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

Abstract

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: $(\lnα)/τ\lesssim E$. 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 $(\ln N)/D\lesssim E$.

Figures (7)

• Figure 1: In a Malament--Hogarth spacetime, a computer travels along a timelike curve $\gamma$ whose proper length is infinite. Its endpoint $y$ is in the causal past of the observer located at $z$, who can then observe the results of a computation that takes infinitely long to compute, e.g. whether a Turing machine ever halts.
• Figure 2: In Minkowski space, due to time dilation, an observer observes a computer moving relative to them as running slower, not faster.
• Figure 3: In order to accelerate computation in Minkowski space, the observer undergoes uniform acceleration along a timelike curve $\gamma_\mathrm{obs}$ while the computer follows a geodesic $\gamma_\mathrm{comp}$ between the same endpoints.
• Figure 4: In a cylindrical spacetime, one can obtain unboundedly high time advantage without any Unruh radiation, but the observer instead encounters relativistic Casimir energy--momentum.
• Figure 5: Time-advantage scheme in anti-de Sitter space, which is Malament--Hogarth. The computer travels along a timelike curve $\gamma_\mathrm{comp}$ with infinite proper time and experiences a finite temperature. The observer remains at a fixed radial coordinate along the timelike curve $\gamma_\mathrm{obs}$. The computer has access to infinite proper time, but the observer only experiences a finite amount of proper time, seemingly leading to an infinite time advantage.