He doesn't assume all that much. In his book The Singularity Is Near Kurzweil looks at the physical limits of computing, to determine how long we can go before Moore's Law comes to an end. He concludes that we've got another fifty years, or possibly seventy if certain technologies prove feasible. As I recall he doesn't count on quantum computing much, but does think we'll manage reversible computing to solve heat dissipation issues.

That's just computation, but computation drives much of our other technology, and will even more once computers get smarter than people. Kurzweil estimates the timeframe for that based on a range of estimates for the computational capacity of the brain.