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For many types of superconducting qubits, magnetic flux noise is a source ofpure dephasing. Measurements on a representative dc superconducting quantuminterference device (SQUID) over a range of temperatures show that $S_\Phi(f) =A^2/(f/1 \hbox{Hz})^\alpha$, where $S_\Phi$ is the flux noise spectral density,$A$ is of the order of 1 $\mu\Phi_0 \, \hbox{Hz}^{-1/2}$ and $0.61 \leq \alpha\leq 0.95$; $\Phi_{0}$ is the flux quantum. For a qubit with an energy levelsplitting linearly coupled to the applied flux, calculations of the dependenceof the pure dephasing time $\tau_\phi$ of Ramsey and echo pulse sequences on$\alpha$ for fixed $A$ show that $\tau_\phi$ decreases rapidly as $\alpha$ isreduced. We find that $\tau_\phi$ is relatively insensitive to the noisebandwidth, $f_1 \leq f \leq f_2$, for all $\alpha$ provided the ultravioletcutoff frequency $f_2 > 1/\tau_\phi$. We calculate the ratio $\tau_{\phi,E} /\tau_{\phi,R}$ of the echo ($E$) and Ramsey ($R$) sequences, and the dependenceof the decay function on $\alpha$ and $f_2$. We investigate the case in which$S_\Phi(f_0)$ is fixed at the "pivot frequency" $f_0 \neq 1$ Hz while $\alpha$is varied, and find that the choice of $f_0$ can greatly influence thesensitivity of $\tau_{\phi,E}$ and $\tau_{\phi,R}$ to the value of $\alpha$.Finally, we present calculated values of $\tau_\phi$ in a qubit correspondingto the values of $A$ and $\alpha$ measured in our SQUID.

Pure dephasing in flux qubits due to flux noise with spectral density scaling as $1/ f^\alpha$