Overcoming the Wall in the Semiclassical Baker's Map

A major barrier in semiclassical calculations for chaotic systems is the exponential increase in the number of terms at long times. Using an analogy with spin-chain partition functions, we overcome this “exponential wall” for the baker's map, reducing to order ${\mathrm{NT}}^{3/2}$ the number of operations needed to evolve an $N-\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e}$ system for $T$ time steps. This method enables us to obtain semiclassical results up to the Heisenberg time and beyond, providing new insight as to the accuracy of the semiclassical approximation. The semiclassical result is often correct; its breakdown is nonuniform.