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Here we ignore the qubit free evolution hamiltonian. On the other hand, when the qubit is subjected to classical noise, the system is modeled as [18, 19] 

where the Ω is the qubit energy splitting and β ( t ) the classical random noise. Let t be the total evolution time, and n pulses are applied at t$_{1}$ < t$_{2}$ < ... < t$_{n}$ in sequence with negligible pulse durations . We use the notation δ$_{j}$ = t$_{j}$ t . This naturally leads to the definition of t$_{0}$ = 0 and t$_{n}$$_{+1}$ = 1. In either (1) or (2), the decay of coher- ence under the dynamical decoupling sequence can be described by the decoherence function [10, 18, 20, 21] e − 2 χ ( t ) with 

where S ( ω ) is environmental noise spectrum. The filter function y$_{n}$ ( t ) is given by 

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Thus minimization of χ ( t ) with respect to δ$_{j}$ gives the optimal decoupling sequence. 

We now consider the case when the noise spectrum is ohmic with a sharp cutoff at ω$_{c}$ , i.e. S ( ω ) = S$_{0}$ω Θ( ω$_{c}$ − ω ). S$_{0}$ is an irrelevant constant factor and Θ is unit step function. Then minimization of (3) turns to minimization of I$_{n}$ with 

where z$_{c}$ = ω$_{c}$t . Since y$_{n}$ (0) = 0, the IR convergence insures the integral converges to a finite value [13]. 

## III. DERIVATION OF OPTIMAL PULSE SEQUENCE 

We follow the approach of Pasini and Uhrig [13] to treat the integral (5). Here we use notation 

and 

from which we get 

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Then the integral I$_{n}$ can be expressed as 

where the integrals I$_{ij}$ ( x ) are 

The limit x → 0 + is carried out because I$_{ij}$ (0) does not exist for arbitrary i, j . Making use of the series represen- tation of exponential function [22] 

where γ is the Euler-Mascheroni constant and the sum converges for all the complex z , I$_{ij}$ ( x ) can be written as 

Since we always have y$_{n}$ (0) = 0 which implies 

we can now proceed by taking the limit x → 0+ in I$_{n}$ 

<img src='content_image/400826.jpg'>

To minimize I$_{n}$ , UDD requires the first n derivatives of y$_{n}$ vanish while OFDD simplifies the optimization process by replacing S ( ω ) by a constant. Here we attempt to minimize I$_{n}$ directly to obtain optimal pulse sequence. We notice that at the optimal pulse locations δ$_{j}$ ( j = 1 , 2 , ..., n ), the gradient of I$_{n}$ vanishes. So we impose the following conditions ∂I$_{n}$ ∂δ$_{m}$ = 0 , for m from 1 to n . Although (10) are complex infinite series, we can still 

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