Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete while retaining local Lorentz invariance. The Benincasa--Dowker action is the causal set equivalent to the Einstein--Hilbert action underpinning Einstein's general theory of relativity. We present a $\tilde{ O}(n^{ 2})$ running-time quantum algorithm to compute the Benincasa--Dowker action in arbitrary spacetime dimensions for causal sets with $n$ elements which is asymptotically optimal and offers a polynomial speed up compared to all known classical or quantum algorithms. To do this, we prepare a uniform superposition over an $O(n^{ 2})$-size arbitrary subset of computational basis states encoding the classical description of a causal set of interest. We then construct $\tilde{ O}(n)$-depth oracle circuits testing for different discrete volumes between pairs of causal set elements. Repeatedly performing a two-stage variant of quantum counting using these oracles yields the desired algorithm.
Categories: quant-ph gr-qc
