4 inspecting the points on the HLODD curve, we expect the HLODD sequence suppresses decoherence in power law n as UDD. At last, we would like to explain the numerical results in another way. If we fixed UV cutoff frequency ω$_{c}$ at the beginning, and compare the HLODD performance for t = 1, t = 5, and t = 10, the numerical results would be the same since z$_{c}$ did not change. So we can also conclude that for the same number of pulses n , HLODD will beat UDD with increasing total evolution time t . In this paper we analytically find the optimal pulse locations to decouple a qubit in an ohmic environment. By deriving the analytical expressions for the derivatives of decoherence function, we obtain a set of non-linear equations which the optimal pulse sequence must obey. These equations are completely different from UDD and are more accurate, because they incorporate the effect of UV cutoff frequency ω$_{c}$ . In our numerical simulation, the analytical results pro- vide an improvement over UDD sequence by an order or two of magnitude, which is consistent with previous re- sults in LODD and OFDD obtained by purely numerical minimization and experimental feedback. We have to mention that the pulse performance is influenced by the sharp UV cutoff frequency ω$_{c}$ greatly. The larger the UV cutoff ω$_{c}$ , the more HLODD deviates from UDD. Early work [13, 14, 18, 20] has pointed out that for soft large UV cutoff, UDD performs even worse and LODD is still a better choice. However, the integral (3) for S ( ω ) with a soft cutoff is hard to analyze. In conclusion, our work provides an analytical solution to optimal dynamical decoupling for ohmic case. Our derivation is based on ohmic spectrum, but we believe it can be extended to super-ohmic case S ( ω ) ∼ ω $^{α}$( α > 1) via slight modification. This work was supported by the National Natu- ral Science Foundation of China (No. 60774099, No. 60821091), the Chinese Academy of Sciences (KJCX3- SYW-S01), and by the CAS Special Grant for Postgrad- uate Research, Innovation and Practice. As shown in Fig. 1, deviation of the pulse locations δ$_{i}$ in HLODD sequence from their UDD counterparts in- creases with ω$_{c}$ . This agrees with our intuition since UDD focuses on suppressing decoherence by minimiz- ing | y$_{n}$ ( z ) | in the neighborhood of y$_{n}$ (0), weakening its ability to maintain small | y$_{n}$ ( z ) | on the other end of the spectrum. For large ω$_{c}$ , UDD sequence is no longer op- timal. In addition, we can see pulse number n plays an important role. By increasing n , UDD can narrow the difference from HLODD. The difference between the two sequences when n = 2 is greatly reduced when we in- crease n to 5, see Fig. 1. Especially for the case ω$_{c}$ = 1, the difference is completely removed. However, for larger ω$_{c}$ this gap can’t be removed by increasing n . Next, to demonstrate the optimal decoupling ability of HLODD sequence, we compute I$_{n}$ versus n while ω$_{c}$ is chosen to be 5. The results are depicted in Fig. 2, and again are compared with UDD. The obtained solutions yield a significant improvement over UDD. For fixed n , the HLODD suppresses decoherence better than UDD by one or two orders of magnitude which is in agreement with the results in [14, 15], where LODD and OFDD sequences are tested for $^{9}$Be+ qubits in a penning ion trap and various spectrum. The qubit error rates are below 10 − 5 when n > 5, and we see that HLODD is capable of suppressing the error rates far below the Fault-Tolerance error threshold [23] by increasing n . Furthermore, by V. CONCLUSIONS FIG. 2: Semi-log plot of I$_{n}$ versus pulse number n . UDD and HLODD sequences are compared. 1 2 3 4 5 6 7 8 9 10 11 10 −15 10 −10 10 −5 10 0 10 5 n I n $_{c}$=5 * UDD + HLODD Acknowledgments [1] L. Viola and S. Lloyd, Phys. Rev. A 58 , 2733 (1998). [2] M. Ban, Journal of Modern Optics 45 , 2315 (1998). [3] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82 , 2417 (1999). [4] G. Gordon, G. Kurizki, and D. A. Lidar, Phys. Rev. Lett. 101 , 010403 (2008). [5] R. R. Ernst, G. Bodenhausen, and A. Wokaun, Prin- ciples of Nuclear Magnetic Resonance in One and Two [6] L. Viola and E. Knill, Phys. Rev. Lett. 94 , 060502 (2005). [7] O. Kern and G. Alber, Phys. Rev. Lett. 95 , 250501 (2005). [8] K. Khodjasteh and D. A. Lidar, Phys. Rev. Lett. 95 , 180501 (2005). [9] K. Khodjasteh and D. A. Lidar, Phys. Rev. A 75 , 062310 (2007). Dimensions (Clarendon Press, Oxford, 1991).