Quantum Manipulations of Single Trapped-Ions Beyond the Lamb-Dicke Limit
2. Dynamics of single trapped ions driven by laser beams
3.3 Simplified approaches for implementing quantum logic gates
In fact, the above approach (with three laser pulses) for generating CNOT gate can be simplified, by using only two laser pulses (28). With the sequential two operations 1) and 2), the system undergoes the following evolutions:
Input state Output state
|0|g −→ |ψa(1) −→ |ψa(2)
|0|e −→ |ψ(1)b −→ |ψ(2)b
|1|g −→ |ψc(1) −→ |ψc(2)
|1|e −→ |ψ(1)d −→ |ψ(2)d
(53)
If α(2)11 =α(2)22 =α(2)33 =α42(2)=1, (54)
i.e.,(see, APPENDIX A):
⎧⎪
⎪⎨
⎪⎪
⎩
1=cos(Ω0,0t1),
1=cos(Ω0,0t1)cos(Ω0,1t2), 1=i−1e−iϑ1sin(Ω1,0t1)cos(Ω1,1t2), 1=i−1eiϑ1sin(Ω1,0t1)cos(Ω0,1t2),
(55)
then|ψa(2)=|0|g,|ψ(2)b =|0|e,|ψc(2)=|1|e, and|ψ(2)d =|1|g. Thus, the desirable CNOT gate (45) is realized.
The above condition could be achieved by properly setting the relevant experimental parameters,t1,t2,ϑ1, andηas
t1=2pπΩ
0,0,t2=2mπΩ
0,1,θ1= (−1)q+1π
2, (56)
withp,q,m=1, 2, 3..., and
η2=1−q−0.5 2p =2−
√2(n−0.5)
m , (57)
with n =1, 2, 3, .... This means that the CNOT gate (45) could be implemented, since all the experimental parameters related above are accurately controllable. This implies that the required laser pulses for implementing a CNOT gate could be really reduced to two ones.
Due to decoherence(i.e., the limit of durationst1+t2), the integers p,q,m,ncould not take arbitrary large values to let (57) be satisfied exactly, although their approximated solutions
η t1Ω t2Ω θ1=±π2 F
0.18 689.6666 993.3470 + 0.9907
0.20 1525.600 1410.200 + 0.9958
0.22 1261.700 1463.000 + 0.9984
0.25 1504.000 1192.800 - 0.9978
0.28 457.4064 326.7189 + 0.9967
0.30 328.6193 438.1591 - 0.9993
0.38 351.1877 284.3625 + 0.9994
0.40 490.0675 170.1623 + 0.9975
0.44 304.5596 283.1649 - 0.9980
0.53 347.0706 190.9979 + 0.9970
0.57 443.4884 492.7649 + 0.9993
0.64 385.5613 481.9516 - 0.9967
0.68 316.7005 395.8756 - 0.9950
0.73 377.2728 292.1111 - 0.9953
0.80 276.8880 281.2143 - 0.9980
0.86 436.5392 486.4535 + 0.9999
0.90 471.0198 460.5527 - 0.9971
0.96 318.7519 394.2894 + 0.9969
Table 4. CNOT gates implemented with sufficiently high fidelities for arbitrarily selected parameters (28).
0.28 130.92 0.99376 0.11153 0.11041 0.99389 0.30 104.95 0.99505 -0.09935 -0.09778 0.99521
0.32 92.35 0.99374 -0.11172 -0.10790 0.99416
0.34 79.49 0.98271 -0.18517 -0.18852 0.98207
0.36 80.64 0.99586 0.09088 0.09407 0.99557
0.38 67.33 0.99541 -0.09572 -0.09377 0.99559
0.40 68.44 0.98505 0.17225 0.16794 0.98580
0.42 54.57 0.98859 -0.15063 -0.15383 0.98810
0.44 55.56 0.99646 0.08411 0.08530 0.99636
0.46 111.30 0.97957 -0.20111 -0.19843 0.98012
0.48 41.83 0.97796 -0.20881 -0.20607 0.97854
0.50 42.72 1.00000 0.00000 0.00000 1.00000
0.52 43.67 0.97497 0.22234 0.21915 0.97570
0.54 87.23 1.00000 0.00000 0.00000 1.00000
0.56 132.93 0.96361 0.26731 0.26592 0.96400
0.58 118.42 0.97544 -0.22027 -0.22025 0.97544
0.60 29.81 0.99320 -0.11640 -0.11358 0.99353
Table 5. CNOT gates implemented with high fidelities for arbitrarily selected parameters(from 0.18 to 0.60) (29).
are still exists. This implies that the desirable CNOT gate is approximately implemented with a fidelity F<1. Similar to that of Sec. 3.2, here the fidelity F of implementing the desirable gate is defined as the minimum among the values ofα(2)11, α(2)22,α(2)33 andα(2)42, i.e., F=min{α(2)11,α(2)22,α33(2),α(2)42}.
In table 4, we present some experimental parameters arbitrarily selected to implement the expected CNOT gate (45), with sufficiently high fidelitiesF>99%. For example, we have F=99.58% for the typical LD parameterη=0.2. It is also seen from the table that the present proposal still works for certain large LD parameters, e.g., 0.90, 0.96, etc.. For the experimental parametersη=0.2 andΩ/(2π)≈500 kHz, the total durationt1+t2for this implementation is about 0.9 ms. By increasing the Rabi frequencyΩvia enhancing the power of the applied laser beams, the durations can be further shorten, and thus the CNOT gates could be realized more efficiently.
In fact, a CNOT gate (apart from certain phase factors) (39) could be still realized by a single laser pulse. By settingϑ1=π/2 and
cos(Ω0,0t1) =sin(Ω1,0t1) =1, (58)
η t1Ω α11(1)=α(1)21 α(1)12 =α22(1) α31(1)=α(1)41 α(1)32 =α(1)42
0.62 30.64 0.99720 0.07473 0.07201 0.99740
0.64 31.52 0.96241 0.27158 0.26906 0.96312
0.66 62.42 0.99952 -0.03096 -0.03027 0.99954
0.68 95.26 0.99509 0.09898 0.09985 0.99500
0.70 353.53 0.99224 0.12437 0.12458 0.99221
0.72 178.61 0.97983 -0.19981 -0.20082 0.97963
0.74 82.49 0.99876 -0.04974 -0.05286 0.99860
0.76 50.12 0.99715 -0.07548 -0.07608 0.99710
0.78 186.67 0.96719 -0.25406 -0.25835 0.96605
0.80 155.88 0.99888 0.04737 0.04576 0.99895
0.82 175.54 0.99258 -0.12162 -0.12311 0.99239
0.84 17.27 0.97693 -0.21356 -0.21395 0.97685
0.86 18.04 0.99867 -0.05149 -0.05191 0.99865
0.88 18.88 0.99205 0.12582 0.12453 0.99221
0.90 19.79 0.95031 0.31129 0.31157 0.95022
0.92 153.85 0.99324 0.11610 0.11507 0.99336
0.94 39.40 0.99517 0.09817 0.09647 0.99534
0.96 60.04 0.99627 0.08632 0.08611 0.99629
0.98 122.12 0.99709 0.07617 0.07482 0.99720
Table 6. CNOT gates implemented with high fidelities for arbitrarily selected parameters(from 0.62 to 0.98) (29).
the operation (46) can realize the following two-qubit quantum operation (see, APPENDIX A)
|0|g −→ |0|g
|0|e −→ |0|e
|1|g −→ −|1|e
|1|e −→ |1|g
(59)
which is equivalent to the standard CNOT gate (45) between the external and internal states of the ion, apart from the phase factors−1.
Obviously, the condition (58) can be satisfied by properly setting the relevant experimental parameters:t1andη, as
t1=2nπΩ
0,0,η2=1−m−34
n , n,m=1, 2, 3...., (60)
with n and m being arbitrary positive integers. Because of the practical existence of decoherence, as we discussed above, the duration of the present pulse should be shorter than the decoherence times of both the atomic and motional states of the ion. This limits that the integersncould not take arbitrary large values to let Eq. (60) be exactly satisfied.
In tables 5 and 6 we present some numerical results for setting proper experimental parameters Ωt1 (all of them 0.1 ms for the experimental Rabi frequency Ω/(2π) ≈ 500 KHz), to implement quantum operation (59) for the arbitrarily selected LD parameters (not limited within the LD regime requiringη1) from 0.18 to 0.98. It is seen that, the
with so short duration is not a great difficulty for the current experimental technology, e.g., the femto-second (10−15s) laser technique. Also, our numerical calculations show that the influence of the possibly-existing fluctuations of the applied durations is really weak. For example, for the Rabi frequencyΩ/(2π)≈500 kHz, the fluctuationδt≈0.1μs of the duration lowers the desirable probability amplitudes, i.e., α(1)11 andα(1)32 presented in tables 5 and 6, just about 5%. Thus, even consider the imprecision of the durations, the amplitude of the desirable elements,α(1)11 andα(1)32, are still sufficiently large, e.g., up to about 0.95. Therefore, the approach proposed here to implement the desirable quantum operation (59) for arbitrary LD parameters should be experimentally feasible.
Finally, we consider how to generate the standard CNOT gate (45) from the quantum operation (59) produced above. This could be achieved by just eliminating the unwanted phase factors in (59) via introducing another off-resonant laser pulse (10). Indeed, a first blue-sideband pulse (of frequencyωL=ωea+νand initial phaseϑ2) induces the following evolution
|1|e −→cos(Ω0,1t2)|1|e −eiϑ2sin(Ω0,1t2)|0|a, (61) but does not evolve the states|0|g,|1|gand|0|e. Above,|ais an auxiliary atomic level, andωea being the transition frequency between it and the excited state |e. Obviously, a
“π-pulse” defined byΩ0,1t2=πgenerates a so-called controlled-Z logic operation (10)
|0|g −→ |0|g
|0|e −→ |0|e
|1|g −→ |1|g
|1|e −→ −|1|e
(62)
For the LD parameters from 0.18 to 0.98, and Ω/(2π)≈500 kHz, the durations for this implementation are numerically estimated as 3.3ì10−3∼1.2ì10−2 ms. Therefore, the standard CNOT gate (45) with a single trapped ion could be implemented by only two sequential operations demonstrated above.