The Economist's View quoted Bloomberg's John Berry that China's Booming Economy Will Never Surpass U.S.. There are a lot of assumption going into how fast the GDP growth in both countries in the next 30-40 years, and other factors such as population and exchange rate.
What I am going to do is look at the historical benchmarking. I feel this is a more accurate measure, as it models the narrowing of gap and increasing difficulty of playing catch up. Below is the chart of OECD nations GDP/cap from 1960 to 2003 (source: demographia), measured in real 2003 PPP converted US$ (PPP so that exchange rate aberration is sort of removed, real so that nominal inflation is adjusted).
China's GDP/cap (in PPP) is about 16% of USA today ($6400 vs $40000), similar to the relative ratio of Korea/US in 1970. It took Korea about 15 years to reach 1/4, and 20 years to reach 1/3 of US level. It is likely to take China the same or longer time given its huge size. Therefore, total GDP of China (in PPP) could be comparable to that of US in 15-20 years, in an optimistic scenario. (There was 9 years of 9-11% growth for Korea from 1960-1990, average CAGR was between 7-8%)
If we assume another big "if", that China will follow Japan's path afterward, i.e. after 15-20 years, China will be at where Japan was at in 1960 relative to US, there will be another 30 years when China joins the developed country pack (70% of US). So the optimistic scenario is that it will around 2055-2065 then. Once China joins the developed country pack, it could follow the path of any of these 14 countries. Whichever path it takes, it will at best reach around 80% of US' GDP/cap. However, no one large/medium country (except small banking havean such as Luxemberg or BVI) has surpassed US at any point of time (*) since 1950 in terms of GDP/cap, and the runners-up are resource rich countries Norway and Canada. It is likely that China will then remain at around 75%-80% of the US level of GDP/cap, like what Japan has been at for the past 20 years.
This is assuming the PPP conversion for Korea/Japan historically and for China today are correct. Alternatively, similar adjustment in exchange rate to converge the PPP gap is assumed during the next 30-50 years.
This is also assuming no major political or economic disruption in the world (not just China, as world recession or war means disruption to China's trading) from now till around 2060. That is why "peaceful development" is the correct, and the only sensible strategy for China. This is also why China cares about a peaceful world environment, and why it has taken great pain in settling border disputes with its neighbors, and is willing to give up its 1.5M sq km claim in dispute with Russian and 90k sq km with India.
Update: (via New Economist) EIU has this forecast, showing China overtake US in PPP terms in total GDP by 2026 already (equality around 2020)
Note (*) Japan did overtake US in the late 1980s briefly in exchange rate GDP per cap, but that was due to a short-lived exchange rate abberation