# Can quantum computers help to solve questions of general relativity theory?

My question is rather straightforward: Can quantum computers be used to solve problem within general relativity theory?

To put more context. As GR is based on solution of rather complicated systems of differentials equation, often numerical approaches are needed (there is even part of GR called numerical relativity). Since numerical solution of differential equations is based on their conversion to algebraic ones, I can imagine application of linear algebra algorithms like HHL.

Gravity is only force for which we do not have quantum theory which can be renormalized. Hence, there is still quest for building up such theory - loop gravity, string theory etc being candidates. As quantum computers were originally proposed for simulation of quantum systems, it would be possible to use them for that task within quantum gravity theories.

So, I would like therefore ask if any body is aware of papers trying to apply quantum computing within general relativity theory.

• The equations of motion for GR are generally non-linear, there doesn't seem to be any proof of exponential quantum advantage in simulating nonlinear PDE's, as quantum mechanics is a linear theory. Commented Apr 16 at 17:42
• @Cuhrazatee: Good point and thanks to show me a right direction. So I tried to google non-linear PDE and quantum computing and found this article by Seth Lloyd: arxiv.org/pdf/2011.06571.pdf. He claims to find an algorithm with exp. speed-up for solving non-linear PDF. And here is article dealing with solving of Navier-Stokes equation on a quantum computer: eprints.gla.ac.uk/227158. And here is another solver of non-linear ODE but employing linearization: quantum-journal.org/papers/q-2023-02-02-913. So maybe, there is a hope... Commented Apr 16 at 19:47
• Yes, they can be simulated, but the difficulty comes from the no-cloning theorem. I would say there are two nice resources to look at. One is Lloyd and Abrams work: arxiv.org/abs/quant-ph/9801041, where showing that generically simulating the nonlinear schrodinger equation with polynomial-time implies polynomial time solutions to NP-complete problems. The second is this work by Liu et al pubmed.ncbi.nlm.nih.gov/34446548, they find simulation cost bounds on the case of "weak nonlinearity" and "weak dissipation", showing that there can be "efficient" algorithms in some cases. Commented Apr 16 at 23:26
• @Cuhrazatee: Thanks, the first article is really fascinating. The second one is more practical and it seems that there is only way to first linearize the non-linear diff. eq. and then solve as linear ones Commented Apr 17 at 6:15