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Quantum Inf Process (2017) 16:294
DOI 10.1007/s11128-017-1747-z
Dressed-state scheme for a fast CNOT gate
Jin-Lei Wu1 · Xin Ji1 · Shou Zhang1
Received: 25 February 2017 / Accepted: 12 October 2017
© Springer Science+Business Media, LLC 2017
Abstract A dressed-state scheme, which aims to speed up the adiabatic population
transfer, is applied for constructing a fast controlled-not (CNOT) gate in a cavity
quantum electrodynamics system. Numerical simulations indicate that the average
fidelity for constructing the CNOT gate is quite high and the gate operation time is
relatively short. Moreover, the effects of the atomic spontaneous emissions and the
photon leakages from the system on the average fidelity are discussed and the results
show the scheme is robust against decoherence.
Keywords CNOT gate · Shortcuts to adiabaticity · Dressed states
1 Introduction
Quantum logic gates are key elements of a quantum computer which possesses stronger
computational power and faster operational speed than a classical computer [1]. As
we all know, all gate operations in quantum computation can be decomposed into a
series of elementary one-qubit unitary gates and two-qubit conditional gates which
are universal for quantum computation [2,3]. Many schemes have been proposed
to implement quantum gate operations in various physical systems, such as ion-trap
systems [4], linear optical systems [5,6], cavity QED systems [7–10], nuclear magnetic
resonance systems [11] and superconducting systems [12,13]. The CNOT gate is an
important two-qubit universal gate which can be used to construct multiqubit gates
combined with single-qubit gates. Up to present, many schemes have been proposed
B
1
Xin Ji
jixin@ybu.edu.cn
Department of Physics, College of Science, Yanbian University, Yanji, Jilin 133002,
People’s Republic of China
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J.-L. Wu et al.
to construct CNOT gates [14–18]. For example, Franson et al. realized a CNOT gate
through the quantum Zeno effect of two optical qubits by two-photon absorbtion [15];
Sangouard et al. implemented a CNOT gate by adiabatic passage with an optical cavity
[16]; Shao et al. proposed two schemes to perform CNOT gates via quantum Zeno
dynamics (QZD) and stimulated Raman adiabatic passage (STIRAP), respectively
[18].
As we can see from the references [14–18], QZD and STIRAP are two techniques
widely used to perform population transfer because of their robustness against decoherence in proper conditions. However, either QZD or STIRAP has its own unavoidable
defects. QZD is sensitive to the atomic spontaneous emission and variation in operation
time. STIRAP usually requires a relatively long interaction time which will accumulate decoherence and destroy the desired dynamics. Therefore, many researchers have
paid more attention to speeding up the adiabatic population transfer (i.e., shortcuts to
adiabaticity) [19–31]. Besides, some remarkable achievements for shortcuts to adiabaticity have been implemented in experiment [32–36]. There are two methods,
transitionless quantum driving and Lewis–Riesenfeld invariant, widely used. Also,
many schemes have been proposed for constructing quantum gates based on the two
methods [37–40]. However, the two methods suffer from some experimental obstacles
in many cases. On one hand, transitionless quantum driving requires either a direct
coupling between the initial and target states [21,41,42] or an unavailable coupling in
the original Hamiltonian [32]. On the other hand, Lewis–Riesenfeld invariant usually
leads to pulse schemes that either need an infinite energy gap to be perfect [23], or do
not smoothly turn on or off [23,43].
Recently, Baksic et al. proposed a new method to speed up adiabatic population
transfer by using dressed states [44]. It is more experimentally feasible than transitionless quantum driving and Lewis–Riesenfeld invariant, because it skillfully escapes
from the experimental obstacles the latter two methods suffer from. By using the
dressed-state method, we have proposed a scheme for fast two-atom quantum state
transfer and entanglement generation in a cavity QED system [45]; Kang et al. [46]
proposed a scheme for generating a three-qubit W state in a superconducting system;
schemes for holonomic quantum computation [47] and quantum state conversion [48]
are also proposed. In this work, we apply the dressed-state method to construct a fast
CNOT gate in a cavity QED system which consists of two identical five-level atoms
trapped, respectively, in two single-mode optical cavities connected by a fiber.
2 Physical model and general process
The schematic setup for constructing the fast CNOT gate is shown in Fig. 1. Two
identical five-level atoms A and B are trapped in two single-mode optical cavities,
respectively. The short fiber limit (l)/(2π c) ≤ 1 ensures that only the resonant
modes of the fibers interact with the cavity modes [49], where l and are the length
of the fiber and the decay rate of the cavity field into a continuum of fiber modes,
respectively. Each atom has an upper level |e and four lower levels |a, |g1 , |g2 and
|g0 . The atomic transitions |eA(B) ↔ |g0 A(B) is resonantly coupled to the mode
of the cavity A(B) with corresponding coupling constant gA(B) , and the transitions
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Dressed-state scheme for a fast CNOT gate
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294
Fig. 1 The diagrammatic sketch of cavity-atom combined system, atomic level configuration and related
transitions
|eB ↔ |aB , |eB ↔ |g1 B and |eB ↔ |g2 B are resonantly driven by classical fields
with time-dependent Rabi frequencies i (t)(i = 11, 1a, 22, 21, 3a, 32) shown at the
right side in Fig. 1 which will be described hereinafter in detail.
The quantum information is encoded in the states
|g1 g1 AB ≡ |00, |g1 g2 AB ≡ |01, |g0 g1 AB ≡ |10, |g0 g2 AB ≡ |11, (1)
where |gi g j AB (i = 0, 1; j = 1, 2) denotes the atoms A and B in the states |gi A
and |g j B , respectively. Assume the system is initially in the superposition state
|0 = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB
+ cos ε sin β|g0 g1 AB + cos ε cos β|g0 g2 AB ,
(2)
where ε and β are chosen as two real parameters for convenience of simulations. After
a CNOT gate operation on the initial state |0 , the outcome state becomes
| = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB
+ cos ε sin β|g0 g2 AB + cos ε cos β|g0 g1 AB .
(3)
Here, atom A acts as the control qubit, and atom B is the target qubit.
Only three steps are needed to achieve such a CNOT gate operation. Firstly, we
implement the complete population transfer |g0 g1 AB → |g0 aAB by using the laser
pulses interacting with atom B to drive the atomic transitions |eB ↔ |g1 B and
|eB ↔ |aB with the corresponding Rabi frequencies 11 (t) and 1a (t), respectively.
Then the initial state becomes
|1 = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB + cos ε sin β|g0 aAB
+ cos ε cos β|g0 g2 AB .
(4)
Secondly, we implement the complete population transfer |g0 g2 AB → |g0 g1 AB by
using the laser pulses interacting with atom B to drive the atomic transitions |eB ↔
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J.-L. Wu et al.
Fig. 2 Schematic representation of the three steps for constructing a CNOT gate
|g2 B and |eB ↔ |g1 B with the corresponding Rabi frequencies 22 (t) and 21 (t),
respectively. Then the state of the system becomes
|2 = sin ε sin β|g1 g1 AB + sin ε cos β|g1 g2 AB + cos ε sin β|g0 aAB
+ cos ε cos β|g0 g1 AB .
(5)
Finally, we implement the complete population transfer |g0 aAB → |g0 g2 AB by using
the laser pulses interacting with atom B to drive the atomic transitions |eB ↔ |aB
and |eB ↔ |g2 B with the corresponding Rabi frequencies 3a (t) and 32 (t), respectively. And thus the state of the system becomes the expected outcome state |. The
process above is equivalent to a CNOT gate operation, which indicates we can construct a CNOT gate by such three steps shown in Fig. 2.
3 Dressed-state scheme for constructing the CNOT gate
For the first step, the time-dependent interaction Hamiltonian of the whole system is
written as (setting h̄ = 1)
H1 (t) = Hal (t) + Hacf ,
Hal (t) = 11 (t)|eB g1 | + 1a (t)|eB a| + H.c.,
(gk ak |ek g0 | + νbak† ) + H.c.,
Hacf =
(6)
k=A,B
where aA(B) is the annihilation operator of the mode of the cavity A(B), b is the
annihilation operator of the fiber mode, and ν is the coupling strength between the
cavities modes and the fiber mode. For simplicity, we assume gA = gB = g. Then
with the initial state |0 in Eq. (2), dominated by the Hamiltonian (6), the whole
system evolves in the Hilbert subspace spanned by
|φ1 = |g1 g1 AB |000, |φ2 = |g1 g2 AB |000, |φ3 = |g0 g1 AB |000,
|φ4 = |g0 g2 AB |000, |φ5 = |g1 eAB |000, |φ6 = |g0 eAB |000,
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Dressed-state scheme for a fast CNOT gate
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294
|φ7 = |g1 aAB |000, |φ8 = |g1 g0 AB |001, |φ9 = |g0 aAB |000,
|φ10 = |g0 g0 AB |001, |φ11 = |g1 g0 AB |010, |φ12 = |g0 g0 AB |010,
|φ13 = |g1 g0 AB |100, |φ14 = |g0 g0 AB |100, |φ15 = |eg0 AB |000, (7)
in which the unsubscripted ket |i jk (i, j, k = 0, 1) denotes i, j and k photon in the
cavity A, fiber and cavity B, respectively.
Then according to the theory of quantum Zeno dynamics [50,51], the eigenstates of the atom–cavity–fiber interaction Hamiltonian Hacf can be split into several
eigenspaces (i.e., quantum Zeno subspaces), and the eigenstates in the same Zeno
subspace are with the same eigenvalue. It is easy to know that the differences between
the eigenvalues of Hacf are positively related to the values of g and v. The atom–laser
interaction Hamiltonian Hal (t) brings a coupling between different eigenspaces with
a coupling strength positively related to 11 (t) and 1a (t). By choosing the quantum
Zeno limit condition 11 (t), 1a (t) g, v, the coupling between different subspaces
can be ignored, which means that the evolution of the system will be limited in the
subspace where the initial state exists in. Because the initial state |0 is the dark
state of Hacf (i.e., Hacf |0 = 0), the whole system will approximatively evolve in
the Zeno subspace consisting of dark states of Hacf
H P = {|φ1 , |φ2 , |φ3 , |φ4 , |φ7 , |φ9 , |φd } ,
(8)
corresponding to the projections
P α = |αα|, (|α ∈ H P ).
(9)
|φd = (ν|φ6 − g|φ12 + ν|φ15 )/ 2ν 2 + g 2 .
(10)
Here,
Therefore, by setting v = g, we can rewrite the system Hamiltonian as the following
form [18,52,53]
P α Hal (t)P α = 1 (t)|φ3 φd | + 2 (t)|φ9 φd | + H.c.,
(11)
H (t) α
√
√
in which 1 (t) = 11 (t)/ 3 and 2 (t) = 1a (t)/ 3. Except |g0 g1 AB , obviously,
|g1 g1 AB , |g1 g2 AB and |g0 g2 AB do not participate in the system evolution governed
by the Hamiltonian (11).
By choosing
1 (t) = −(t) sin θ (t), 2 (t) = (t) cos θ (t),
(12)
with (t) = 1 (t)2 + 2 (t)2 and θ (t) = − arctan[1 (t)/ 2 (t)], we can easily
obtain the time-dependent eigenstates of H (t)
|ϕd (t) = cos θ (t)|φ3 + sin θ (t)|φ9 ,
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J.-L. Wu et al.
√
|ϕ± (t) = [sin θ (t)|φ3 ∓ |φd − cos θ (t)|φ9 ]/ 2,
(13)
with the eigenvalues E d = 0 and E ± = ±(t), respectively. Assume that the system
evolution is started at t = ti and finished at t = t f . The state transfer from |φ3 to |φ9 can be achieved via√|ϕd (t) by setting θ (ti ) = 0 and θ (t f ) = π/2 under the adiabatic
criterion |θ̇(t)| 2(t) [54,55], which requires a relatively long operation time.
In the following, we use the dressed-state scheme to speed up the state transfer.
For convenience, we transform the time-dependent Hamiltonian
(11) to the timeindependent eigenstates frame by the unitary operator U (t) =
j=d,± |ϕ j ϕ j (t)|.
In the time-independent eigenstates frame the Hamiltonian (11) becomes
Had (t) = (t)Mz + θ̇ (t)M y ,
(14)
√
where Mz = |ϕ+ ϕ+ | − |ϕ− ϕ− | and M y = i(|ϕ+ + |ϕ− )ϕd |/ 2 + H.c.. In order
to protect the system evolution from the second term of the Hamiltonian (14) which
leads to an imperfect population transfer, we introduce the modified Hamiltonian
Hmod (t) = H (t) + Hc (t) to govern a perfect population transfer with the addition of
a correction Hamiltonian Hc (t). Hc (t) can be given by the general form
Hc (t) = U † (t)[gx (t)Mx + gz (t)Mz ]U (t),
(15)
√
with Mx = (|ϕ− − |ϕ+ )ϕd |/ 2 + H.c.. gx (t) and gz (t) are two undetermined
parameters. Thus the Hamiltonian (11) becomes
Hmod (t) = H (t) + Hc (t) = 1 (t)|φ3 φd | + 2 (t)|φ9 φd | + H.c.,
(16)
with the modified pulses
1 (t) = gx (t) cos θ (t) − [gz (t) + (t)] sin θ (t),
2 (t) = gx (t) sin θ (t) + [gz (t) + (t)] cos θ (t),
(17)
and then the Hamiltonian (6) becomes
H1mod (t) = 11 (t)|eB g1 | + 1a (t)|e B a|
+
gk ak |ek g0 | + νbak† + H.c.,
(18)
k=A,B
√
√
with 11 (t) = 31 (t) and 1a (t) = 32 (t).
With reference to Ref. [44], we introduce a set of dressed states |ϕ̃±,d (t) by the
unitary operation |ϕ̃±,d (t) = V † (t)|ϕ±,d . We choose the unitary operator
V (t) = exp[iμ(t)Mx ],
123
(19)
Dressed-state scheme for a fast CNOT gate
Page 7 of 15
294
with an Euler angle μ(t). After transforming the modified Hamiltonian (16) to the
time-independent dressed-state frame defined by V (t), the Hamiltonian (16) becomes
Hnew (t) = V (t)Had (t)V † (t) + V (t)U (t)Hc (t)U † (t)V † (t) + i
= η(t)(|ϕ̃+ ϕ̃+ | − |ϕ̃− ϕ̃− |)
+ [ξ1 (t)|ϕ̃+ ϕ̃d | + ξ2 (t)|ϕ̃− ϕ̃d | + H.c.],
d V (t) †
V (t)
dt
(20)
with |ϕ̃±,d = V (t)|ϕ̃±,d (t) and three time-dependent parameters
η(t) = [gz (t) + (t)] cos μ(t) − θ̇ (t) sin μ(t),
√
ξ1 (t) = {i[gz (t) + (t)] sin μ(t) + i θ̇ (t) cos μ(t) + μ̇(t) − gx (t)}/ 2,
√
ξ2 (t) = {i[gz (t) + (t)] sin μ(t) + i θ̇ (t) cos μ(t) − μ̇(t) + gx (t)}/ 2. (21)
After simple calculations, we choose
gx (t) = μ̇(t), gz (t) = −(t) −
θ̇(t)
,
tan μ(t)
(22)
to remove the second term of the Hamiltonian (20), which prevents the Hamiltonian (20) from driving transitions between different dressed states. Then back to the
original frame, the dark dressed state, which corresponds to the zero eigenvalue of
Hnew (t), is written as
|ϕ0 (t) = U † (t)V † (t)|ϕ̃d = cos μ(t) [cos θ (t)|φ3 + sin θ (t)|φ9 ]
− i sin μ(t)|φd .
(23)
If the parameters satisfy θ (ti ) = 0, θ (t1 f ) = π/2 and μ(ti ) = μ(t1 f ) = 0, where
ti(1 f ) is the initial (final) time of the first step for constructing the CNOT gate, the
desired population transfer |φ3 → |φ9 will be achieved by the system evolution
along the dark dressed state |ϕ0 (t).
Based on the process above, we achieve the transfer |g0 g1 AB → |g0 aAB , but
the states |g1 g1 AB , |g1 g2 AB and |g0 g2 AB remain unchanged. Therefore, the first
step is achieved for constructing the CNOT gate. Besides, the evolution process is not
necessarily slow, and there is not a direct coupling between the initial and target states.
Similar to the first step, the modified Hamiltonian of the second step for constructing
the CNOT gate is written as
H2mod (t) = 22 (t)|eB g2 | + 21 (t)|eB g1 |
+
(gk ak |ek g0 | + νbak† ) + H.c.,
(24)
k=A,B
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J.-L. Wu et al.
in which
22 (t) =
√
3{gx (t − t1 f ) cos θ (t − t1 f ) − [gz (t − t1 f )
+ (t − t1 f )] sin θ (t − t1 f )},
√
21 (t) = 3{gx (t − t1 f ) sin θ (t − t1 f ) + [gz (t − t1 f )
+ (t − t1 f )] cos θ (t − t1 f )}.
(25)
If the parameters satisfy θ (t1 f ) = 0, θ (t2 f ) = π/2 and μ(t1 f ) = μ(t2 f ) = 0, where
t1 f (2 f ) is the initial (final) time of the second step for constructing the CNOT gate,
the desired transfer |g0 g2 AB → |g0 g1 AB will be achieved.
Similarly, the modified Hamiltonian of the third step for constructing the CNOT
gate is written as
H3mod (t) = 3a (t)|eB a| + 32 (t)|eB g2 |
(gk ak |ek g0 | + νbak† ) + H.c.,
+
(26)
k=A,B
in which
3a (t) =
32 (t)
√
3{gx (t − t2 f ) cos θ (t − t2 f )
− [gz (t − t2 f ) + (t − t2 f )] sin θ (t − t2 f )},
√
= 3{gx (t − t2 f ) sin θ (t − t2 f )
+ [gz (t − t2 f ) + (t − t2 f )] cos θ (t − t2 f )}.
(27)
If the parameters satisfy θ (t2 f ) = 0, θ (t3 f ) = π/2 and μ(t2 f ) = μ(t3 f ) = 0, where
t2 f (3 f ) is the initial (final) time of the third step for constructing the CNOT gate,
the desired transfer |g0 aAB → |g0 g2 AB will be achieved. By now, the transform
|0 → | is achieved and we implement the CNOT gate successfully.
4 Numerical simulations
First of all, 1 (t) and 2 (t) can be chosen as the Gaussian pulses [54,55]
1 (t) = 0 exp[−(t − t f /2 − t0 )2 /τ 2 ],
2 (t) = 0 exp[−(t − t f /2 + t0 )2 /τ 2 ],
(28)
in which the related parameters are chosen as t f = 20/0 and τ = t0 = 0.1t f .
Besides, there exist the relations t1 f = t f , t2 f = 2t f and t3 f = 3t f . μ(t) is defined
by
μ(t) = − arctan
123
θ̇ (t)
,
G(t)/τ + (t)
(29)
Dressed-state scheme for a fast CNOT gate
θ(t)
(a)
294
π/2
1
(b)
μ(t)
2
Page 9 of 15
0
−0.2
−0.4
0
Step 1
Step 2
10
20
Step 3
30
40
50
60
t/Ω−1
0
Fig. 3 θ (t) and μ(t) versus t/ −1
0 . The parameters used here are {t f = 20/ 0 , τ = t0 = 0.1t f }
where G(t) = sech(t/τ ) is chosen to regularize μ(t) such that it can meet the
condition μ(ti ) = μ(t f ) = 0 and make sin2 μ(t), the population of |φd [see
Eq. (23)], as small as possible. Then, in order to check whether or not the parameters we choose are suitable, we plot θ (t) and μ(t) versus t/−1
0 in Fig. 3. Obviously,
Fig. 3 shows that the chosen parameters are in accord with all of the boundary conditions {θ (ti ) = 0, θ (t1 f ) = π/2, μ(ti ) = μ(t1 f ) = 0}, {θ (t1 f ) = 0, θ (t2 f ) =
π/2, μ(t1 f ) = μ(t2 f ) = 0} and {θ (t2 f ) = 0, θ (t3 f ) = π/2, μ(t√
2 f ) = μ(t3 f ) = 0}
for constructing the CNOT gate. Based on the relations {11 (t) = 31 (t), 1a (t) =
√ 32 (t), Eqs. (25), (27)}, we can determine the six time-dependent modified pulses
11 (t), 1a (t), 22 (t), 21 (t), 3a (t) and 32 (t). In Fig. 4a, we plot the six modified pulses versus t/−1
0 . And in Fig. 4b, we plot the time-dependent population
conversions of the three steps for constructing the CNOT gate, respectively. Here, for
the Zeno limit condition i (t) g, v, we choose g = v = 100 . In Fig. 4a, the six
modified pulses can be replaced by complete Gaussian pulses by curve fitting and can
be smoothly turned on and off. According to the recent experiment [35], the values
and plus or minus signs of the pulses can be easily controlled by modulating their
amplitudes and phases, respectively. From Fig. 4a, we know that the modified pulses
amplitude 0 ≈ 2.50 , and then g = v ≈ 40 . However, Fig. 4b shows that all
of the expected population conversions are achieved near perfectly, which shows that
the CNOT gate can also be constructed even when the condition i (t) g, v is not
strictly satisfied.
Next, we show the average fidelity of each step in Fig. 5a and that of the whole gate
process in Fig. 5b, respectively. The average fidelity is defined by
F=
1
(2π )2
2π
0
2π
|ideal |(t)|2 dεdβ.
(30)
0
Here, |ideal = |1 for the step 1; |ideal = |2 for the step 2; |ideal = | for
the step 3 and the whole gate process. |(t) is the state of the system governed by the
Hamiltonian (18) for the step 1, Hamiltonian (24) for the step 2, or Hamiltonian (16)
for the step 3. As shown in Fig. 5a, the average fidelity of each step is near unit at
each finial time. Correspondingly, in Fig. 5b, the average fidelity of the whole gate
123
J.-L. Wu et al.
(a)
3
0
Page 10 of 15
1.5
Ω’22
Ω’11
Ω’3a
0
i
Ω’ (t)/Ω
294
Ω’1a
−1.5
−3
Populations
(b)
Step 1
1
Ω’21
Step 2
Ω’
32
Step 3
|g a〉
0
|g0g1〉
|g g 〉
|g g 〉
|g0g2〉
|g a〉
0 2
0.5
0
0
0 1
10
20
30
t/Ω−1
0
0
40
50
60
Fig. 4 a The six time-dependent modified pulses versus t/ −1
0 ; b the time-dependent population conversions of the three steps for constructing the CNOT gate versus t/ −1
0 . The parameters used here are
{t f = 20/ 0 , τ = t0 = 0.1t f , g = v = 100 }
Fidelity
(a)
Fidelity
(b)
1 Step 1
0.9
0.8
0.7
Step 2
Step 3
1
0.8
0.6
0.4
0
10
20
30
40
50
60
−1
0
t/Ω
Fig. 5 a The average fidelity of each step versus t/ −1
0 ; b the average fidelity of the whole gate process
versus t/ −1
0 . The parameters used here are the same as in Fig. 4
process is F = 0.994 at t = 60/0 , which indicates our scheme is highly feasible
within a very short gate operation time. For a clearer illustration, in Fig. 6, we give a
truth table of the CNOT gate constructed by the above dressed-state scheme, which
also indicates our scheme is highly feasible.
In the following, we take the effect of decoherence on the dressed-state scheme into
account. The whole system is dominated by the master equation
ρ̇(t) = −i[H mod (t), ρ(t)]
γj σe j ,e j ρ(t) − 2σk j ,e j ρ(t)σe j ,k j + ρ(t)σe j ,e j
−
2
j=A,B k=a,g1 ,g2 ,g0
κj †
a j a j ρ(t) − 2a j ρ(t)a †j + ρ(t)a †j a j
−
2
j=A,B
κf †
−
b bρ(t) − 2bρ(t)b† + ρ(t)b† b ,
(31)
2
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Dressed-state scheme for a fast CNOT gate
Page 11 of 15
294
Fig. 6 The truth table of the
CNOT gate. The parameters
used here are the same as in
Fig. 4
1
0.994
1.000
0.994
1.000
0.5
0
|g g 〉
1 1
|g1g2〉
|g g 〉
0 1
|g g 〉
1 2
|g g 〉
|g0g2〉
|g g 〉
0 2
|g0g1〉
1 1
0.99
Fidelity
1
0.98
0.98
0.97
0.96
0
0
0.05
0.05
κ/Ω0
0.1 0.1
γ/Ω 0
Fig. 7 The fidelity as a function of γ / 0 and κ/ 0 . The parameters used here are the same as in Fig. 4
where H mod (t) = Himod (t) for the ith step (i = 1, 2, 3). γA(B) is the spontaneous emission rate of atom A(B) from the excited state |eA(B) to the ground state
|kA(B) (k = a, g1 , g2 , g0 ); κA(B) denotes the photon leakage rate from the cavity A(B);
σe j ,k j = |e j k|. For simplicity, we assume γA = γB = γ /4 and κA = κB = κ f = κ.
Then the average fidelity is rewritten as
F=
1
(2π )2
0
2π
2π
|ideal |ρ(t)|ideal |dεdβ.
(32)
0
Based on the above master equation, we plot the effect of decoherence on the average
fidelity of the whole process for constructing the CNOT gate in Fig. 7. As we can see
from Fig. 7, we learn that the influence of the photon leakages from the cavities on the
fidelity is obviously greater than that of the atomic spontaneous emissions. It is not
difficult to understand it. As we all know, QZD can effectively restrain the influence
of the photon leakages from the cavities if the Zeno limit condition is met. But in our
scheme, the amplitude of the six modified pulses is near 2.50 (see Fig. 4a) and the
chosen parameters g = v = 100 do not satisfy the Zeno limit condition i g, v
strictly. Even so, however, the fidelity of the whole process for constructing the CNOT
gate is over 0.96 even when κ = γ = 0.10 . Therefore, the dressed-state scheme
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(a)
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J.-L. Wu et al.
(b)
1
1
0.99
Fidelity
0.95
0.98
γ=κ=0
γ=κ=0.05Ω0
0.97
γ=κ=0.1Ω0
(γ,κ)=(2.62,3.5)×2π MHz
Ω0=75×2π MHz
0.96
0.95
0
0.9
20
40
60
g/Ω
0
80
100
0
50
100
t /Ω−1
f 0
150
0.85
200
Fig. 8 a The fidelity as a function of g/ 0 with t f = 20/ 0 and different values of γ and κ; b the fidelity
as a function of t f / −1
0 with g = 100 and different values of γ and κ. Other parameters used here are
the same as in Fig. 4
for constructing the CNOT gate is robust against the decoherence induced by the
atomic spontaneous emissions and the photon leakages from the cavity-fiber system.
Considering the current experimental conditions, by using cesium atoms [56] and a set
of predicted cavity QED parameters (g, κ, γ )/2π = (750, 3.3, 2.62) MHz [57], the
CNOT gate can be constructed with a fidelity F = 0.99. Hence, even in the presence
of decoherence, the dressed-state scheme for constructing a fast CNOT gate is also
highly feasible.
Finally, in order to explore the effect of the two main parameters g and t f on
constructing the CNOT gate with different values of γ and κ, we plot the average
fidelities of the whole process versus g/0 and t f /−1
0 in Figs. 8a, b, respectively.
In Fig. 8a, with one pair of the same (zero or non-zero) values of γ and κ, a larger
g will lead to a higher average fidelity, because a larger g can satisfy the Zeno limit
condition better. In Fig. 8b, in the absence of decoherence (blue dashed line), the
average fidelity will continuously rise up to near 1 with the increase of t f . For nonzero γ and κ, however, when t f is over a certain value (t f = 20/0 ), the average
fidelities will be lower and lower as t f increases. The reason is that a larger t f leads to
a greater decoherence accumulation which definitely destroys the desired dynamics.
For the predicted cavity QED parameters (g, κ, γ )/2π = (750, 3.3, 2.62) MHz (red
solid lines with 0 = g/10), g ≥ 100 and 10/0 < t f < 100/0 can guarantee a
high average fidelity (F > 0.95).
5 Conclusion
The dressed-state scheme does not require a direct coupling between the target and
initial states, and all of the driving pulses are smoothly turned on or off, which ensures
the high feasibility of the scheme in experiment. During the whole process of constructing the CNOT gate, the adiabatic condition need not be satisfied, and thus the
gate operation is fast. Besides, the results of the numerical simulations indicate that
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Dressed-state scheme for a fast CNOT gate
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294
the construction of the CNOT gate is robust against the decoherence induced by the
atomic spontaneous emissions and the photon leakages from the system.
Acknowledgements Authors would like to thank Xing Ri Jin’s team (Yanbian University), Shi-Lei
Su (Zhengzhou University), Xiao-Qiang Shao (Northeast Normal University) and Zheng-Yuan Xue (South
China Normal University) for their great help. This work was supported by the National Natural Science
Foundation of China under Grants Nos. 11464046 and 61465013.
References
1. Grover, L.K.: Quantum computers can search rapidly by using almost any transformation. Phys. Rev.
Lett. 80, 4329–4332 (1998)
2. DiVincenzo, D.P.: Two-bit gates are universal for quantum computation. Phys. Rev. A 51, 1015–1022
(1995)
3. Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin,
J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)
4. Cirac, J.I., Zoller, P.: Quantum computations with cold trapped ions. Phys. Rev. Lett. 74, 4091–4094
(1995)
5. Bai, C.H., Wang, D.Y., Hu, S., Cui, W.X., Jiang, X.X., Wang, H.F.: Scheme for implementing multitarget
qubit controlled-NOT gate of photons and controlled-phase gate of electron spins via quantum dotmicrocavity coupled system. Quantum Inf. Process. 15, 1485–1498 (2016)
6. Kang, Y.H., Xia, Y., Lu, P.M.: Two-photon phase gate with linear optical elements and atomcavity
system. Quantum Inf. Process. 15, 4521–4535 (2016)
7. Barenco, A., Deutsch, D., Ekert, A.: Conditional quantum dynamics and logic gates. Phys. Rev. Lett.
74, 4083–4086 (1995)
8. Shao, X.Q., Zhu, A.D., Zhang, S., Chung, J.S., Yeon, K.H.: Efficient scheme for implementing an
N-qubit Toffoli gate by a single resonant interaction with cavity quantum electrodynamics. Phys. Rev.
A 75, 034307 (2007)
9. Su, S.L., Liang, E.J., Zhang, S., Wen, J.J., Sun, L.L., Jin, Z., Zhu, A.D.: One-step implementation of
the Rydberg–Rydberg-interaction gate. Phys. Rev. A 93, 012306 (2016)
10. Su, S.L., Gao, Y., Liang, E.J., Zhang, S.: Fast Rydberg antiblockade regime and its applications in
quantum logic gates. Phys. Rev. A 95, 022319 (2017)
11. Gershenfeld, N.A., Chuang, I.L.: Bulk spin-resonance quantum computation. Science 275, 350–356
(1997)
12. Yang, C.P., Chu, S.I., Han, S.: Possible realization of entanglement, logical gates, and quantuminformation transfer with superconducting-quantum-interference-device qubits in cavity QED. Phys.
Rev. A 67, 042311 (2003)
13. Yang, C.P., Han, S.: Realization of an n-qubit controlled-U gate with superconducting quantum interference devices or atoms in cavity QED. Phys. Rev. A 73, 032317 (2006)
14. Beige, A., Braun, D., Tregenna, B., Knight, P.L.: Quantum computing using dissipation to remain in
a decoherence-free subspace. Phys. Rev. Lett. 85, 1762 (2000)
15. Franson, J.D., Jacobs, B.C., Pittman, T.B.: Quantum computing using single photons and the Zeno
effect. Phys. Rev. A 70, 062302 (2004)
16. Sangouard, N., Lacour, X., Guérin, S., Jauslin, H.R.: CNOT gate by adiabatic passage with an optical
cavity. Eur. Phys. J. D 37, 451–456 (2006)
17. Sugny, D., Bomble, L., Ribeyre, T., Dulieu, O., Desouter-Lecomte, M.: Rovibrational controlled-NOT
gates using optimized stimulated Raman adiabatic passage techniques and optimal control theory.
Phys. Rev. A 80, 042325 (2009)
18. Shao, X.Q., Chen, L., Zhang, S., Yeon, K.H.: Fast CNOT gate via quantum Zeno dynamics. J. Phys.
B 42, 165507 (2009)
19. Demirplak, M., Rice, S.A.: Adiabatic population transfer with control fields. J. Phys. Chem. A 107,
9937–9945 (2003)
20. Berry, M.V.: Transitionless quantum driving. J. Phys. A 42, 365303 (2009)
21. Chen, X., Lizuain, I., Ruschhaupt, A., Guéry-Odelin, D., Muga, J.G.: Shortcut to adiabatic passage in
two- and three-level atoms. Phys. Rev. Lett. 105, 123003 (2010)
123
294
Page 14 of 15
J.-L. Wu et al.
22. Chen, X., Torrontegui, E., Muga, J.G.: Lewis-Riesenfeld invariants and transitionless quantum driving.
Phys. Rev. A 83, 062116 (2011)
23. Chen, X., Muga, J.G.: Engineering of fast population transfer in three-level systems. Phys. Rev. A 86,
033405 (2012)
24. Ibáñez, S., Chen, X., Torrontegui, E., Muga, J.G., Ruschhaupt, A.: Multiple Schrödinger pictures and
dynamics in shortcuts to adiabaticity. Phys. Rev. Lett. 109, 100403 (2012)
25. Torrontegui, E., Ibáñez, S., Martínez-Garaot, S., Modugno, M., del Campo, A., Guéry-Odelin, D.,
Ruschhaupt, A., Chen, X., Muga, J.G.: Shortcuts to adiabaticity. Adv. At., Mol. Opt. Phys. 62, 117
(2013)
26. del Campo, A.: Shortcuts to adiabaticity by counterdiabatic driving. Phys. Rev. Lett. 111, 100502
(2013)
27. Torosov, B.T., Valle, G.D., Longhi, S.: Non-Hermitian shortcut to adiabaticity. Phys. Rev. A 87, 052502
(2013)
28. Song, X.K., Ai, Q., Qiu, J., Deng, F.G.: Physically feasible three-level transitionless quantum driving
with multiple Schrodinger dynamics. Phys. Rev. A 93, 052324 (2016)
29. Chen, Y.H., Xia, Y., Wu, Q.C., Huang, B.H., Song, J.: Method for constructing shortcuts to adiabaticity
by a substitute of counterdiabatic driving terms. Phys. Rev. A 93, 052109 (2016)
30. Benseny, A., Kiely, A., Zhang, Y., Busch, T., Ruschhaupt, A.: Spatial non-adiabatic passage using
geometric phases. EPJ Quantum Technol. 4, 3 (2017)
31. Song, X.K., Deng, F.G., Lamata, L., Muga, J.G.: Robust state preparation in quantum simulations of
Dirac dynamics. Phys. Rev. A 95, 022332 (2017)
32. Bason, M.G., Viteau, M., Malossi, N., Huillery, P., Arimondo, E., Ciampini, D., Fazio, R., Giovannetti,
V., Mannella, R., Morsch, O.: High-fidelity quantum driving. Nat. Phys. 8, 147–152 (2012)
33. Du, Y.X., Liang, Z.T., Li, Y.C., Yue, X.X., Lv, Q.X., Huang, W., Chen, X., Yan, H., Zhu, S.L.: Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms. Nat. Commun.
7, 12479 (2016)
34. An, S., Lv, D., del Campo, A., Kim, K.: Shortcuts to adiabaticity by counterdiabatic driving for trappedion displacement in phase space. Nat. Commun. 7, 12999 (2016)
35. Zhang, Z., Wang, T., Xiang, L., Yao, J., Wu, J., Yin, Y.: Measuring the Berry phase in a superconducting
phase qubit by a shortcut to adiabaticity. Phys. Rev. A 95, 042345 (2017)
36. Zhou, B.B., Baksic, A., Ribeiro, H., Yale, C.G., Heremans, F.J., Jerger, P.C., Auer, A., Burkard, G.,
Clerk, A.A., Awschalom, D.D.: Accelerated quantum control using superadiabatic dynamics in a solidstate lambda system. Nat. Phys. 13, 330–334 (2017)
37. Chen, Y.H., Xia, Y., Chen, Q.Q., Song, J.: Fast and noise-resistant implementation of quantum phase
gates and creation of quantum entangled states. Phys. Rev. A 91, 012325 (2015)
38. Liang, Y., Wu, Q.C., Su, S.L., Ji, X., Zhang, S.: Shortcuts to adiabatic passage for multiqubit controlledphase gate. Phys. Rev. A 91, 032304 (2015)
39. Zhang, J., Kyaw, T.H., Tong, D.M., Sjöqvist, E., Kwek, L.C.: Fast non-Abelian geometric gates via
transitionless quantum driving. Sci. Rep. 5, 18414 (2015)
40. Song, X.K., Zhang, H., Ai, Q., Qiu, J., Deng, F.G.: Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm. New J. Phys. 18,
023001 (2016)
41. Giannelli, L., Arimondo, E.: Three-level superadiabatic quantum driving. Phys. Rev. A 89, 033419
(2014)
42. Masuda, S., Rice, S.A.: Fast-Forward assisted STIRAP. J. Phys. Chem. A 119, 3479–3487 (2015)
43. Kiely, A., Ruschhaupt, A.: Inhibiting unwanted transitions in population transfer in two- and three-level
quantum systems. J. Phys. B 47, 115501 (2014)
44. Baksic, A., Ribeiro, H., Clerk, A.A.: Speeding up adiabatic quantum state transfer by using dressed
states. Phys. Rev. Lett. 116, 230503 (2016)
45. Wu, J.L., Ji, X., Zhang, S.: Fast adiabatic quantum state transfer and entanglement generation between
two atoms via dressed states. Sci. Rep. 7, 46255 (2017)
46. Kang, Y.Y., Chen, Y.H., Shi, Z.C., Song, J., Xia, Y.: Fast preparation of W states with superconducting
quantum interference devices by using dressed states. Phys. Rev. A 94, 052311 (2016)
47. Liu, B.J., Huang, Z.H., Xue, Z.Y., Zhang, X.D.: Superadiabatic holonomic quantum computation in
cavity QED. Phys. Rev. A 95, 062308 (2017)
48. Zhou, X., Liu, B.J., Shao, L.B., Zhang, X.D., Xue, Z.Y.: Quantum state conversion in opto-electromechanical systems via shortcut to adiabaticity. Laser Phys. Lett. 14, 095202 (2017)
123
Dressed-state scheme for a fast CNOT gate
Page 15 of 15
294
49. Serafini, A., Mancini, S., Bose, S.: Distributed quantum computation via optical fibers. Phys. Rev. Lett.
96, 010503 (2006)
50. Facchi, P., Pascazio, S.: Quantum Zeno subspaces. Phys. Rev. Lett. 89, 080401 (2002)
51. Facchi, P., Marmo, G., Pascazio, S.: Quantum Zeno dynamics and quantum Zeno subspaces. J. Phys:
Conf. Ser. 196, 012017 (2009)
52. Shao, X.Q., Wu, J.H., Yi, X.X.: Dissipation-based entanglement via quantum Zeno dynamics and
Rydberg antiblockade. Phys. Rev. A 95, 062339 (2017)
53. Shao, X.Q., Li, D.X., Ji, Y.Q., Wu, J.H., Yi, X.X.: Ground-state blockade of Rydberg atoms and
application in entanglement generation. Phys. Rev. A 96, 012328 (2017)
54. Bergmann, K., Theuer, H., Shore, B.W.: Coherent population transfer among quantum states of atoms
and molecules. Rev. Mod. Phys. 70, 1003 (1998)
55. Vitanov, N.V., Halfmann, T., Shore, B.W., Bergmann, K.: Laser-induced population transfer by adiabatic passage techniques. Annu. Rev. Phys. Chem. 52, 763 (2001)
56. Liang, Y., Song, C., Ji, X., Zhang, S.: Fast CNOT gate between two spatially separated atoms via
shortcuts to adiabatic passage. Opt. Express 23, 23798–23810 (2015)
57. Spillane, S.M., Kippenberg, T.J., Vahala, K.J., Goh, K.W., Wilcut, E., Kimble, H.J.: Ultrahigh-Q
toroidal microresonators for cavity quantum electrodynamics. Phys. Rev. A 71, 013817 (2005)
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