To sum up, we calculate the position fluctuation of the ions as indication of cooling effi-ciency under sympathetic cooling together with optical tweezers. We find that for longitu-dinal mode, the optical tweezers together with sympathetic cooling significantly improve the cooling efficiency. For transverse mode, the optical tweezers play the role of the “heat
block”. We show that for the local gate operation, we only need to cool the ions near computational ions rather than the whole ion chain. Finally, we discuss the relaxation dy-namics. We find that the cooling time is shorter when we apply the optical tweezers than that when we do not.
0.980 0.985 0.990 0.995 1.000 1.005 1.010 1.015 1.020
µ/ω x
0.990 0.992 0.994 0.996 0.998 1.000
F
(a)
τ = 500 τ
0200 τ
0100 τ
00 1
t/τ
0.06 0.04 0.02 0.00 0.02 0.04 0.06
η x Ω /ω x
(b)
Figure 3.14: Quantum gate design for N = 121. Cooling ions: 51, 61. Pinned ions: 50, 62. Gate ions: 55, 57. (a)The fidelity with different gate time and laser frequency. (b) The laser shape with τ = 500τ0and µ = 0.982ωx (denoted by the arrow in (a)). δF = 10−13. The other parameters are same as Figure 3.5.
Chapter 4 Conclusion
In this thesis, we propose a scalable ion trap quantum computer scheme. We use optical tweezers to stabilize the ion chain, and use sympathetic cooling method to suppress the background heating. For an ion array arrange in the z-direction, the transverse trapping frequency ωx is much greater than the longitudinal trapping frequency ωz. The trapping frequency ωzgoes to 0 when the number of ion increases, causing the difficulty to scale up the ion trap system. By applying the optical tweezers, we can enhance the mode frequency and hence improve the efficient of cooling.
For large number of ions, the optical tweezers dominate the motional mode frequency.
The upper bond of the mode frequency corresponds to the mode frequency when the degree of freedom of the node ions are removed.
Here we conclude our findings and the proposed architecture for large-scale quantum computing. As a linear Paul trap is still regarded as the simplest configuration of imple-mentation compared to other schemes such as ion shuttling and quantum network. Tra-ditionally, a large-scale ion array uses DC voltages to provide longitudinal confinement, which, in order for finite ion separation for individual addressability, must be very weak so that its corresponding motional excitation (phonon) is very hard to be removed. The significance of applying optical tweezers has three folds. First, it raises the frequency of the ground mode from zero, thus stabilizing the ion array and making possible for further laser cooling. Secondly, optical tweezers pin some of the ions so that these pinned ions can serve as “heat blocks.” Therefore, parallel computation is possible by means of the
supposedly collective motion: Gate operations are separated and also protected by optical tweezers. Thirdly, the scheme can be repeatedly extended with only linear increase of the cost. In the following, we summarize in more detail the effects of the optical tweezers on the collective motion including longitudinal and transverse modes as well as the cooling and heating dynamics.
We have shown that the longitudinal and transverse modes behave differently because their characteristic frequencies are separated. Typically, we have ωx ≫ ω0 ? ωz, where ωxand ωzare transverse and longitudinal trapping frequencies of a Paul trap, respectively, and ω0 =√
e2 4πϵ0
1
md30 accounts for residual Coulomb interaction between ions. Therefore, we can expect that the longitudinal mode is more collective and good for heat exchange.
A major source of error comes from the longitudinal motion, where the large displace-ment lets the ion see spatial variation of the field due to the finite beam size of a Gaussian beam. By applying optical tweezers (? 2π×500 kHz for the longitudinal mode), as N gets large (∼ 500) while the spacing is kept about d0 (≈ 10 µm), the longitudinal frequency (∼ 2π × 10 kHz) is mostly determined by the potential assuming that the “pinned ions”
fixed in space instead of the global trap. Thus, the optical tweezers reduce the position fluctuation through stronger confinement than the case without tweezers. Combining the technique of sympathetic cooling, the long ion array can then be stabilized and continu-ously cooled.
As for the transverse mode, if the applied optical tweezers provide additional confine-ment for the transverse mode, it becomes a negative factor, which can make the position fluctuation even larger. This is due to the heat blocking effect of the transversely pinned ions. Since the pinned ion has a higher frequency than its neighbors, the heat exchange is less likely to occur. Note that the main source of error associated with the transverse mode is due to the anharmonicity that contributes to the gate fidelity. Unlike the longitudinal part, it has nothing to do with the structural stability of the ion array so the scalability is assured. Such transverse position fluctuation can be suppressed by applying sympathetic cooling within a region sandwiched by two optical tweezers. It is expected that the heat sink in such arrangement can be very efficient so also scalable.
Further, we propose the concept of “local traps” in a large-scale ion chain. We consider only a small segment of the ion array defined by two optical tweezer beams. In other words, the segment is “locally trapped” by two pinned ions. When a quantum gate is operated within the segment, we can just cool this segment instead of cooling the whole ion array. It helps us to reduce the overhead of laser cooling and optical tweezers. For the longitudinal motion, the position fluctuations of the ions in the local trap is nearly halved compared to the case by distributing sympathetically cooled ions without optical tweezers. However, for the transverse mode, the position fluctuation is slightly larger but no more than two times. Note that the actual distribution depends on which ions are chosen to be pinned or sympathetically cooled when the size of the ion chain is finite. Its trend shown here is representative. We also study the relaxation dynamics under the local trap architecture. For both longitudinal and transverse modes, the cooling time is shorter by about an order of magnitude than the case without tweezers.
As future work, we plan to explicitly formulate the local trap idea by employing the open system theory and language, and derive the effective motional spectrum. This can further help us understand and explore the possibility of “local trap sideband cooling.”
Appendix A Gate design
In this appendix, we describe the pulse shaping scheme [28] to design a two-qubit gate.
The procedures have been explicitly discussed in Reference [37]. For completeness of this thesis, we here summarize the method in the following.
Since this scheme is based on the transverse mode, we apply on the two qubit ions bichromatic laser field that generates the spin-dependent forces along the transverse di-rection. The Hamiltonian of the laser-ion interaction under the Lamb-Dicke limit in the interaction picture is given by
∑
n,k
¯
hΩn(t) sin(µt)Gknηk(a†keiωkt+ ake−iωk)σnz, (A.1)
where Ωn(t) is the Rabi frequency of the laser applied on the nth ion, µ is the beatnote frequency of bichromatic field, Gknis the mode function coupling the nth ion and the kth mode, ηkn = |∆k|√
¯ h
2mωk is the Lamb-Dicke parameter with the wavevector difference of the two beams|∆k|, ak (a†k) is the phonon annihilation (creation) operator of the kth mode, σnz is the Pauli-z matrix for the nth ion.
Using the second-order Magnus expansion, we find the evolution operator
U (τ ) = exp [
i∑
n
ϕn(τ )σzn+ i∑
l<n
ϕln(τ )σzlσnz ]
, (A.2)
where
To construct a CPF gate, we need to find appropriate Ω(t) that satisfies the constraints
ϕi(j) = 0, (A.6a)
ϕij(τ ) = π
4. (A.6b)
A straightforward strategy is to chop Ω(t) into M equal segments. We have Ω(t) = Ωm when τm−1 < t < τm with τm = mτ /M . Equations (A.6) give 2N + 1 constraints. To exactly fulfill A.6, we need 2N + 1 independent parameters so that M = 2N + 1.
Instead of exactly eliminating ϕi(j), we can use M ≤ 2N + 1 segments and maximize the fidelity. The fidelity is defined by
F = ⟨ψf|Trm{U(τ)(|ψ0⟩⟨ψ0| ⊗ ρth)U†(τ )}|ψf⟩ , (A.7)
where the final qubit state|ψf⟩ = eiπσizσjz/4|ψ0⟩ with the initial qubit state |ψ0⟩, ρth is the density matrix of the phonon, and Trmdenotes the trace over all phonon Fock states.
Choose a typical initial qubit state|ψ0⟩ = (|0⟩i+|1⟩i)⊗ (|0⟩j+|1⟩j), and assume that
the infidelity becomes minimize Equation (A.8) under the constraints Equation (A.6b).
We can use the quadratic minimization method to simplify the calculation. We focus on the infidelity δF ∼ 0, which corresponds to αi(j)k ∼ 0. We take Taylor expansion e−x≈ 1 − x + O(x2) for Γ’s. Then the infidelity becomes Hik·X, where · denotes the inner product. Further,∑
kβ¯k αki 2can be rewritten as X⊤AiX with the matrix element Aimm′ =∑
kβ¯kHimk Himk∗. Define A = 14(Ai+ Aj). Finally we get
δF = X⊤AX. (A.11)
The conditional phase ϕij (Equation (A.5)) can be also expressed in the matrix form
ϕij = X⊤BX, (A.12)
where the matrix Bmm′ = Pmm′/2 + Qmδmm′ with the definition
Pmm′ =
∫ τm
τm−1
dt2
∫ τm′
τm′−1
dt1∑
k
ηklηknGklGknsin µt2sin µt1sin ωk(t2− t1), (A.13a)
Qm =
∫ τm
τm−1
dt2
∫ t2
τm−1
dt1∑
k
ηklηknGklGknsin µt2sin µt1sin ωk(t2− t1). (A.13b)
To minimize Equation (A.11) with constraint Equation (A.12), we use Lagrange unde-termined multiplier method. The modified infidelity δF⋆ = X⊤AX + λX⊤BX. To find minimum value of δF⋆, we make ∂δF∂X⊤⋆ = 0, which gives a generalized eigenvalue equa-tion AX = λBX. By solving the eigenvectors and checking the associated infidelity, we can obtain the proper X, which will be further re-scaled such that ϕij = π/4.
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