The short answer
This question is the subject of intense and ongoing research, and at this point there are too many connections between gravity and quantum information to enumerate here. However the most relevant concept to your question is probably the ER=EPR conjecture, which was posited by Susskind (and Maldacena) in order to resolve the so-called 'black hole firewall paradox.' See Cool horizons for entangled black holes. This conjecture posits that entanglement (an essential concept in quantum information) leads to connectivity of spacetime (like wormholes and the smoothness of space). While I don't think anyone yet knows exactly what this statement means, there is substantial evidence that something resembling it is true:
The long answer
It's probably fair to say that line of research started when Stephen Hawking discovered that black holes evaporate, and in doing so recognized the black hole information paradox. His calculation showed that black holes (which are gravitational objects in nature) appear to erase information, which actually violates quantum mechanics. Work connecting quantum information and quantum gravity has been driven in large part by attempts to resolve this and related paradoxes.
The first connection between spacetime and entanglement is probably Mark Van Raamsdonk's paper Building up spacetime with quantum entanglement. Here, he points out the significance of an earlier discovery, that physically separated 'universes' can be connected by a wormhole if they are entangled with each other. Later, the Ryu-Takayanagi conjecture provided a more quantitative connection, this time relating spacetime to entanglement within a single region, instead of entanglement between separate ones. The ER=EPR conjecture took it into even broader and more speculative territory.
Speculative as ER=EPR may be, it appears to be a key ingredient to resolving the black hole information paradox. At this point in the story, there are way too many papers to give even the highest-level summary, but a major breakthrough of the last few years (as of 2021) is the resolution of this paradox via a mechanism similar to that proposed in the original ER=EPR paper. These Quanta articles give a decent account of recent developments.
A few more miscellaneous connections between gravity and quantum information:
- Black holes set a limit on the amount of information that can be stored or processed in a given volume of spacetime, because cramming too much information into a bounded volume eventually forms a black hole (see Wikipedia: Bekenstein bound)
- The connection between traversal of wormholes and quantum teleportation (Quanta article)
- Black holes are posited to be the fastest 'scramblers' of quantum information (A bound on chaos), which causes old black holes to rapidly reemit any information that is thrown into them (Black holes as mirrors: quantum information in random subsystems)
- The circuit complexity, which is a sort of measure of how hard a task is for a quantum computer to perform, is conjectured (also by Susskind) to relate to some properties of spacetime, like the length of a wormhole. It may also prevent casualty violations; operations that could in principle violate casualty, such as those that manipulate the structure of spacetime in really weird ways, are too hard to implement in practice.
- Lots of other stuff I'm not aware of. The field is constantly changing and growing.