Any quantum circuit can be decomposed into single-qubit rotation gates and two-qubit entangling gates, e.g. control-NOT (CNOT) gates or control-phase-flip (CPF) gates [20].
This is called the universality of the quantum gates. In this section, we will discuss how to implement the universal gates with the trapped ions.
Suppose the information is encoded in an ion’s hyperfine state |g⟩ (ground state) and
|e⟩ (excited state). The energy separation between the two levels is ¯hωeg. Consider a vibrational mode in a harmonic trap with an angular frequency ν, which can be described by the Fock basis|n⟩. The energy configuration of the joint states of an ion is shown in Figure 1.3.
ωeg
ν
|�, 0⟩
|�, 1⟩
|�, 2⟩
|�, 0⟩
|�, 1⟩
|�, 2⟩
Figure 1.3: The energy level of a trapped ion. The green, blue, and red arrows correspond to the carrier, blue sideband and red sideband transitions respectively.
We now apply laser beams to drive transitions between these states. The Hamiltonian of the laser-ion interaction in the interaction picture reads [21]
HI = ¯hΩσ+e−i(δt−ϕ)exp(
iη(ae−iνt+ a†eiνt))
+ H.c., (1.1)
where Ω is the Rabi frequency of the laser, δ = ω− ωegis the laser-ion detuning, ϕ is the phase offset of the laser. σ+ = |e⟩⟨g| and σ− =|g⟩⟨e| are raising and lowering operators of the atomic states; a and a†are annihilation and creation operators of the phonon states;
η = k
√
¯ h
2mν is the Dicke parameter with k the wavevector of the laser. The Lamb-Dicke parameter characterizes the ratio between the oscillation amplitude of the ion to
the wavelength of the laser. Taking the Lamb-Dicke limit η√
n ≪ 1, which means the displacement of the ion is much smaller than the wavelength so that it does not feel the spatial dependence of the field, the Hamiltonian becomes
HI = ¯hΩ{σ+e−i(δt−ϕ)+σ−ei(δt−ϕ)+iη(σ+e−i(δt−ϕ)−σ−ei(δt−ϕ))(ae−iνt+a†eiνt)}. (1.2)
Now we discuss the three cases of interest: δ = 0, and δ = ±ν in the following, where we ignore the fast oscillating terms:
1. For δ = 0 (carrier transition, shown by the green arrow in Figure 1.3), the laser couples|g, n⟩ and |e, n⟩. The Hamiltonian is
H = ¯hΩ(σ+eiϕ+ σ−e−iϕ). (1.3)
2. For δ = ν (blue sideband transition, shown by the blue arrow in Figure 1.3), the laser couples|g, n⟩ and |e, n + 1⟩. The Hamiltonian is
H = i¯hΩη(σ+a†eiϕ− σ−ae−iϕ), (1.4)
and we get the effective Rabi frequency
Ωn,n+1 =√
n + 1ηΩ. (1.5)
3. For δ = −ν (red sideband transition, shown by the red arrow in Figure 1.3), the laser couples|g, n⟩ and |e, n − 1⟩. The Hamiltonian is
H =−i¯hΩη(σ+aeiϕ− σ−a†e−iϕ), (1.6)
and the effective Rabi frequency is
Ωn,n−1 =√
nηΩ. (1.7)
When we set δ = 0, the evolution of the states is
where θ = Ωτ , with the laser pulse duration τ . By tuning the duration time and the phase offset of the laser pulse, the single-qubit rotation can be achieved.
For a two-qubit gate, we use vibration as the quantum bus to communicate two qubits.
At the beginning and the end of the operation, the internal states and the motional states are desired to be disentangled. In the following subsections, we will discuss three types of the two-qubit gates.
1.3.1 Cirac-Zoller gate
The first two-qubit gate operation of ion trap was proposed by J. I. Cirac and P. Zoller in 1995 [8]. It requires the motional mode to be cooled to the ground state|n = 0⟩. The scheme has three steps, as illustrated in Figure 1.4.
1. Shine a laser beam with detuning δ =−ν on the mth ion. The laser couples |e⟩m|0⟩
and|g⟩m|1⟩. The evolution of the states is given by
|e0⟩ → cos(θ remains unchanged. If the mth ion is at |e⟩m initially, the population is driven to
|g⟩m|1⟩ and gains a phase −i (see Equation (1.9)).
2. Apply a 2π-pulse (θ = 2π) that couples|g⟩n|1⟩ and an auxiliary state |a⟩n|0⟩ on the nth ion. If the nth ion is at|e⟩ initially, the state remains unchanged. If the nth ion is at|g⟩ initially, the state gains a phase −1.
3. Apply again a π-pulse with δ = −ν and ϕ = 0 on the mth ion to take the phonon
state back to|0⟩.
|�⟩�|0⟩
|�⟩�|0⟩
|�⟩�|1⟩
|�⟩�|1⟩
|�⟩�|0⟩
|�⟩�|0⟩
|�⟩�|1⟩
|�⟩�|1⟩
|�⟩�|0⟩
|�⟩�|0⟩
|�⟩�|1⟩
|�⟩�|1⟩
|�⟩�|0⟩
Step 1: on mth ion Step 2: on nth ion Step 3: on mth ion
Figure 1.4: The concept of the Cirac-Zoller gate.
The overall evolution of the states then becomes
|g⟩m|g⟩n|0⟩ → |g⟩m|g⟩n|0⟩ → |g⟩m|g⟩n|0⟩ → |g⟩m|g⟩n|0⟩
|g⟩m|e⟩n|0⟩ → |g⟩m|e⟩n|0⟩ → |g⟩m|e⟩n|0⟩ → |g⟩m|e⟩n|0⟩
|e⟩m|g⟩n|0⟩ → −i |g⟩m|g⟩n|1⟩ → i|g⟩m|g⟩n|1⟩ → |e⟩m|g⟩n|0⟩
|e⟩m|e⟩n|0⟩ → −i |g⟩m|e⟩n|1⟩ → −i |g⟩m|e⟩n|1⟩ → − |e⟩m|e⟩n|0⟩
.
(1.10) Define|g⟩ = |0⟩ and |e⟩ = |1⟩, then we get the CPF gate. Because the effective Rabi frequency Rabi frequency Ωn,n−1 is different with different phonon state|n⟩, we need to cool the motional state to the ground one|n = 0⟩.
Cirac-Zoller gate was first realized by Schmit-Kaler et al. in 2003 [22].
1.3.2 Mølmer-Sørensen gate
K. Mølmer and A. Sørensen proposed the gate scheme that can be operated in thermal motion [9]. In this scheme, we apply bichromatic laser beams on both ions. The laser frequencies are ωeg ± δ, which are close to the blue and red sidebands, respectively. The energy levels and the transition paths are shown in Figure 1.5.
In the weak-field regime ηΩ ≪ ν − δ, the phonon number n only changes by ±1.
� �
Figure 1.5: The concept of the Mølmer-Sørensen gate. Bichromatic laser beams are app-lied on the ions. The intermediate states are not populated because of the off resonance.
The overall transitions are|ggn⟩ ↔ |een⟩ (left panel) and |egn⟩ ↔ |gen⟩. These paths interfere destructively and eliminate the dependence of phonon number n [10].
In the left panel of Figure 1.5, the path |ggn⟩ ↔ {|egn ± 1⟩ , |gen ± 1⟩} ↔ |een⟩ is the cascade-type Raman transition with multi intermediate levels. Since the intermediate states do not fulfill the resonance condition, these states are not populated in the process.
The evolution is then
with the effective Rabi frequency
eΩ =∑
m
ΩnmΩmn
∆m , (1.12)
where m denotes the intermediate states|eg ± 1⟩ and |ge ± 1⟩, Ωnm(Ωmn) is the effective Rabi frequency driving|ggn⟩ ↔ |m⟩ (|m⟩ ↔ |een⟩) (shown in Equation (1.5) and (1.7)), and ∆m = ωm − (Em − Eggn) with ωm the laser frequency driving |ggn⟩ ↔ |m⟩ and (Em− Eggn) the energy spacing between|m⟩ and |ggn⟩.
We get
eΩ = η2Ω2
ν− δ, (1.13)
which is independent of the phonon number n. The independence of n is because the transitions|ggn⟩ ↔ {|egn − 1⟩ , |gen − 1⟩} ↔ |een⟩ gives a factor of n, and the transi-tion|ggn⟩ → {|gen + 1⟩ , |egn + 1⟩} ↔ |een⟩ gives a factor of n + 1. These paths have opposite detunings, eliminating the dependence of n. As a result, the Mølmer-Sørensen gate can be operated when the phonon state is under thermal distribution. K. Mølmer and A. Sørensen also showed that this gate scheme has high fidelity even during heating.
Similarly, The evolution in the right panel of Figure 1.5 is
|gen⟩ → cos
with the same effective Rabi frequency eΩ shown in Equation (1.13).
Setting eΩτ = π/2, we get the maximal entanglement of the qubits. Combining with single-qubit rotation, we can achieve CNOT gate.
The experiment of the the Mølmer-Sørensen gate was first realize by C. Sackett et al. in 2000 [23]. In 2011, T. Monz et al. created 14-qubit entanglement by using this gate scheme [24].
1.3.3 Geometrical phase gate
K. Mølmer and A. Sørensen also proved that their gate scheme can be achieved without the restriction of being in the weak-field regime [10]. A similar idea was proposed indepen-dently by G. Milburn et al. in 2000 [25]. In Milburn’s scheme, we apply spin-dependent forces on the ions, and drive clockwise or counterclockwise trajectories in the phase space depending on the internal states. At the end of the gate operation, the ions return to the original motional state, obtaining a phase which equals to the area of the close loop.
To study the effect of a spin-dependent force, first we consider a forced oscillator. For an oscillator with an angular frequency ν pushed by a force F sin ωt, the Hamiltonian in the interaction picture is [26]
H = F x0(aeiδt+ a†e−iδt), (1.15)
where δ = ω− ν, and x0 =√
¯
h/(2mν). The evolution operator of the forced oscillator is [26]
U (τ ) = eiϕ(τ )D(α(τ )), (1.16) where D(α) = eαa†−α∗ais the displacement operator with α(τ ) = −¯hi ∫τ
0 F x0eiδtdt, and ϕ(τ ) = Im{∫τ
0 α∗dα} .
The additional phase ϕ can be understood by the the relation of displacement ope-rator D(α + β)ei Im{α∗β} = D(α)D(β). Equation (1.16) can be regraded as a series of infinitesimal displacement.
After a period τ = 2π/δ, the displacement operator D(α) = 1 makes the motional state back to the origin, and the state gains a phase ϕ which equals to the area of the closed loop.
For spin-dependent forces acting on the ions, the Hamiltonian is
H =∑
i
F x0(aeiδt+ a†e−iδt)σzi, (1.17)
where σzi =|e⟩⟨e| − |g⟩⟨g| is the Pauli matrix, and the subscript i = 1, 2 denotes the ion’s index. Equation (1.17) shows that the ion feels a force F when the internal state is|e⟩, and−F when the internal state is |g⟩. When both ions are at the same internal states (|ee⟩
or|gg⟩), the ions move back and force together (center-of-mass mode). When the two ions have different internal states (|eg⟩ or |ge⟩), the ions move in the opposite directions (breathing mode). Consider that only the breathing mode is excited by applying the laser with frequency close to the mode frequency. If the ions’ internal state is|ee⟩ or |gg⟩, it remains unchanged. If the ions’ internal state is|eg⟩ or |ge⟩, the motional state displaces in the phase space.
When the ions’ motional state goes back to the origin, the evolution of the internal
states is described by
|ee⟩ → |ee⟩
|gg⟩ → |gg⟩
|eg⟩ → eiϕ|eg⟩
|ge⟩ → eiϕ|ge⟩
, (1.18)
which is illustrated in Figure 1.6. By choosing appropriate laser frequency and intensity, we can make ϕ = π/2. Then Equation (1.18) is equivalent to the CFP gate up to single-qubit rotation. The experiment of the geometrical phase gate is realized by Leibried et al. in 2003 [27].
x p
|��⟩ → ���|��⟩
|��⟩ → ���|��⟩
Figure 1.6: The concept of the geometrical phase gate. If the ions’ internal state is|eg⟩ or
|ge⟩, the motional state displaces in the phase space. At the end of the gate operation, the motional state goes back to the origin and gains a phase corresponding to the area of the closed loop.