Implementation of holonomic quantum computation through engineering and
  manipulating environment
Implementation of holonomic quantum computation through engineering and
manipulating environment
Zhang-qi Yin, Fu-li Li,∗ and Peng Peng
Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China
We consider an atom-field coupled system, in which two pairs of four-level atoms are respectively
driven by laser fields and trapped in two distant cavities that are connected by an optical fiber. First,
we show that an effective squeezing reservoir can be engineered under appropriate conditions. Then,
we show that a two-qubit geometric CPHASE gate between the atoms in the two cavities can be
implemented through adiabatically manipulating the engineered reservoir along a closed loop. This
scheme that combines engineering environment with decoherence-free space and geometric phase
quantum computation together has the remarkable feature: a CPHASE gate with arbitrary phase
shift is implemented by simply changing the strength and relative phase of the driving fields.
PACS numbers: 03.67Lx, 03.65.Vf, 03.65.Yz, 42.50.Dv
I. INTRODUCTION
Quantum computation, attracting much current inter-
est since Shor’s algorithm [1] was proposed, depends on
two key factors: quantum entanglement and precision
control of quantum systems. Unfortunately, quantum
systems are inevitably coupled to their environment so
that entanglement is too fragile to be retained. This
makes the realization of quantum computation extremely
difficult in the real world. In order to overcome this dif-
ficulty, one proposed the decoherence-free space concept
[2, 3]. It is found that when qubits involved in quantum
computation collectively interact with a same environ-
ment there exists a “protected” subspace in the entire
Hilbert space, in which the qubits are immune to the
decoherence effects induced by the environment. This
subspace is called decoherence-free space (DFS). To per-
form quantum computation in a DFS, one has to design
the specific Hamiltonian containing controlling parame-
ters, which eigenspace is spanned by DFS states and the
state-unitary manipulation related to quantum compu-
tation goal is implemented by changing the controlling
parameters [4].
As well known, instantaneous eigenstates of a quantum
system with the time-dependent Hamiltonian may ac-
quire a geometric phase when the time-dependent param-
eters adiabatically undergo a closed loop in the parameter
space [5]. The phase depends only on the swept solid an-
gle by the parameter vector in the parameter space. This
feature can be utilized to implement geometric quantum
computation (GQC) which is resilient to stochastic con-
trol errors [6, 7, 8]. On combining the DFS approach
with the GQC scheme, one may build quantum gates
which may be immune to both the environment-induced
decoherence effects and the control-led errors [9]. In the
scheme, quantum logical bits are represented by degener-
ate eigenstates of the parameterized Hamiltonian. These
∗Email: flli@mail.xjtu.edu.cn
states have the features: they belong to DFS, and unitar-
ily evolve in time and acquire a geometric phase when the
controlling parameters adiabatically vary and undergo a
closed loop.
In the recent paper [10], Carollo and coworkers showed
that a cascade three-level atom interacting with a broad-
band squeezed vacuum bosonic bath can be prepared in a
state which is decoupled to the environment. This state
depends on the reservoir parameters such as squeezing
degree and phase angle. As the squeezing parameters
smoothly vary, the atomic state can unitarily evolve in
time and always be in the manifold of the DFS. More-
over, after a cyclic evolution of the squeezing parameters,
the state acquires a geometric phase. This investigation
has been generalized to cases where both quantum sys-
tems and manipulated reservoir under consideration are
not restricted to cascade three-level atoms and squeezed
vacuum [11]. These results strongly inspire us that in-
stead of engineering Hamiltonian one may implement the
decoherence-free GQC by engineering and manipulating
reservoir.
In this paper, we propose a scheme in which the
quantum-reservoir engineering [12, 13, 14] is combined
with DFS and Berry phase together to realize a two-
qubit CPHASE gate [15]. We show that atomic states
can unitarily evolve in time in a DFS if the change rate
of reservoir parameters is much smaller than the charac-
teristic relaxation time of an atom-reservoir coupled sys-
tem. Moreover, we find that as the reservoir parameters
adiabatically change in time along an appropriate closed
loop, the atomic state in the DFS acquires a Berry phase
and a CPHASE gate with arbitrary phase shift can be
realized. To our knowledge, it is the first proposal for the
realization of quantum gates by engineering and steering
the environment.
This paper is organized as follows. In Sec. II, we in-
troduce a cavity-atom coupling model in which two pairs
of four-level atoms are respectively trapped in two dis-
tant cavities that are connected by an optical fiber. In
the model, each of pairs of the atoms are simultaneously
driven by laser fields and coupled to the local cavity
http://arxiv.org/abs/0704.0482v3
mailto:flli@mail.xjtu.edu.cn
modes through the double Raman transition configura-
tion. Under large detuning and bad cavity limits, we
investigate to engineer an effective broadband squeezing
reservoir for the atoms. In Sec. III, we analyze how to re-
alize controlling gates between the atoms trapped in the
two cavities by steering the squeezing reservoir. Section
IV contains conclusions of our investigations.
laserlaser
fibre
FIG. 1. Atom-field coupling scheme.
|gjn〉
|rjn〉
|sjn〉
δjn|ejn〉
FIG. 2. Atomic level configuration for atom j in cavity
II. ENGINEERING A SQUEEZING
ENVIRONMENT AND GENERATING A
DECOHERENCE-FREE SUBSPACE
Our scheme is shown in Fig.1. A pair of four-level
atoms are trapped in each of two distant cavities, respec-
tively, which are connected through an optical fiber. In
the short fiber limit [16, 17, 18], only one fiber mode b
is excited and coupled to cavity modes a1 and a2 with
strength ν [19]. We assume that the cavity modes and
the fiber mode have the same frequency ω. The level
scheme of atoms is shown in Fig.2. Atom j in cavity n
is labeled by the index jn with j, n = 1, 2. The distance
between the atoms in the same cavity is assumed to be
large enough that there is no direct interaction between
the atoms. The levels |gjn〉 and |ejn〉 of atom j in cavity
n, with j, n = 1, 2 are stable with a long life time. The
energy of the level |gjn〉 is taken to be zero as the energy
reference point. The lower lying level |ejn〉, and upper
levels |rjn〉 and |sjn〉 have the energy δjn, and ωrjn and
ωsjn, respectively, in the unit with ~ = 1. Transitions
|gjn〉 ↔ |sjn〉 and |ejn〉 ↔ |rjn〉 are driven by laser fields
of frequencies ωLsjn and ω
jn with Rabi frequencies Ω
and Ωrjn and relative phase ϕ, respectively. Transitions
|gjn〉 ↔ |rjn〉 and |ejn〉 ↔ |sjn〉 are coupled to the cav-
ity mode an with the strengths g
jn and g
jn, respectively.
Here, we set ∆rjn = ω
jn − ω = ωrjn − ω
jn − δjn, and
∆sjn = ω
jn − ω − δjn = ωsjn − ω
Under the Markovian approximation, the master equa-
tion of the density matrix for the whole system under
consideration can be written as [14]
ρ̇T = −i[H, ρT ] + Lcav1ρT + Lcav2ρT + LfiberρT , (1)
where H = H0 +Hd +Hac +Hcf with
j,n=1
ωrjn|rjn〉〈rjn|+ ωsjn|sjn〉〈sjn|+ δjn|ejn〉〈ejn|)
a†nan + b
j,n=1
(Ωsjn
t|sjn〉〈gjn|
e−i(ω
t+ϕ)|rjn〉〈ejn|+H.c.
Hac =
j,n=1
(grjn|rjn〉〈gjn|an + gsjn|sjn〉〈ejn|an +H.c.),
Hcf =ν
1 + a
2) + H.c.
Here, H0 is the free energy of atoms and cavity fields,
Hd is the interaction energy between the atoms and laser
fields, Hac is the interaction energy between the atoms
and the cavity fields, and Hcf describes the interaction
between the cavity modes and the fiber mode. The last
three terms in (1) describe the relaxation processes of
the cavity and fibre modes in the usual vacuum reservoir,
taking the forms
LcavnρT =κn(2anρTa
n − a†nanρT − ρT a†nan),
LfiberρT =κf (2bρT b
† − b†bρT − ρT b†b),
where κn is the leakage rate of photons from cavity n,
and κf is the decay rate of the fiber mode.
Let’s introduce collective basis: |a〉n = (|g1n〉|e2n〉 −
|e1n〉|g2n〉)/
2, | − 1〉n = |g1n〉|g2n〉, |0〉n = (|g1n〉|e2n〉+
|e1n〉|g2n〉)/
2, |1〉n = |e1n〉|e2n〉. The states |a〉n and
| − 1〉n are taken as a qubit n for quantum computation.
In the large detuning limit, adiabatically eliminating the
excited states and setting
= βrn and
= βsn, from (2), we obtain the effective
interaction Hamiltonian
Heff =
iϕS+n + β
nSn) +H.c.
+Hcf , (4)
where S+n = |0〉nn〈−1|+|1〉nn〈0|. In the derivation of (4),
we have assumed the resonant condition
〈a†nan〉 +
〈a†nan〉+ δ′jn. In order to satisfy the
condition with the flexible choice of Ωrjn, Ω
jn, ∆
jn and
∆sjn, we have introduced additional ac-Stark shifts δ
jn to
states |gjn〉, which can be generated by using a laser field
to couple the level |gjn〉 to an ancillary level.
We now introduce three normal modes c and c± with
frequencies ω and ω±
2ν by use of the unitary transfor-
mation a1 =
(c+ + c− +
2c), a2 =
(c+ + c− −
b = 1√
(c+ − c−) [17, 18]. In the limit ν ≫ |βrj |, |βsj |,
neglecting the far off-resonant modes c± and setting
1 = −β
2 = β
p with p = r, s, we can approximately
write the effective Hamiltonian (4) as
Heff = (β
reiϕS+ + βsS)c+H.c., (5)
where S+ = S+1 + S
Since the modes c± are nearly not excited and decou-
pled with the resonant mode c, the fiber mode b is mostly
in the vacuum state, therefore, LfiberρT can be neglected,
and Lcav1ρT + Lcav2ρT can be approximated as
LcavρT = κ(2cρT c
† − c†cρT − ρT c†c), (6)
where κ = (κ1 + κ2)/2.
In the bad cavity limit, κ≫ β, adiabatically eliminat-
ing the mode c [12, 14], from Eq. (1) with the replace-
ment of the Hamiltonian (2) and the relaxation terms (3)
by the effective Hamiltonian (5) and the relaxation term
(6), respectively, we can obtain the master equation for
the density matrix of the atoms
ρ̇ = −Γ
(R+Rρ+ ρR+R− 2RρR+), (7)
where ρ = Trf (ρT ), R = S cosh r + e
iϕS† sinh r, r =
cosh−1(βr/
βr2 − βs2) and Γ = 2(βr2−βs2)/κ. Eq. (7)
describes the collective interaction of two cascade three-
level atoms with the effective squeezed vacuum reservoir
[10]. The parameters βr, βs and ϕ are easily changed and
controlled at will by varying the strength and phase of the
driving lasers [8]. We will show that a geometric phase
gate can be realized through changing these parameters.
The DFS of the atomic system is spanned by the
states which satisfy the equation R(r, ϕ)|ψDF(r, ϕ)〉 = 0
[10]. In terms of basis states |e1〉 = |a〉1|a〉2, |e2〉 =
|a〉1| − 1〉2, |e3〉 = | − 1〉1|a〉2, |e4〉 = |a〉1|0〉2, |e5〉 =
|0〉1|a〉2, |e6〉 = |a〉1|1〉2, |e7〉 = |1〉1|a〉2, |e8〉 =
|1〉1|1〉2, |e9〉 = 1√2 (|1〉1|0〉2 + |0〉1|1〉2), |e10〉 = | −
1〉1| − 1〉2, |e11〉 = 1√2 (|0〉1| − 1〉2 + | − 1〉1|0〉2), |e12〉 =
(|1〉1|−1〉2+ |−1〉1|1〉2)+ 2√6 |0〉1|0〉2), the DFS states
can be written as
|ψDF(r, ϕ)〉1 =|e1〉,
|ψDF(r, ϕ)〉j =
cosh r√
cosh 2r
|ej〉 − eiϕ
sinh r√
cosh 2r
|ej+4〉, j = 2, 3,
|ψDF(r, ϕ)〉4 =
e2iϕ(tanh r)2|e8〉 −
eiϕ tanh r|e12〉+ |e10〉
(tanh r)4 + 2
(tanh r)2 + 1
Let’s introduce a unitary transformation O(r, ϕ) by
|φi〉 =
i=1Oij(r, ϕ)|ej〉, where |φi〉 = |ψDF〉i for i =
1, 2, 3, 4. For the transformed density matrix ρ̄ = O†ρO,
we have
= i[G, ρ̄] +O†
O, (9)
where G(r, ϕ) = iO† dO
= iO†[ṙ dO
+ ϕ̇dO
]. To solve
Eq. (9) in the DFS, let’s define the time-independent
projector Π(0) = O†Π(r, ϕ)O =
†|φi〉〈φi|O =
j=1 |ej〉〈ej | + |e10〉〈e10| onto the DFS. From (9), we
obtain the equation of motion for ρ̄DF = Π(0)ρ̄Π(0)
dρ̄DF
=i[GDF, ρ̄DF] + iΠ(0)GΠ⊥(0)ρ̄Π(0)
− iΠ(0)ρ̄Π⊥(0)GΠ(0) + Π(0)O†
OΠ(0),
where Π⊥(0) = 1− Π(0) and GDF = Π(0)GΠ(0). In the
limit of ṙ, ϕ̇≪ Γ, the last three terms in Eq. (10) can be
neglected [11]. In this way, Eq. (10) is reduced to
dρ̄DF
= i[GDF, ρ̄DF]. (11)
Therefore, in the frame dragged adiabatically by the
reservoir, the state of the atoms in the DFS unitarily
evolves in time.
III. REALIZING CONTROLLING PHASE
GATES THROUGH MANIPULATING THE
SQUEEZING ENVIRONMENT
In this section, we investigate how to realize a
CPHASE gate through manipulating the engineered
reservoir. Suppose that at the initial time the laser
field driving the transition |g〉 ↔ |s〉 is switched off
but the laser field driving the transition |r〉 ↔ |e〉 is
switched on and the atoms are in the DFS state |Ψ(0)〉a =
(|a〉1|a〉2 + |a〉1| − 1〉2 + | − 1〉1|a〉2 + | − 1〉1| − 1〉2) =
j=1 |ψDF(0, 0)〉j/2. To generate a geometric phase for
the atomic state, we smoothly change the parameters of
the engineered reservoir along a closed loop, which is di-
vided into the following three steps: (1) From time 0 to
T1, hold on ϕ = 0, and adiabatically increase the parame-
ter r from 0 to r0; (2) From time T1 to T2, hold on r = r0,
and adiabatically change the phase ϕ from 0 to ϕ0; (3)
From time T2 to T3, hold on ϕ = ϕ0, and adiabatically
decrease r from r0 to 0. When the cyclic evolution ends,
the atomic state becomes
|Ψ(T3)〉a =
(|e1〉+ eiχ1 |e2〉+ eiχ1 |e3〉+ eiχ12 |e10〉),
where geometric phases χ1 = −ν1ϕ0, χ12 = −ν12ϕ0 with
sinh2 r0
sinh2 r0+cosh
, ν12 =
2 tanh4 r0+
tanh2 r0
tanh4 r0+
tanh2 r0+1
. By per-
forming local transformations U1 = e
−iχ1 |−1〉11〈−1| and
U2 = e
−iχ1 | − 1〉22〈−1|, the state (12) can be written as
|Ψ′(T3)〉a = U1U2|Ψ(T3)〉a = 12 (|a〉1|a〉2 + |a〉1| − 1〉2 +
| − 1〉1|a〉2 + ei∆| − 1〉1| − 1〉2), where ∆ = χ12 − 2χ1 =
(2ν1 − ν12)ϕ0. Thus, the CPHASE gate with the phase
shift ∆ is realized. If both the atoms in cavity 1 and
the atoms in cavity 2 “see” different environments, |χ12|
must be equal to |2χ1| and ∆ = 0. Therefore, the phase
shift ∆ results from the collective coupling of the atoms
in both cavities with the same engineered environment.
If r0 = atanh(
4/3− 1) ≃ 0.4157, |ν12| = |ν1|. Under
this condition with ϕ0 = π/ν1, the state of the atoms at
the time T3 is |Ψ′′(T3)〉a = − 12 (−|a〉1|a〉2 + |a〉1| − 1〉2 +
| − 1〉1|a〉2 + | − 1〉1| − 1〉2). In this case, the Controlled-
Z gate between the two qubits is realized without local
transformations.
0 0.2 0.4 0.6 0.8
0.9965
0.997
0.9975
0.998
0.9985
0.999
0.9995
T=100/Γ
T=200/Γ
T=400/Γ
FIG. 3. Fidelity Fr of the atomic state.
0 0.2 0.4 0.6 0.8
0.965
0.975
0.985
0.995
T = 200/Γ
T = 400/Γ
T = 1000/Γ
FIG. 4. Fidelity Fp of the atomic state.
The above results depend on the adiabatical approxi-
mation. To check the adiabatical condition, we numer-
ically simulate the following two examples. In the first
example, we suppose that at the initial time the atoms
are in the state |Ψ1〉a = (|a〉1|a〉2+ |ψDF(0, 0)〉2)/
2 and
the laser field driving the transition |e〉 ↔ |r〉 are turned
on. Then, by slowly switching the laser field driving the
transition |g〉 ↔ |s〉, we increase the parameter r from 0
to r0 according to the linear function r(t) = r0t/T . In the
adiabatical limit (T ≫ Γ−1), the atomic state becomes
|Ψ′1〉a = (|a〉1|a〉2 + |ψDF(r0, 0)〉2)/
2 at the time T . On
the other hand, in the Hilbert space spanned by the ba-
sis states {|ei〉} for i = 1, 2, · · · , 12 , we can numerically
solve Eq. (7) and obtain the density matrix ρ1(T ) of the
atoms. Let’s define Fr = a〈Ψ′1|ρ1(T )|Ψ′1〉a as the fidelity
for this process. As shown in Fig. 3, if T > 100/Γ, Fr is
always bigger than 0.997 if r ∈ (0, 0.8), corresponding to
the almost perfect evolution.
In the second example, we suppose that the atoms are
initially in the state |Ψ2〉a = (|a〉1|a〉2+ |ψDF(r, 0)〉4)/
and all the driving fields are turned on to hold the pa-
rameters r = r0 and ϕ = 0. By adiabatically changing
the phase ϕ from 0 to 2π at the rate ϕ̇ = 2π/T , the
atomic state at the time T becomes |Ψ′2〉a = (|a〉1|a〉2 +
eiχ12 |ψDF(r, 2π)〉4)/
2. Let’s define the fidelity for this
example as Fp = a〈Ψ′2|ρ(T )|Ψ′〉a, where ρ(T ) is the nu-
merical solution of Eq. (7). As shown in Fig. 4, Fp
increases as T increases but decreases as the parameter
r0 increases. If T > 1000/Γ, Fp is larger than 0.992 for
0 < r0 < 0.8. From these two examples, we find that to
fulfill the adiabatical condition the time used in the step
2 should be much longer than in the steps 1 and 3.
A controlled-Z gate has been numerically simulated
by directly solving Eq. (7) with r0 = 0.5, and ϕ0 =
π/|2ν1 − ν12|. In the simulation, we set ṙ = r0/T1 in the
steps 1 and 3, and ϕ̇ = ϕ0/(T2 − T1) in the step 2 with
T1 = 0.05T3 and T2 − T1 = 0.90T3. If T3 > 1100/Γ, we
find that the fidelity F = a〈Ψ(T3)|ρ(T3)|Ψ(T3)〉a is larger
than 0.95. For an almost perfect controlled-Z gate with
F > 0.99, we find that T3 must be longer than 6000/Γ.
Now let’s briefly discuss the effects of the atomic spon-
taneous emission, the fiber mode decay and cavity photon
leakage. For simplicity but without the loss of generality,
we suppose that atomic spontaneous emission rates of the
excited levels are equal to γ. In the large detunig limit,
the characteristic spontaneous emission rate of the atoms
is γeff = γ(Ω
2/2∆2) [14, 16] and the effective decay rate
of the fiber mode is κeff = κfΩ
2g2/(4∆2ν2). If κf ≤ γ
and g2 ≪ ν2, κeff can be much smaller than γeff . Under
this condition, the present scheme is feasible if Γ ≫ γeff .
In the current cavity quantum dynamic (CQED) ex-
periment, the parameters (g, κ, γ) = (2000, 10, 10) MHz
could be available [20]. If setting Ω/(2∆) = 1√
× 10−3,
we have Γ ≃ 4 × 104γeff . The condition is held. In
the present scheme, the large cavity decay rate is re-
quired to ensure that the cavity modes are in a broad-
band squeezed vacuum reservoir and then the atoms al-
ways ”see” the broadband squeezed vacuum reservoir
during the dynamic evolution. For an arbitrary small but
nonzero value of the squeezing degree of the reservoir, a
CPHASE gate with arbitrary high fidelity can always be
realized in the represent scheme. The cavity decay does
not directly affect the fidelity of the realized CPHASE
gates. However, the larger the decay rate is, the longer
the operation time of the CPHASE gates is. Thus, we
have the condition κ >> β, γeff for realizing the reliable
CPHASE gates. Based on the parameters quoted above,
this condition can be well satisfied. With the parameters
of the current CQED experiment, we find that the op-
eration time of the controlled-Z gate, with fidelity larger
than 0.95, is about 2.8 ms. It is much shorter than both
1/γeff and the single-atom trapping time in cavity [21].
On the other hand, the present scheme needs a strong
coupling between the cavity and the fiber. This could be
realized at the current experiment [22]. Therefore, the
requirement for the realization of the present scheme can
be satisfied with the current technology.
IV. CONCLUSIONS
We propose a cavity-atom coupled scheme for the real-
ization of quantum controlling gates, in which each of two
pairs of four-level atoms in two distant cavities connected
by a short optical fibre are simultaneously driven by laser
fields and coupled to the local cavity modes through the
double Raman transition configuration. We show that
an effective squeezing reservoir coupled to the multilevel
atoms can be engineered under appropriate driving con-
dition and bad cavity limit. We find that in the scheme
a CPHASE gate with arbitrary phase shift can be im-
plemented through adiabatically changing the strength
and phase of driving fields along a closed loop. It is also
noticed that the larger the effective coupling strength be-
tween the environment and the atoms is, the more reli-
able the realized CPHASE gate is.
Acknowledgments
We thank Yun-feng Xiao and Wen-ping He for valuable
discussions and suggestions. This work was supported
by the Natural Science Foundation of China (Grant Nos.
10674106, 60778021 and 05-06-01).
[1] P. W. Shor, in Proceeings, 35th Annual Symposium on
Foundations of Computer Science, edited by S. Gold-
wasser (IEEE Press, Los Almitos, CA, 1994), p. 124.
[2] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. 79, 1953
(1997).
[3] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306
(1997).
[4] D. A. Lidar and et al., Phys. Rev. Lett. 81, 2594 (1998).
[5] M. V. Berry, Proc. R. Soc. A 392, 45 (1984).
[6] P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 (1999).
[7] S.-L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301
(2005).
[8] L.-M. Duan and et al., Science 292, 1695 (2001).
[9] L.-A. Wu and et al., Phys. Rev. Lett. 95, 130501 (2005).
[10] Angelo Carollo et al., Phys. Rev. Lett. 96, 150403 (2006).
[11] Angelo Carollo et al., Phys. Rev. Lett. 96, 020403 (2006).
[12] J. I. Cirac, Phys. Rev. A 46, 4354 (1992).
[13] C. J. Myatt and et al., Nature (London) 403, 269 (2000).
[14] S. G. Clark and A. S. Parkins, Phys. Rev. Lett. 90,
047905 (2003).
[15] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995).
[16] T. Pellizzari, Phys. Rev. Lett. 79, 5242 (1997).
[17] A. Serafini and et al., Phys. Rev. Lett. 96, 010503 (2006).
[18] Zhang-qi Yin and Fu-li Li, Phys. Rev. A 75, 012324
(2007).
[19] S. J. van Enk and et al., Phys. Rev. A 59, 2659 (1999).
[20] S. M. Spillane et al., Phys. Rev. A 71, 013817 (2005).
[21] Stefan Nuβmann and et al., Nature Phys. 1, 122 (2005).
[22] S. M. Spillane et al., Phys. Rev. Lett. 91, 043902 (2003).
