# 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

Based on comments by Cuhrazatee and my further searching, I figured out that QC can be used in solving problems arising within general relativity. However, as pointed out by Cuhrazatee, differential equations of GR are rather non-linear whereas quantum mechanics, and hence quantum computers, is governed by linear equations. To tackle this difficulty, equations of GR have to be firstly linearized. Fortunately, there are several articles dealing with that and presenting quantum algorithms for solving non-linear differential equations. As these algorithms are general, they can be used also outside GR, e.g. in fluid mechanics (Navier-Stokes equations), plasma physics (somehow related to fluid mechanics) and many other ares.

Here is a brief list of articles dealing with solution of non-linear differential equations:

Interestingly, my attention was also brought to an article speculating what would happen if quantum mechanics were non-linear and hence quantum computers allowed to solve non-linear differential equations without necessity of linearization (i.e. natively). Here is a link to question dealing with this (rather speculative) possibility and its consequences for complexity theory.

• I also want to note, every one of those papers is referring to a digital quantum simulation, where the quantum computer performs operations perfectly from a finite gate set to approximate the desired operations. The best one can really do at this point is concrete resource estimations for particular problems. The real question one needs to ask before performing such a simulation is: what observable quantities do I wish to estimate from the solution state? Is there no efficient classical algorithm that can perform similarly for estimating those quantities? Commented Apr 20 at 22:02