Benincasa-Dowker causal set actions by quantum counting

TL;DR

This paper introduces a quantum algorithm that efficiently computes the Benincasa-Dowker causal set action, a discrete analogue of the Einstein-Hilbert action, offering a polynomial speed-up over classical methods in quantum gravity research.

Contribution

The authors develop an asymptotically optimal quantum algorithm with $ ilde{O}(n^{2})$ runtime for calculating the causal set action in any spacetime dimension, improving computational efficiency.

Findings

Quantum algorithm runs in $ ilde{O}(n^{2})$ time, asymptotically optimal.

Provides polynomial speed-up over classical algorithms.

Enables efficient computation of causal set actions in quantum gravity simulations.

Abstract

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.