When the environmental disturbace to a quantum system has a wavelength much
larger than the system size, all qubits localized within a small area are under
action of the same error operators. Noiseless subsystem and decoherence free
subspace are known to correct such collective errors. We construct simple
quantum circuits, which implement these collective error correction codes, for
a small number $n$ of physical qubits. A single logical qubit is encoded with
$n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive
relations among the subspaces employed in noiseless subsystem and decoherence
free subspace play essential r\^oles in our implementation. The recursive
relations also show that the number of gates required to encode $m$ logical
qubits increases linearly in $m$.Comment: 9 pages, 3 figure

Chi-Kwong Li

D. W. Kribs

Hiroyuki Tomita

M. A. Neilsen

M. Nakahara

Mikio Nakahara

Nung-Sing Sze

Yiu-Tung Poon

Physical Review A

English

arXiv

When an environmental disturbance to a quantum system has a wavelength much larger than the system size, all qubits in the system are under the action of the same error operator. The noiseless subsystem and decoherence-free subspace are immune to such collective noise. We construct simple quantum circuits that implement these error-avoiding codes for a small number n of physical qubits. A single logical qubit is encoded with n=3 and 4, while two and three logical qubits are encoded with n=5 and 7, respectively. Recursive relations among subspaces employed in these codes play essential roles in our implementation.Department of Applied Mathematic

Li, CK

Nakahara, M

Poon, YT

Sze, NS

Tomita, H

The Hong Kong Polytechnic University Pao Yue-kong Library

Recursive encoding and decoding of the noiseless subsystem and decoherence-free subspace

When an environmental disturbance to a quantum system has a wavelength much larger than the system size, all qubits in the system are under the action of the same error operator. The noiseless subsystem and decoherence-free subspace are immune to such collective noise. We construct simple quantum circuits that implement these error-avoiding codes for a small number n of physical qubits. A single logical qubit is encoded with n = 3 and 4, while two and three logical qubits are encoded with n = 5 and 7, respectively. Recursive relations among subspaces employed in these codes play essential roles in our implementation

CiteSeerX

PHYSICAL REVIEW A 84, 044301 (2011)
Recursive encoding and decoding of the noiseless subsystem and decoherence-free subspace
Chi-Kwong Li,1,* Mikio Nakahara,2,† Yiu-Tung Poon,3,‡ Nung-Sing Sze,4,§ and Hiroyuki Tomita5,‖
1Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187-8795, USA
2Research Center for Quantum Computing and Department of Physics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka 577-8502, Japan
3Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA
4Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
5Research Center for Quantum Computing, Kinki University, 3-4-1 Kowakae, Higashi-Osaka 577-8502, Japan
(Received 4 July 2011; revised manuscript received 14 July 2011; published 5 October 2011)
When an environmental disturbance to a quantum system has a wavelength much larger than the system
size, all qubits in the system are under the action of the same error operator. The noiseless subsystem and
decoherence-free subspace are immune to such collective noise. We construct simple quantum circuits that
implement these error-avoiding codes for a small number n of physical qubits. A single logical qubit is encoded
with n = 3 and 4, while two and three logical qubits are encoded with n = 5 and 7, respectively. Recursive
relations among subspaces employed in these codes play essential roles in our implementation.
DOI: 10.1103/PhysRevA.84.044301 PACS number(s): 03.67.Pp
I. INTRODUCTION
A quantum system is vulnerable to external noise and the
system must be protected from it in quantum information
processing and quantum computation. The majority of quan-
tum systems employed for these purposes are microscopic
in size, typically on the order of a few micrometers. In
contrast, environmental noise, such as electromagnetic waves,
has a wavelength on the order of a few centimeters or more.
Therefore, it is natural to assume that all the qubits in the
register suffer from the same error operator. We call such
an error the collective error. The n-qubit quantum states ρ
are represented as 2n × 2n density matrices and a quantum
channel is realized as a completely positive linear map  with
an operator sum representation
(ρ) =
r∑
j=1
EjρE
†
j (1)
with error operators {Ej } [1,2]. The error operators in  can
be expressed as multiples of an operator of the form W⊗n ∈
2⊗n, where 2 is the two-dimensional (fundamental) irreducible
representation (IR) of SU(2).
The decoherence-free subspace (DFS) [3–6] and noiseless
subsystem (NS) [7–10] are two standard methods of cod-
ing used to correct collective errors [10,11]. The scheme
is explained using the operator sum representation of the
quantum channel Eq. (1). Suppose the finite-dimensional C∗-
algebra An generated by the error operators admits the unique
decomposition into IRs up to a unitary equivalence (similarity)
as ⊕j (Irj ⊗ Mnj ) with
∑
j rjnj = N , where N = 2n, nj is
the dimension of the IR, and rj is its multiplicity. Then every
error operator Ei has the form ⊕j (Irj ⊗ Bj ) with Bj ∈ Mnj .
For every index j , if we regard MN = (Irj ⊗ Mnj ) ⊕ Mq with
*ckli@math.wm.edu
†nakahara@math.kindai.ac.jp
‡ytpoon@iastate.edu
§raymond.sze@inet.polyu.edu.hk
‖tomita@alice.math.kindai.ac.jp
q = N − rjnj and if we apply the channel to a quantum state
ρ = (ρ̂ ⊗ σ ) ⊕ Oq with ρ̂ ∈ Mrj and σ ∈ Mnj according to
this decomposition, then (ρ) = (ρ̂ ⊗ σE) ⊕ Oq because of
the special form of the error operators in this decomposition.
Here σE is the ancilla state after  is applied and Oq is a null
matrix of order q. Thus the state ρ̂ encoded as above will not
be affected by the errors (noise) and can be easily recovered.
This gives rise to a NS. The situation is particularly pleasant
if nj = 1, i.e., we use the one-dimensional IRs of An, so that
(ρ̂ ⊕ Oq) = ρ̂ ⊕ Oq. In such a case, we get a DFS.
We are interested in simple implementation of DFSs and
NSs for the channels with a common error on each qubit in the
register. From the discussion in the preceding paragraph we
have that the DFS employs one-dimensional IRs of algebra An
generated by 2⊗n for encoding while the NS encodes logical
qubits by making use of the multiplicity of an IR.
The purpose of this paper is to investigate the imple-
mentation of these ideas in terms of quantum circuits. We
consider the DFS with n = 4, which implements a single
logical qubit, and the NS with n = 3 and 5, which encodes
a single logical qubit and two logical qubits, respectively.
Viola et al. [12] worked out the circuit implementation of the
n = 3 NS and demonstrated its validity by using an ion-trap
quantum computer. We are unaware of further work having
been conducted to date for n  4. Our implementation, starting
with n = 3 NS, is recursive so that the n = 4 DFS and the
n = 5 NS are implemented with the quantum circuit for n = 3.
Moreover, our circuit for n = 3 is simpler than that obtained
in Refs. [9,12].
We construct a quantum circuit for the n = 3 NS in
Sec. II. We analyze the n = 4 DFS and the n = 5 NS in
Secs. III and IV, respectively, by making use of the result of
Sec. II. Our analysis is concrete and the encoding basis vectors
and quantum circuits are explicitly constructed. We conclude
in Sec. V with a summary and discussion.
We use the fact [10] that the algebra An generated by
2⊗n has the unique decomposition
⊕
0jn/2(Irj ⊗ Mnj ) with
(r0,n0) = (1,n + 1) and (rj ,nj ) = (
(
n
j
) − ( n
j−1
)
,n + 1 − 2j )
for 0 < j  n/2. We also employ the Lie theoretic notation
and regard a qubit belonging to the representation space of
044301-11050-2947/2011/84(4)/044301(4) ©2011 American Physical Society
BRIEF REPORTS PHYSICAL REVIEW A 84, 044301 (2011)
the fundamental representation 2 of SU(2) while the product
operator W⊗n acts as a reducible representation 2⊗n.
II. THE 3-QUBIT NOISELESS SUBSYSTEM
Let us consider a 3-qubit system and see how it can be used
to encode a logical qubit that is robust against any noise of
the form W⊗3, where W is an arbitrary element of 2. We first
consider the algebra A3 of 2⊗3 in which A3 is decomposed
into the sum of IRs as 2⊗3 = 4 ⊕ (I2 ⊗ 2), where In is the unit
matrix of dimension n. Corresponding to this decomposition,
any unitary matrix V ∈ 2⊗3 can be decomposed as V = V4 ⊕
(I2 ⊗ V2) for a proper choice of basis vectors. Here V4 belongs
to 4 and V2 belongs to 2 of SU(2). It should be noted that I2 is
immune to any collective noise of the form W⊗3,W ∈ 2, and
the corresponding vector space forms the NS.
The success of our schemes depends on a judicious choice
of orthonormal basis for the decomposition of the algebra A3
generated by 2⊗3. Let {|e4,1〉,|e4,2〉,|e4,3〉,|e4,4〉|} be a basis of
4 and {|ea1〉,|ea2〉} and {|eb1〉,|eb2〉} be bases of the two 2’s
defined as
|e4,1〉 = |000〉,
|e4,2〉 = 1√
3
[|0〉(|01〉 + |10〉) + |1〉|00〉],
(2)
|e4,3〉 = (σx)⊗3|e4,2〉,
|e4,4〉 = |111〉 = (σx)⊗3|e4,1〉;
|ea1〉 = 1√
2
|0〉(|10〉 − |01〉),
(3)
|ea2〉 = −(σx)⊗3|ea1〉;
|eb1〉 = 1√
6
[|0〉(|01〉 + |10〉) − 2|1〉|00〉],
(4)
|eb2〉 = −(σx)⊗3|eb1〉,
where σk is the kth Pauli matrix. We implement a NS from two
2 IRs.
Suppose U (3)E is an encoding matrix that generates the above
basis vectors from the binary basis vectors |i1i2i3〉 (ik ∈ {0,1}).
We choose U (3)E to have columns
(|ea1〉,|eb1〉,|ea2〉,|eb2〉,|e4,4〉,|e4,2〉, − |e4,1〉, − |e4,3〉)
in this order. The basis vectors {|e4,k〉} are irrelevant for our
purposes and their order and signs have been chosen so as to
make the circuit implementation as simple as possible.
Figure 1 shows an example of the encoding circuit, in which
G1 = 1√
3
(
1
√
2
−√2 1
)
, G2 = 1√
2
(
1 1
−1 1
)
.
Note that our circuit is simpler than that found in Refs. [9,12]
regarding the number of gates.
Now we prove that collective errors are corrected by
employing basis vectors Eqs. (3)and (4) as a logical qubit
basis.
Theorem. Let α, β, and γ be any real numbers and let
Xα = (eiασx )⊗3, Yβ = (eiβσy )⊗3, Zγ = (eiγ σz )⊗3.
FIG. 1. Encoding circuit U (3)E of the NS with n = 3. It encodes a
single-qubit state |ψ̂〉. The part surrounded by the dashed line can be
omitted if the initial state |v〉 of the central qubit is |0〉. The recovery
operation is given by U (3)†E .
Consider a quantum channel  : M8 → M8 given by
(ρ) = p0ρ + p1XαρX†α + p2YβρY †β + p3Zγ ρZ†γ
for some pi ∈ R such that
∑3
i=0 pi  1. Then for any data
state ρ̂ ∈ M2, U (3)E and  satisfy the identity
U
(3)†
E 
(
U
(3)
E (|0〉〈0| ⊗ ρa ⊗ ρ̂)U (3)†E
)
U
(3)
E
= |0〉〈0| ⊗
⎛
⎝ 3∑
j=0
pjUjρaU
†
j
⎞
⎠ ⊗ ρ̂, (5)
that is, the initial data state is recovered in the output state with
no entanglement with the ancilla qubits. Here ρa is an initial
single-qubit ancilla state and U0 = I2, U1 = eiασx , U2 = eiβσy ,
and U3 = eiγ σz .
Proof. We show that the 2 ⊕ 2 IRs form a NS by explicit
evaluation. Let the logical |0〉L lie in the span of {|ea1〉,|ea2〉}
and the logical |1〉L lie in the span of {|eb1〉,|eb2〉}. We show that
noise operators Xα , Yβ , and Zγ leave each subspace invariant.
Let Pa =
∑2
i=1 |eai〉〈eai | and Pb =
∑2
i=1 |ebi〉〈ebi |. Then it is
easy to show that XαPkX†α = YβPkY †β = Zγ PkZ†γ = Pk (k =
a,b).
Now we prove the identity. We use a pure-state notation to
simplify the expressions. The general case with mixed initial
states ρa and ρ̂ is obtained by simply mixing the pure-state
results using linearity. Let |ψ̂〉 = a|0〉 + b|1〉 be a data qubit
state to be encoded and |v〉 = v0|0〉 + v1|1〉 be the initial state
of the first ancilla qubit, while that of the first qubit is set to
|0〉. The encoding under the action of U (3)E yields
|	〉 = U (3)E |0〉|v〉|ψ̂〉 = v0(a|ea1〉 + b|eb1〉)
+ v1(a|ea2〉 + b|eb2〉).
Let us consider a noise operator Xα . Its action on |	〉 yields
Xα|	〉 = (v0 cos α + iv1 sin α)(a|ea1〉 + b|eb1〉)
+ (v1 cos α + iv0 sin α)(a|ea2〉 + b|eb2〉).
The action of the recovery operator U (3)†E recovers the initial
state, except for the second qubit, as
U
(3)†
E Xα|	〉 = |0〉
(
eiασx |v〉) |ψ̂〉,
which shows that data qubit state is immune to Xα . It is shown
similarly that the data qubit is immune to other error operators
as well. Since each error is in action with the probability pi ,
we have proved the identity in Eq. (5). 
044301-2
BRIEF REPORTS PHYSICAL REVIEW A 84, 044301 (2011)
In contrast with an ordinary quantum error-correcting code,
the scheme corrects multiple actions of the error operators.
It was shown in the theorem that the central qubit can be
any superposition state or mixed state initially and its output
state is another superposition or mixed state under an action
of a single error operator in Xα,Yβ , and Zγ . Note that the
error channel leaves the encoded word unchanged, namely,
given any initial ancilla state ρa , there exists an ancilla state
ρ ′a such that (U
(3)
E (|0〉〈0| ⊗ ρa ⊗ ρ̂)U (3)†E ) = U (3)E (|0〉〈0| ⊗
ρ ′a ⊗ ρ̂)U (3)†E . Then the error correction may be repeated as
many times as required. This implies that it corrects any error
operator of the form W⊗3, where W ∈ 2. This is because any
element W ∈ 2 is decomposed as W = eiθ1σx eiθ2σy eiθ3σx . This
is clear since W⊗3 is expressed as a product Xθ1Yθ2Xθ3 , each
factor of which leaves the NS invariant.
III. THE 4-QUBIT DECOHERENCE FREE SUBSPACE
We design the 4-qubit DFS, which is robust against
collective noise of the form W⊗4 (W ∈ 2), by taking advantage
of the NS analyzed in the preceding section. The algebra A4
obtained from 2⊗4 is decomposed as 2⊗4 = 5 ⊕ (I3 ⊗ 3) ⊕
(I2 ⊗ 1). Correspondingly, any V ∈ 2⊗4 is decomposed as
V = V5 ⊕ (I3 ⊗ V3) ⊕ (I2 ⊗ V1) for a proper choice of basis
vectors. Here Vk is an element of a k-dimensional IR of
SU(2)⊗4. The singlet IR is immune to any operator V = W⊗4,
where W ∈ 2, and two of them form a single logical qubit that
is immune to any noise of the form V . This (reducible) vector
space is the DFS that is robust against the collective noise.
We generate basis vectors of two one-dimensional IRs of
SU(2) from {|eai〉,|ebi〉} as
|0〉L = 1√
2
(|1〉|ea1〉 − |0〉|ea2〉)
= 1√
2
[|1〉|ea1〉 + |0〉(σx)⊗3|ea1〉], (6)
|1〉L = 1√
2
(|1〉|eb1〉 − |0〉|eb2〉)
= 1√
2
[|1〉|eb1〉 + |0〉(σx)⊗3|eb1〉].
It is important in the implementation of the encoding circuit
to realize that
|0〉L = (X ⊗ I8)(CNNN)(H ⊗ I8)|0〉 ⊗ |ea1〉
= (X ⊗ I8)(CNNN)
(
H ⊗ U (3)E
)|0〉 ⊗ |000〉,
|1〉L = (X ⊗ I8)(CNNN)(H ⊗ I8)|0〉 ⊗ |eb1〉
= (X ⊗ I8)(CNNN)
(
H ⊗ U (3)E
)|0〉 ⊗ |001〉,
where CNNN is a controlled-NOT gate with three target bits.
Figure 2 shows an example of the encoding circuit for the
DFS. In contrast with the three-qubit NS, all the ancilla qubits
must be initially set to |0〉.
IV. THE 5-QUBIT NOISELESS SUBSYSTEM
The NS using five qubits encodes two data qubits. It is
recursively implemented by employing the encoding circuit
U
(3)
E for the three-qubit NS.
FIG. 2. Encoding circuit U (4)E of the DFS with n = 4. It encodes
a single-qubit state |ψ̂〉.
The algebra A5 obtained from 2⊗5 is decomposed as 2⊗5 =
6 ⊕ (I4 ⊗ 4) ⊕ (I5 ⊗ 2). Correspondingly, any unitary matrix
V ∈ 2⊗5 is decomposed as V = V6 ⊕ (I4 ⊗ V4) ⊕ (I5 ⊗ V2)
under a proper choice of basis vectors. We implement a NS by
employing four of five two-dimensional IRs.
Let {|eai〉,|ebi〉} be basis vectors introduced for n = 3 in
Sec. II. We generate eight basis vectors from them as
|00〉L = 1√
2
|ea1〉(|01〉 − |10〉),
|01〉L = 1√
6
[|ea1〉(|01〉 + |10〉) − 2|ea2〉|00〉],
(7)
|10〉L = 1√
2
|eb1〉(|01〉 − |10〉),
|11〉L = 1√
6
[|eb1〉(|01〉 + |10〉) − 2|eb2〉|00〉]
and their bit-flipped basis vectors, which are obtained by
applying σ⊗5x to them in the same manner that Eqs. (3) and
(4) are obtained from the basis vectors |0〉 and |1〉 of n = 1.
{
{
FIG. 3. (a) Encoding circuit U (5)E of the 5-qubit NS, which
encodes a two-data-qubit state |ψ̂〉. The third and fourth qubits in
Eq. (7) are exchanged to make the recursive symmetry manifest in
this diagram. (b) Encoding circuit U (7)E for the 7-qubit NS, which
encodes a three-data-qubit state |ψ̂〉.
044301-3
BRIEF REPORTS PHYSICAL REVIEW A 84, 044301 (2011)
We emphasize the similar structure between Eqs. (7) and
Eqs. (3) and (4). This observation makes implementation of the
encoding-decoding circuit almost a trivial matter. Note that we
do not need to worry about the rest of the basis vectors as long
as they are orthogonal to the above basis vectors spanning the
NS. This orthogonalization is automatically taken into account
by the unitarity of the encoding circuit.
Figure 3(a) shows an example of the encoding circuit U (5)E
of the NS. The central qubit can be any state while all other
ancilla qubits must be in |0〉. Each U (3)E acts on the three qubits
numbered 1, 2, and 3, which are fed into the input ports 1,
2, and 3, respectively, in Fig. 1. The n = 7 NS encoding
circuit can be also constructed recursively, as shown in
Fig. 3(b).
V. CONCLUSION
The DFS and NS make use of vector subspaces that are
immune to noise of the form W⊗n, where W belongs to 2 of
SU(2). We have constructed simple encoding and decoding
quantum circuits for the NS for n = 3 and 5 and the DFS for
n = 4. Our strategy is to use the encoding-decoding circuit
U
(3)
E for n = 3 recursively in the implementation for n = 4
and 5. We have constructed the bases of 1’s for n = 4 and
the bases of 2’s for n = 5 from the bases of two 2’s for
n = 3, as given in Eqs. (6) and (7). We can then generalize
this construction to find the bases of 1’s for n = 2m + 2
and the basis of 2’s for n = 2m + 3 from the 2m bases
of 2’s for n = 2m + 1. We have implemented DFS and
NS encoding-decoding circuits by taking advantage of these
recursive relations among basis vectors of different n. The
number of logical qubits in the limit of large n is asymptotically
n/2 for both even n and odd n. It should be clear from our
construction that m logical qubits are implemented by use of
m U
(3)
E modules, which shows that the circuit complexity for
our encoding and decoding circuits increases merely linearly
in m.
Note, however, that our construction does not give the
maximum number of correctable qubits for the channel. There
are
(
n
m
) − ( n
m−1
)
basis vectors in two-dimensional IRs for n =
2m + 1, which encode k = log2[
(
n
m
) − ( n
m−1
)
] qubits. This
number k is greater than m for n  9 and actually k/n → 1 as
n → ∞. This asymptotic behavior is also observed in Ref. [10]
for the DFS.
Full details of the recursion scheme are beyond the scope
of the present paper. Moreover, we plan to present an in-depth
discussion of the decomposition of the algebra 2n and the
construction of other noiseless subsystems of channels with
error operators in the algebra elsewhere [13].
It was shown that the central qubit in Figs. 1
and 3 can be any state. Although the entropy of the
qubit system generally increases, it remains constant if
the central qubit is maximally mixed initially as ρa = 12I2.
This behavior is somewhat analogous to that of the DFS
with ρa = |0〉〈0|, in which the entropy does not change at
all.
ACKNOWLEDGMENTS
C.-K.L. was supported by a US NSF grant, a HK RGC grant,
and the 2011 Shanxi 100 Talent Program. M.N. and H.T. were
supported by the “Open Research Center” Project for Private
Universities with a matching fund subsidy from MEXT. M.N.
would like to thank the JSPS (Grant No. 23540470) for partial
support of Grants-in-Aid for Scientific Research. Y.-T.P. was
supported by a US NSF grant. N.-S.S. was supported by a HK
RGC grant.
[1] M. A. Neilsen and I. L. Chuang, Quantum Computation and
Quantum Information (Cambridge University Press, Cambridge,
2000).
[2] M. Nakahara and T. Ohmi, Quantum Computing, From Linear
Algebra to Physical Realization (CRC, New York, 2008).
[3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997).
[4] P. Zanardi and M. Rasetti, Mod. Phys. Lett. B 11, 1085 (1997).
[5] P. Zanardi, Phys. Rev. A 57, 3276 (1998).
[6] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett.
81, 2594 (1998).
[7] E. Knill, R. Laflamme, and L. Viola, Phys. Rev. Lett. 84, 2525
(2000).
[8] S. De Filippo, Phys. Rev. A 62, 052307 (2000).
[9] C.-P. Yang and J. Gea-Banacloche, Phys. Rev. A 63, 022311
(2001).
[10] J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, Phys. Rev.
A 63, 042307 (2001).
[11] D. W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky, Quantum
Inf. Comput. 6, 382 (2006).
[12] L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill,
R. Laflamme, and D. G. Cory, Science 293, 2059
(2001).
[13] C.-K. Li, M. Nakahara, Y.-T. Poon, N.-S. Sze, and H. Tomita
(unpublished).
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Recursive encoding and decoding of noiseless subsystem and decoherence free subspace

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PHYSICAL REVIEW A 84, 044301 (2011)Recursive encoding and decoding of the noiseless subsystem and decoherence-free subspaceChi-Kwong Li,1,* Mikio Nakahara,2,† Yiu-Tung Poon,3,‡ Nung-Sing Sze,4,§ and Hiroyuki Tomita5,‖1Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187-8795, USA2Research Center for Quantum Computing and Department of Physics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka 577-8502, Japan3Department of Mathematics, Iowa State University, Ames, Iowa 50011, USA4Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong5Research Center for Quantum Computing, Kinki University, 3-4-1 Kowakae, Higashi-Osaka 577-8502, Japan(Received 4 July 2011; revised manuscript received 14 July 2011; published 5 October 2011)When an environmental disturbance to a quantum system has a wavelength much larger than the systemsize, all qubits in the system are under the action of the same error operator. The noiseless subsystem anddecoherence-free subspace are immune to such collective noise. We construct simple quantum circuits thatimplement these error-avoiding codes for a small number n of physical qubits. A single logical qubit is encodedwith n = 3 and 4, while two and three logical qubits are encoded with n = 5 and 7, respectively. Recursiverelations among subspaces employed in these codes play essential roles in our implementation.DOI: 10.1103/PhysRevA.84.044301 PACS number(s): 03.67.PpI. INTRODUCTIONA quantum system is vulnerable to external noise and thesystem must be protected from it in quantum informationprocessing and quantum computation. The majority of quan-tum systems employed for these purposes are microscopicin size, typically on the order of a few micrometers. Incontrast, environmental noise, such as electromagnetic waves,has a wavelength on the order of a few centimeters or more.Therefore, it is natural to assume that all the qubits in theregister suffer from the same error operator. We call suchan error the collective error. The n-qubit quantum states ρare represented as 2n × 2n density matrices and a quantumchannel is realized as a completely positive linear map  withan operator sum representation(ρ) =r∑j=1EjρE†j (1)with error operators {Ej } [1,2]. The error operators in  canbe expressed as multiples of an operator of the form W⊗n ∈2⊗n, where 2 is the two-dimensional (fundamental) irreduciblerepresentation (IR) of SU(2).The decoherence-free subspace (DFS) [3–6] and noiselesssubsystem (NS) [7–10] are two standard methods of cod-ing used to correct collective errors [10,11]. The schemeis explained using the operator sum representation of thequantum channel Eq. (1). Suppose the finite-dimensional C∗-algebra An generated by the error operators admits the uniquedecomposition into IRs up to a unitary equivalence (similarity)as ⊕j (Irj ⊗ Mnj ) with∑j rjnj = N , where N = 2n, nj isthe dimension of the IR, and rj is its multiplicity. Then everyerror operator Ei has the form ⊕j (Irj ⊗ Bj ) with Bj ∈ Mnj .For every index j , if we regard MN = (Irj ⊗ Mnj ) ⊕ Mq with*ckli@math.wm.edu†nakahara@math.kindai.ac.jp‡ytpoon@iastate.edu§raymond.sze@inet.polyu.edu.hk‖tomita@alice.math.kindai.ac.jpq = N − rjnj and if we apply the channel to a quantum stateρ = (ρˆ ⊗ σ ) ⊕ Oq with ρˆ ∈ Mrj and σ ∈ Mnj according tothis decomposition, then (ρ) = (ρˆ ⊗ σE) ⊕ Oq because ofthe special form of the error operators in this decomposition.Here σE is the ancilla state after  is applied and Oq is a nullmatrix of order q. Thus the state ρˆ encoded as above will notbe affected by the errors (noise) and can be easily recovered.This gives rise to a NS. The situation is particularly pleasantif nj = 1, i.e., we use the one-dimensional IRs of An, so that(ρˆ ⊕ Oq) = ρˆ ⊕ Oq. In such a case, we get a DFS.We are interested in simple implementation of DFSs andNSs for the channels with a common error on each qubit in theregister. From the discussion in the preceding paragraph wehave that the DFS employs one-dimensional IRs of algebra Angenerated by 2⊗n for encoding while the NS encodes logicalqubits by making use of the multiplicity of an IR.The purpose of this paper is to investigate the imple-mentation of these ideas in terms of quantum circuits. Weconsider the DFS with n = 4, which implements a singlelogical qubit, and the NS with n = 3 and 5, which encodesa single logical qubit and two logical qubits, respectively.Viola et al. [12] worked out the circuit implementation of then = 3 NS and demonstrated its validity by using an ion-trapquantum computer. We are unaware of further work havingbeen conducted to date forn  4. Our implementation, startingwith n = 3 NS, is recursive so that the n = 4 DFS and then = 5 NS are implemented with the quantum circuit for n = 3.Moreover, our circuit for n = 3 is simpler than that obtainedin Refs. [9,12].We construct a quantum circuit for the n = 3 NS inSec. II. We analyze the n = 4 DFS and the n = 5 NS inSecs. III and IV, respectively, by making use of the result ofSec. II. Our analysis is concrete and the encoding basis vectorsand quantum circuits are explicitly constructed. We concludein Sec. V with a summary and discussion.We use the fact [10] that the algebra An generated by2⊗n has the unique decomposition⊕0jn/2(Irj ⊗ Mnj ) with(r0,n0) = (1,n + 1) and (rj ,nj ) = ((nj)− ( nj−1),n + 1 − 2j )for 0 < j  n/2. We also employ the Lie theoretic notationand regard a qubit belonging to the representation space of044301-11050-2947/2011/84(4)/044301(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 84, 044301 (2011)the fundamental representation 2 of SU(2) while the productoperator W⊗n acts as a reducible representation 2⊗n.II. THE 3-QUBIT NOISELESS SUBSYSTEMLet us consider a 3-qubit system and see how it can be usedto encode a logical qubit that is robust against any noise ofthe form W⊗3, where W is an arbitrary element of 2. We firstconsider the algebra A3 of 2⊗3 in which A3 is decomposedinto the sum of IRs as 2⊗3 = 4 ⊕ (I2 ⊗ 2), where In is the unitmatrix of dimension n. Corresponding to this decomposition,any unitary matrix V ∈ 2⊗3 can be decomposed as V = V4 ⊕(I2 ⊗ V2) for a proper choice of basis vectors. Here V4 belongsto 4 and V2 belongs to 2 of SU(2). It should be noted that I2 isimmune to any collective noise of the form W⊗3,W ∈ 2, andthe corresponding vector space forms the NS.The success of our schemes depends on a judicious choiceof orthonormal basis for the decomposition of the algebra A3generated by 2⊗3. Let {|e4,1〉,|e4,2〉,|e4,3〉,|e4,4〉|} be a basis of4 and {|ea1〉,|ea2〉} and {|eb1〉,|eb2〉} be bases of the two 2’sdefined as|e4,1〉 = |000〉,|e4,2〉 = 1√3[|0〉(|01〉 + |10〉) + |1〉|00〉],(2)|e4,3〉 = (σx)⊗3|e4,2〉,|e4,4〉 = |111〉 = (σx)⊗3|e4,1〉;|ea1〉 = 1√2|0〉(|10〉 − |01〉),(3)|ea2〉 = −(σx)⊗3|ea1〉;|eb1〉 = 1√6[|0〉(|01〉 + |10〉) − 2|1〉|00〉], (4)|eb2〉 = −(σx)⊗3|eb1〉,where σk is the kth Pauli matrix. We implement a NS from two2 IRs.Suppose U (3)E is an encoding matrix that generates the abovebasis vectors from the binary basis vectors |i1i2i3〉 (ik ∈ {0,1}).We choose U (3)E to have columns(|ea1〉,|eb1〉,|ea2〉,|eb2〉,|e4,4〉,|e4,2〉, − |e4,1〉, − |e4,3〉)in this order. The basis vectors {|e4,k〉} are irrelevant for ourpurposes and their order and signs have been chosen so as tomake the circuit implementation as simple as possible.Figure 1 shows an example of the encoding circuit, in whichG1 = 1√3(1√2−√2 1), G2 = 1√2( 1 1−1 1).Note that our circuit is simpler than that found in Refs. [9,12]regarding the number of gates.Now we prove that collective errors are corrected byemploying basis vectors Eqs. (3)and (4) as a logical qubitbasis.Theorem. Let α, β, and γ be any real numbers and letXα = (eiασx )⊗3, Yβ = (eiβσy )⊗3, Zγ = (eiγ σz )⊗3.FIG. 1. Encoding circuit U (3)E of the NS with n = 3. It encodes asingle-qubit state | ˆψ〉. The part surrounded by the dashed line can beomitted if the initial state |v〉 of the central qubit is |0〉. The recoveryoperation is given by U (3)†E .Consider a quantum channel  : M8 → M8 given by(ρ) = p0ρ + p1XαρX†α + p2YβρY †β + p3Zγ ρZ†γfor some pi ∈ R such that∑3i=0 pi  1. Then for any datastate ρˆ ∈ M2, U (3)E and  satisfy the identityU(3)†E (U(3)E (|0〉〈0| ⊗ ρa ⊗ ρˆ)U (3)†E)U(3)E= |0〉〈0| ⊗⎛⎝ 3∑j=0pjUjρaU†j⎞⎠⊗ ρˆ, (5)that is, the initial data state is recovered in the output state withno entanglement with the ancilla qubits. Here ρa is an initialsingle-qubit ancilla state and U0 = I2, U1 = eiασx , U2 = eiβσy ,and U3 = eiγ σz .Proof. We show that the 2 ⊕ 2 IRs form a NS by explicitevaluation. Let the logical |0〉L lie in the span of {|ea1〉,|ea2〉}and the logical |1〉L lie in the span of {|eb1〉,|eb2〉}. We show thatnoise operators Xα , Yβ , and Zγ leave each subspace invariant.Let Pa =∑2i=1 |eai〉〈eai | and Pb =∑2i=1 |ebi〉〈ebi |. Then it iseasy to show that XαPkX†α = YβPkY †β = ZγPkZ†γ = Pk (k =a,b).Now we prove the identity. We use a pure-state notation tosimplify the expressions. The general case with mixed initialstates ρa and ρˆ is obtained by simply mixing the pure-stateresults using linearity. Let | ˆψ〉 = a|0〉 + b|1〉 be a data qubitstate to be encoded and |v〉 = v0|0〉 + v1|1〉 be the initial stateof the first ancilla qubit, while that of the first qubit is set to|0〉. The encoding under the action of U (3)E yields|	〉 = U (3)E |0〉|v〉| ˆψ〉 = v0(a|ea1〉 + b|eb1〉)+ v1(a|ea2〉 + b|eb2〉).Let us consider a noise operator Xα . Its action on |	〉 yieldsXα|	〉 = (v0 cosα + iv1 sinα)(a|ea1〉 + b|eb1〉)+ (v1 cosα + iv0 sinα)(a|ea2〉 + b|eb2〉).The action of the recovery operator U (3)†E recovers the initialstate, except for the second qubit, asU(3)†E Xα|	〉 = |0〉(eiασx |v〉) | ˆψ〉,which shows that data qubit state is immune to Xα . It is shownsimilarly that the data qubit is immune to other error operatorsas well. Since each error is in action with the probability pi ,we have proved the identity in Eq. (5). 044301-2BRIEF REPORTS PHYSICAL REVIEW A 84, 044301 (2011)In contrast with an ordinary quantum error-correcting code,the scheme corrects multiple actions of the error operators.It was shown in the theorem that the central qubit can beany superposition state or mixed state initially and its outputstate is another superposition or mixed state under an actionof a single error operator in Xα,Yβ , and Zγ . Note that theerror channel leaves the encoded word unchanged, namely,given any initial ancilla state ρa , there exists an ancilla stateρ ′a such that (U (3)E (|0〉〈0| ⊗ ρa ⊗ ρˆ)U (3)†E ) = U (3)E (|0〉〈0| ⊗ρ ′a ⊗ ρˆ)U (3)†E . Then the error correction may be repeated asmany times as required. This implies that it corrects any erroroperator of the form W⊗3, where W ∈ 2. This is because anyelement W ∈ 2 is decomposed as W = eiθ1σx eiθ2σy eiθ3σx . Thisis clear since W⊗3 is expressed as a product Xθ1Yθ2Xθ3 , eachfactor of which leaves the NS invariant.III. THE 4-QUBIT DECOHERENCE FREE SUBSPACEWe design the 4-qubit DFS, which is robust againstcollective noise of the form W⊗4 (W ∈ 2), by taking advantageof the NS analyzed in the preceding section. The algebra A4obtained from 2⊗4 is decomposed as 2⊗4 = 5 ⊕ (I3 ⊗ 3) ⊕(I2 ⊗ 1). Correspondingly, any V ∈ 2⊗4 is decomposed asV = V5 ⊕ (I3 ⊗ V3) ⊕ (I2 ⊗ V1) for a proper choice of basisvectors. Here Vk is an element of a k-dimensional IR ofSU(2)⊗4. The singlet IR is immune to any operator V = W⊗4,where W ∈ 2, and two of them form a single logical qubit thatis immune to any noise of the form V . This (reducible) vectorspace is the DFS that is robust against the collective noise.We generate basis vectors of two one-dimensional IRs ofSU(2) from {|eai〉,|ebi〉} as|0〉L = 1√2(|1〉|ea1〉 − |0〉|ea2〉)= 1√2[|1〉|ea1〉 + |0〉(σx)⊗3|ea1〉], (6)|1〉L = 1√2(|1〉|eb1〉 − |0〉|eb2〉)= 1√2[|1〉|eb1〉 + |0〉(σx)⊗3|eb1〉].It is important in the implementation of the encoding circuitto realize that|0〉L = (X ⊗ I8)(CNNN)(H ⊗ I8)|0〉 ⊗ |ea1〉= (X ⊗ I8)(CNNN)(H ⊗ U (3)E)|0〉 ⊗ |000〉,|1〉L = (X ⊗ I8)(CNNN)(H ⊗ I8)|0〉 ⊗ |eb1〉= (X ⊗ I8)(CNNN)(H ⊗ U (3)E)|0〉 ⊗ |001〉,where CNNN is a controlled-NOT gate with three target bits.Figure 2 shows an example of the encoding circuit for theDFS. In contrast with the three-qubit NS, all the ancilla qubitsmust be initially set to |0〉.IV. THE 5-QUBIT NOISELESS SUBSYSTEMThe NS using five qubits encodes two data qubits. It isrecursively implemented by employing the encoding circuitU(3)E for the three-qubit NS.FIG. 2. Encoding circuit U (4)E of the DFS with n = 4. It encodesa single-qubit state | ˆψ〉.The algebra A5 obtained from 2⊗5 is decomposed as 2⊗5 =6 ⊕ (I4 ⊗ 4) ⊕ (I5 ⊗ 2). Correspondingly, any unitary matrixV ∈ 2⊗5 is decomposed as V = V6 ⊕ (I4 ⊗ V4) ⊕ (I5 ⊗ V2)under a proper choice of basis vectors. We implement a NS byemploying four of five two-dimensional IRs.Let {|eai〉,|ebi〉} be basis vectors introduced for n = 3 inSec. II. We generate eight basis vectors from them as|00〉L = 1√2|ea1〉(|01〉 − |10〉),|01〉L = 1√6[|ea1〉(|01〉 + |10〉) − 2|ea2〉|00〉], (7)|10〉L = 1√2|eb1〉(|01〉 − |10〉),|11〉L = 1√6[|eb1〉(|01〉 + |10〉) − 2|eb2〉|00〉]and their bit-flipped basis vectors, which are obtained byapplying σ⊗5x to them in the same manner that Eqs. (3) and(4) are obtained from the basis vectors |0〉 and |1〉 of n = 1.{{FIG. 3. (a) Encoding circuit U (5)E of the 5-qubit NS, whichencodes a two-data-qubit state | ˆψ〉. The third and fourth qubits inEq. (7) are exchanged to make the recursive symmetry manifest inthis diagram. (b) Encoding circuit U (7)E for the 7-qubit NS, whichencodes a three-data-qubit state | ˆψ〉.044301-3BRIEF REPORTS PHYSICAL REVIEW A 84, 044301 (2011)We emphasize the similar structure between Eqs. (7) andEqs. (3) and (4). This observation makes implementation of theencoding-decoding circuit almost a trivial matter. Note that wedo not need to worry about the rest of the basis vectors as longas they are orthogonal to the above basis vectors spanning theNS. This orthogonalization is automatically taken into accountby the unitarity of the encoding circuit.Figure 3(a) shows an example of the encoding circuit U (5)Eof the NS. The central qubit can be any state while all otherancilla qubits must be in |0〉. Each U (3)E acts on the three qubitsnumbered 1, 2, and 3, which are fed into the input ports 1,2, and 3, respectively, in Fig. 1. The n = 7 NS encodingcircuit can be also constructed recursively, as shown inFig. 3(b).V. CONCLUSIONThe DFS and NS make use of vector subspaces that areimmune to noise of the form W⊗n, where W belongs to 2 ofSU(2). We have constructed simple encoding and decodingquantum circuits for the NS for n = 3 and 5 and the DFS forn = 4. Our strategy is to use the encoding-decoding circuitU(3)E for n = 3 recursively in the implementation for n = 4and 5. We have constructed the bases of 1’s for n = 4 andthe bases of 2’s for n = 5 from the bases of two 2’s forn = 3, as given in Eqs. (6) and (7). We can then generalizethis construction to find the bases of 1’s for n = 2m + 2and the basis of 2’s for n = 2m + 3 from the 2m basesof 2’s for n = 2m + 1. We have implemented DFS andNS encoding-decoding circuits by taking advantage of theserecursive relations among basis vectors of different n. Thenumber of logical qubits in the limit of largen is asymptoticallyn/2 for both even n and odd n. It should be clear from ourconstruction that m logical qubits are implemented by use ofm U(3)E modules, which shows that the circuit complexity forour encoding and decoding circuits increases merely linearlyin m.Note, however, that our construction does not give themaximum number of correctable qubits for the channel. Thereare(nm)− ( nm−1)basis vectors in two-dimensional IRs for n =2m + 1, which encode k = log2[(nm)− ( nm−1)] qubits. Thisnumber k is greater than m for n  9 and actually k/n → 1 asn → ∞. This asymptotic behavior is also observed in Ref. [10]for the DFS.Full details of the recursion scheme are beyond the scopeof the present paper. Moreover, we plan to present an in-depthdiscussion of the decomposition of the algebra 2n and theconstruction of other noiseless subsystems of channels witherror operators in the algebra elsewhere [13].It was shown that the central qubit in Figs. 1and 3 can be any state. Although the entropy of thequbit system generally increases, it remains constant ifthe central qubit is maximally mixed initially as ρa = 12I2.This behavior is somewhat analogous to that of the DFSwith ρa = |0〉〈0|, in which the entropy does not change atall.ACKNOWLEDGMENTSC.-K.L. was supported by a US NSF grant, a HK RGC grant,and the 2011 Shanxi 100 Talent Program. M.N. and H.T. weresupported by the “Open Research Center” Project for PrivateUniversities with a matching fund subsidy from MEXT. M.N.would like to thank the JSPS (Grant No. 23540470) for partialsupport of Grants-in-Aid for Scientific Research. Y.-T.P. wassupported by a US NSF grant. N.-S.S. was supported by a HKRGC grant.[1] M. A. Neilsen and I. L. Chuang, Quantum Computation andQuantum Information (Cambridge University Press, Cambridge,2000).[2] M. Nakahara and T. Ohmi, Quantum Computing, From LinearAlgebra to Physical Realization (CRC, New York, 2008).[3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997).[4] P. Zanardi and M. Rasetti, Mod. Phys. Lett. B 11, 1085 (1997).[5] P. Zanardi, Phys. Rev. A 57, 3276 (1998).[6] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett.81, 2594 (1998).[7] E. Knill, R. Laflamme, and L. Viola, Phys. Rev. Lett. 84, 2525(2000).[8] S. De Filippo, Phys. Rev. A 62, 052307 (2000).[9] C.-P. Yang and J. Gea-Banacloche, Phys. Rev. A 63, 022311(2001).[10] J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, Phys. Rev.A 63, 042307 (2001).[11] D. W. Kribs, R. Laflamme, D. Poulin, and M. Lesosky, QuantumInf. Comput. 6, 382 (2006).[12] L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill,R. Laflamme, and D. G. Cory, Science 293, 2059(2001).[13] C.-K. Li, M. Nakahara, Y.-T. Poon, N.-S. Sze, and H. Tomita(unpublished).044301-4

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Recursive Encoding and Decoding of Noiseless Subsystem and Decoherence Free Subspace

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