Limits to Computational Acceleration Imposed by Quantum Field Theory and Quantum Gravity

A computer, in order to perform a given computation, requires a certain amount of space (memory) and a certain amount of time (runtime). This leaves certain computations beyond reach due to technological limits on processing speed and memory density. Some computations, such as the halting problem, are not possible even in principle. However, curved spacetimes and exotic fields appear to provide avenues to accelerate computation, for instance by exploiting time dilation. Impossible computations seemingly become tractable, butting up against intuition. However, we show that such schemes are consistently thwarted by physical effects from quantum gravity (including swampland conjectures) and quantum field theory in curved space. More precisely, we show that an observer and a computer able to withstand energy scales up to order $E$ can, by using relativistic effects, accelerate computation at a rate of at most $\mathcal O(1)E$ e-folds per unit time in natural units: $(\ln\alpha)/\tau\lesssim E$. The Bekenstein bound for entropy can then be understood as the space (memory) analogue to (run)time: if a computer of length scale $D$, operating at energies up to order $E$, has access to $N$ different memory states, then $(\ln N)/D\lesssim E$.