
e0 ­ 6Xsa0dZsb0d. Thene ande0 commute or anticom-
mute accordingly as̄e, ē0d ­ 0 or 1.
A subspaceS̄ of Ē is said to be totally singular if
Qss̄d ­ 0 for all s̄ [ S̄. It follows that for s̄, s̄0 in S̄
the inner productss̄, s̄0d ­ 0. If M̄ is a maximal totally
singular subspace, then the groupM̄ has2n distinct linear© 1997 The American Physical Society 405
VOLUME 78, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 20 JANUARY 1997
m
-
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e
.
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h
a
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as
,
it
o
e
clic
r
ns,
r
characters, and the corresponding eigenvectors deter
a coordinate frameF sM̄d (an orthonormal basis ofR2n ).
For example, the image ofZsV d in Ē determines the co
ordinate framejyl, y [ V , and the image ofXsV d deter-
mines the coordinate frame22ny2
P
ys21du?y jyl, u [ V .
If S̄ # M̄ is a k-dimensional totally singular subspac
then the group̄S has2k distinct linear characters. The2n
vectors inF sM̄d are partitioned into2k sets of size2n2k
with each set corresponding to a different eigenspace.
view each eigenspace as a quantum error-correcting c
which mapsn 2 k qubits inton qubits. The2n2k vectors
from F sM̄d in that eigenspace constitute the code word
More generally, we may use complex space, and so
define a quantum error-correcting code encodingk qubits
into n qubits to be any2k-dimensional subspaceC of C2
n
.
This code will protect against errors in a certain error
E , which we take to be a subset of the extraspecial gr
E. As remarked earlier, there is no loss of generality
restricting to error sets inE.
For such a codeC to protect against all errors in an erro
setE , it is necessary and sufficient [7,12] that for any tw
vectorsjc1l and jc2l in C with kc1jc2l ­ 0, and any two
transformationse1 ande2 from E , we have
kc1je211 e2jc2l ­ 0 , (4)
kc1je211 e2jc1l ­ kc2je
21
1 e2jc2l. (5)
Note that since we are assuming thatE is contained in
E, we can replace211 e2 by e1e2 in the above equations
since e1 ­ 6e
21
1 . An interesting special case occu
when both sides of Eq. (5) are always equal to 0
e1 fi e2 [ E . This implies that there is a measureme
which will uniquely determine the error without affectin
the encoded subspace. After this measurement, the
can subsequently be corrected by a unitary operation
both sides of Eq. (5) are not always 0, then there can
two errorse1 and e2 between which it is impossible to
distinguish. However, these two errors are guarantee
have identical effects on vectors within the subspaceC, and
so need not be distinguishable in order to be correctab
We now can show the connection between orthogo
geometry and quantum error-correcting codes.
Theorem 1:Suppose thatS̄ is a k-dimensional to-
tally singular subspace of̄E. Let S̄' be thes2n 2 kd-
dimensional subspace orthogonal toS̄ with respect to the
inner product (3). Further suppose that for any two v
tors e1 ande2 in an error setE # E, either ē1ē2 [ S̄ or
ē1ē2 ” S̄'. Then the eigenspaceC corresponding to any
character of the group̄S is an error-correcting code whic
will correct any errore [ E .
Proof. We first show that ife [ E, then e permutes
the 2k spacesCi which are generated by the2k different
linear characters of̄S. Consider an element̄s [ S̄ with
eigenvaluels. We writes for the associated represent
tion. Then for anyjcl [ C we havesjcl ­ lsjcl, and
sejcl ­ s21dss̄,ēdesjcl ­ s21dss̄,ēdlsejcl, where ss̄, ēd is
the inner product (3). Sinces21dss̄,ēdls is independent of406ine
,
e
de
s.
we
et
up
in
r
o
s
r
t
rror
If
be
to
e.
al
c-
-
jcl, this shows that the action ofe permutes the eigenspace
generated by the characters ofS.
We next divide the proof into two cases, accordingly
ē1ē2 [ S̄ or ē1ē2 ” S̄'.
Case 1:Supposēe1ē2 [ S̄. It follows that forjc1l and
jc2l [ C with kc1jc2l ­ 0,
kc1je1e2jc2l ­ le1e2kc1jc2l ­ 0 , (6)
and for alljcl [ C,
kcje1e2jcl ­ le1e2 kcjcl ­ le1e2 , (7)
establishing Eqs. (4) and (5).
Case 2:Supposēe1ē2 ” S̄'. It follows that for some
s̄ [ S̄, se1e2 ­ 2e1e2s. Thus, forjcl [ C,
se1e2jcl ­ 2e1e2sjcl ­ 2lse1e2jcl, (8)
soe1e2jcl ” C. Thuse1e2jcl is in a different eigenspace
so
kc1je1e2jc2l ­ 0 (9)
for all c1, c2 [ C (including c1 ­ c2). Again Eqs. (4)
and (5) hold. QED
Example 1:We first describe the code mapping 1 qub
into 5 qubits presented in Ref. [7], which contains tw
code words,
jc0l ­ j00000l
1j11000l 1 j01100l 1 j00110l 1 j00011l 1 j10001l
2j10100l 2 j01010l 2 j00101l 2 j10010l 2 j01001l
2j11110l 2 j01111l 2 j10111l 2 j11011l 2 j11101l,
jc1l ­ j11111l
1j00111l 1 j10011l 1 j11001l 1 j11100l 1 j01110l
2j01011l 2 j10101l 2 j11010l 2 j01101l 2 j10110l
2j00001l 2 j10000l 2 j01000l 2 j00100l 2 j00010l.
Essentially the same code is given in Ref. [5], but w
use the above presentation since it is fixed under cy
permutations. It is easily verified thatXs11000dZs00101d
fixes jc0l and jc1l. Thus we may take the vecto
s11000j00101d [ Ē to be in the subspacēS. Using the
fact that the code is closed under cyclic permutatio
we find that S̄ is the four-dimensional totally singula
subspace generated by the vectors
11000j00101
01100j10010
00110j01001
00011j10100
[The fifth cyclic shift, s10001j01010d, is also in this
subspace.] The dual̄S' is generated byS̄ and the
VOLUME 78, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 20 JANUARY 1997
t
is
.
e
in
4]
s:
t
nd
m
c
m
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d
s
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ds
de
fts.
o
its
ode
t
ing
line
of
at
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ates
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ry
n
his
n
two additional vectorss11111j00000d ands00000j11111d.
It is straightforward to verify that the minimal weigh
vectors in S̄' have weight three [one such vector
s00111j00101d] and thus the code can correct one error
Example 2:Suppose we have a classical linearfn, k, dg
binary error-correcting codeC [i.e., it is over Zn2 , it is
k dimensional, and it has minimal distanced, so that
it correctst ­ bsd 2 1dy2c errors]. Suppose furthermor
that C' , C. We can define a subspaceS̄ to consist
of all vectors sy1jy2d [ Ē, where y1, y2 [ C'. The
dual S̄' consists of all vectorsy1jy2d with y1, y2 [ C,
showing that the corresponding quantum error-correct
code correctst errors. The subspacēS is 2sn 2 kd
dimensional, so the quantum code mapsn 2 2k qubits
into n qubits. This is the method described in Refs. [3,
Example 3: Consider the subspacēS obtained by
modifying the classical [8,4,4] Hamming code as follow
01110100j00111010
00111010j00011101
00011101j01001110
11111111j00000000
00000000j11111111. (11)
It is straightforward to verify that these vectors genera
a five-dimensional totally singular subspaceS̄ which is
invariant under cyclic permutations of the last 7 bits, a
that S̄' has minimal weight 3. This gives a quantu
code mapping 3 qubits into 8 qubits which can corre
one error. The same code was discovered by Gottes
[8], who used similar group-theoretic techniques, and
Steane [9].
Example 4:By duplicating the 5-qubit code (10) an
adding two vectors, we can obtain a code which map
qubits into 10 qubits and corrects one error:
01100 11110j10010 01100
00110 01111j01001 00110
00011 10111j10100 00011
10001 11011j01010 10001
11111 11111j00000 00000
00000 00000j11111 11111 . (12)
Example 5:The following construction is a genera
ization of the 5-qubit code (10) inspired by classic
quadratic residue codes. It works for any primep of the
form 8j 1 5. We have not found good theoretical boun
on the minimal distance, but for small primes these co
are excellent. To construct the first vectorsajbd, put
aj ­ 1 whenj is a nonzero quadratic residue modp (that
is, j ­ k2 mod p for somek) and putbj ­ 1 whenj is a
quadratic nonresidue. To obtainp 2 1 vectors that gen-
erate the subspacēS, takep 2 2 cyclic shifts of the first
vector. Forp ­ 13, the first basis vector is
0101100001101j0010011110010g
.
e
t
an
y
4
l
s
and the remaining vectors are obtained by cyclic shi
The minimal weight ofS̄' was calculated by computer t
be 5. This gives a code mapping one qubit into 13 qub
which corrects 2 errors. Forp ­ 29, this subspacēS'
has minimal distance 11, so this construction gives a c
mapping 1 qubit to 29 which corrects 5 errors.
We now give the construction of the groupL , OsR2n d
from its subgroupE. The groupL is the normalizer of
E in the real orthogonal groupOsR2n d; that is, it is the
subgroup of elementsg [ OsR2n d such thatg21Eg ­ E.
This normalizer acts onE by conjugation, fixing the
centerJsEd (g [ L acts onE as the permutatione !
g21eg). Hence there is a well-defined action ofL on the
binary vector spacēE that preserves the quadratic formQ.
The quotientLyE is the orthogonal groupO1s2n, 2d, a
finite classical group [14]. The groupL appears in recen
connections between classical Kerdock error-correct
codes, orthogonal geometry, and extremal Euclidean
sets [13]. This group also appears [7] as the group
Bell-state-preserving bilateral local transformations th
two experimenters (A and B) can jointly perform onn
pairs of particles (each pair being in a Bell state). Hen
there is a one-to-one correspondence between Bell st
and elements of̄E [cf. Eqs. (39) and (67) of Ref. [7] ].
The quadratic formQsēd is 0 or 1 accordingly as the
Bell states are symmetric or antisymmetric under t
interchange ofA andB.
The following are group elements that generateL from
E, shown together with their induced action on the bina
vector spacēE.
(1) H ­ 22ny2fs21du?ygu,y[V , which interchanges
Xsad and Zsbd. This applies the transformatio
R ­
1
p
2
s 1 11 21 d to every qubit.
(2) Ĥ2 ­
1
p
2
s 1 11 21 d ≠ I2n21 , which appliesR to the
first qubit and leaves the other qubits unchanged. T
acts onĒ by interchanginga1 andb1.
(3) Every matrixA in the general linear groupGLsV d
determines a permutation matrixjyl ! jyAl in OsR2n d.
The action on Ē induced by conjugation isXsad !
XsaAd, Zsbd ! ZsbA2Td. For example, the quantum
XOR taking jq1q2l ! jq1sq1 © q2dl is represented by
The back action of theXOR on the phases is evident i
its effect onb1 andb2. Any orthogonal matrix inOsR2
n d
that normalizes bothXsV d and ZsV d is of this type for
someA [13], Lemma 3.14.407
VOLUME 78, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 20 JANUARY 1997
.
e
e
a
f
tha
.
r
nt
ha
es
or
e
ra
s
-
er,
.
J.(4) Diagonal matricesdM ­ diagfs21dQM sydg, where
QM is a binary quadratic form onV for which the
associated bilinear formQMsu 1 yd 2 QMsud 2 QM syd
is uMyT . Note thatM is symmetric with zero diagonal
The induced action on̄E is given by
These matrices are precisely the elements ofL that
induce the identity on the subgroupZsV d. In terms of
their effect on qubits, these are the transformations inL
that change the phases of the qubits while fixing th
values.
Remark: The group E0 is generated byE and iI
in the unitary groupUs2nd. Now we cannot define
Qsēd ­ e2 becausesied2 fi e2. However, we still have
the nonsingular alternating binary form
sX̄sadZ̄sbd, X̄sa0dZ̄sb0dd ­ a ? b0 1 a0 ? bsmod2d.
(15)
The groupL0 is the normalizer ofE0 and is generated byL
and by diagonal transformationsdP ­ diag fiTPsydg where
TP is a Z4-valued quadratic form [13], Section 4. Th
induced action on̄E0 ­ E0yJsE0d is described by
whereP is symmetric and may have a nonzero diagon
For example, applying thepy2 rotation around thez
axis s 1 00 i d to each qubit corresponds toP ­ I. The
quotient L0yE0 is the symplectic groupSps2n, 2d [14].
As the quadratic formQ played no role in the proof o
Theorem 1, it is only necessary for a quantum code
S̄ satisfy sS̄, S̄d ­ 0 with respect to the alternating form
This means the complex groupsE0 andL0 can be used for
code construction.
The groupL acts transitively on the totally singula
subspaces̄S of dimensionk, so there is some elementg [
L taking any particulark-dimensional totally singular
subspace to the subspace corresponding to a qua
code generated as in Theorem 1. This implies t
someg [ L takes the canonical2n2k-dimensional Hilbert
space generated by the firstn 2 k qubits to the encoded
subspace. SinceL can be generated byNOT’s, XOR’s,
andpy2 rotations around thex axis, these three quantum
gates are therefore sufficient for encoding any of th
quantum codes.
We next give a Gilbert-Varshamov lower bound f
the asymptotic rate of this class of codes. It match
the Gilbert-Varshamov lower bounds known for gene
quantum error-correcting codes [10].408ir
l.
t
um
t
e
s
l
Theorem 2:There exist quantum error-correcting code
with asymptotic rate
R ­ 1 2 2d log2 3 2 H2s2dd, (17)
whered is the fraction of qubits that are subject to deco
herence andH2sdd ­ 2d log2 d 2 s1 2 dd log2s1 2 dd
is the binary entropy function.
Proof: Let Nk denote the number ofk-dimensional
totally singular subspaces. We count pairsse, S̄d where
e [ E is in E 2 (i.e., e ­ e1e2 with e1, e2 in the error
setE ) andS̄ is a k-dimensional totally singular subspace
with ē [ S̄'nS̄. Transitivity of L on singular points
[ē fi 0, Qsēd ­ 0], and on nonsingular points [Qsēd fi 0]
implies that eache [ E satisfies ē [ S̄'nS̄ for mNk
subspaces̄S, where the fractionm ø 22k. If jE 2j , 2k
then there exists ak-dimensional totally singular subspace
S̄ that satisfiesē ” S̄'nS̄ for all e [ E 2. Hence the
achievable rateR satisfies
1 2 R ­ log2 jE
2jyn
­ log2f3
2dns n2dn dgyn
­ 2d log2 3 1 H2s2dd. (18)
We would like to thank David DiVincenzo for discus-
sions about the groupL as presented in [7].
[1] P. W. Shor, Phys. Rev. A52, 2493 (1995); A. Steane,
Phys. Rev. Lett.77, 793 (1996).
[2] C. H. Bennett, G. Brassard, S. Popescu, B. Schumach
J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett.76,
722 (1996).
[3] A. R. Calderbank and P. W. Shor, Phys. Rev. A54, 1098
(1996).
[4] A. Steane, Proc. R. Soc. London A452, 2551 (1996).
[5] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Phys
Rev. Lett.77, 198 (1996).
[6] M. B. Plenio, V. Vedral, and P. L. Knight, LANL Report
No. quant-phy9603022; L. Vaidman, L. Goldenberg, and
S. Wiesner, Phys. Rev. A54, 1745 (1996);P. W. Shor and
J. A. Smolin, LANL Report No. quant-phy9604006; S. L.
Braunstein, LANL Report No. quant-phy9604036.
[7] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K.
Wootters, Phys. Rev. A54, 3824 (1996).
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Quantum Error Correction and Orthogonal Geometry

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