2 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 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]. We follow the approach of Pasini and Uhrig [13] to treat the integral (5). Here we use notation and from which we get 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$_{n}$ 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 H = 1 2 [Ω + β ( t )] σ$_{z}$, (2) χ ( t ) = ∫ ∞ 0 S ( ω ) ω 2 | y$_{n}$ ( ωt ) | $^{2}$dω, (3) y$_{n}$ ( t ) = 1 + ( − 1) n $^{+1}$e i ωt + 2 n ∑ j =1 ( − 1) $^{j}$e i $^{ωtδ$_{j}$}$. (4) I$_{n}$ = ∫ z$_{c}$ 0 | y$_{n}$ ( z ) | 2 z dz, (5) q$_{j}$ = { 0 if j = 0 , n + 1, 1 if j ∈ { 1 , 2 , ..., n } , I$_{n}$ = lim x → 0 $_{+}$I$_{n}$ ( x ) , I$_{n}$ ( x ) = n +1 ∑ i,j =0 2 q$_{i}$ + $^{q$_{j}$}$( − 1) i + $^{j}$I$_{ij}$ ( x ) , (6) I$_{ij}$ ( x ) = ∫ z$_{c}$ x e Δ$_{ij}$ z z dz = ∫ − Δ$_{ij}$ z$_{c}$ − Δ$_{ij}$ x e − z z dz. (7) E$_{1}$ ( z ) = ∫ ∞ z e − t t dt = − γ − ln z + ∞ ∑ k =1 ( − 1) k +1 k ! k z $^{k}$, I$_{ij}$ ( x ) = E$_{1}$ ( − Δ$_{ij}$ x ) − E$_{1}$ ( − Δ$_{ij}$ z$_{c}$ ) = ln ( z$_{c}$/x ) + ∞ ∑ k =1 Δ k ij k ! k ( z k c − x $^{k}$) . (8) | y$_{n}$ (0) | 2 = n +1 ∑ i,j =0 2 q$_{i}$ + $^{q$_{j}$}$( − 1) i + j = 0 , (9) I$_{n}$ = lim x → 0 + n +1 ∑ i,j =0 2 q$_{i}$ + $^{q$_{j}$}$( − 1) i + $^{j}$[ln ( z$_{c}$/x ) + ∞ ∑ k =1 Δ k ij k ! k ( z k c − x $^{k}$)] = lim x → 0 + n +1 ∑ i,j =0 2 q$_{i}$ + $^{q$_{j}$}$( − 1) i + j ∞ ∑ k =1 Δ k ij k ! k ( z k c − x $^{k}$) = n +1 ∑ i,j =0 ∞ ∑ k =1 2 q$_{i}$ + $^{q$_{j}$}$( − 1) i + $_{j}$Δ k ij k ! k z k $_{c}$. (10) | y$_{n}$ ( z ) | 2 = n +1 ∑ i,j =0 2 q$_{i}$ + $^{q$_{j}$}$( − 1) i + $^{j}$ez $^{Δ$_{ij}$}$. Δ$_{ij}$ = i( δ$_{i}$ − δ$_{j}$ ) , III. DERIVATION OF OPTIMAL PULSE SEQUENCE