With the vanishingly few fossils we have of ancient animals, and the forces of Subduction, volcanism, continental drift etc, what would we see today of a civilisation that died out a billion or even 2 billion years ago, that was at or above todays level of tech? I've seen answers that we would see evidence of their mining of resources and farming practices, but anything mined or farmed at that time would be way out of reach by now (subduction etc)
As mentioned your best bet would be objects off Earth, though they would likely have to be probes anchored on some stable body. After a billion years almost anything around the earth should be lost to orbital decay or to deep space from the steady push radiation pressure from the sun and orbital decay from the exosphere drag. This is a known issue with satellites in GEO due to the earth itself, interactions with the sun and moon, and radiation pressure. Station Keeping is required of nearly all spacecraft in orbit around the earth due to these interactions. Without constant upkeep of satellites and over this timescale it's unlikely that there would be anything to find in orbit around the earth.
On earth actually, you are in luck; there are places on/near the surface that have remained stable for that period of time. It would be possible to find layers of strata that include unnatural minerals or even artifacts from that era. We have left an impression on the earth's geological record even now with the creation of plastics, even minerals, refinement of metals, movement of materials, and interfering with the ratios of radioactive isotopes.
What subduction? The continental cratons have never been submerged, much less subducted. On the other hand, one billion years of sedimentation and glacial scraping...
However, some traces would remain. Unexplicable traces soda-lime glass dust, which should not be there. Traces of unexplicable radioactive elements such as technetium-98 or plutonium-244, which should not exist on Earth.
But there was no such civilisation. For example, we have used all the readily available cryolite; there is no more natural cryolite and we have to manufacture it because we need it in the process of smelting aluminium. A previous civilization would have done the same, leaving us with widely dispersed traces of cryolite (in all sorts of unsuitable places) but no sizeable deposits. We have used up all the readily available native copper; a previous civilization would have done the same, leaving us with widely dispersed pieces of pure copper (in unexplicably large quantities and located in most unexpected places) but no sizeable deposits. Then there are the iron deposits. We have used tremedous amounts of iron ore deposits; but those deposits appear to have been completely undisturbed for one or two billion years: they appear in the right geological sequence with nothing mysterious going on below or above.
A bigger problem is that one billion years ago there was no wood, no coal, no petroleum and no natural gas. How did this ancient civilization go directly from the stone age to atomic power? There was very little lime: what did they use for cement?
And the biggest problem is that one billion years ago there was no life on land, except maybe some micro-organisms...
In conclusion, one billion years is not enough to erase all traces, and at the same time much too early in the history of life on this green Earth. Green, that is, today; one billion years ago there were no land plants, so the continents were gray...
1-2 billion years ago puts us in the middle of the Proterozoic Eon, which was marked by glaciation, very active geological processes, and a sharp rise in oxygen in the atmosphere. The fossils from around that time really only show evidence of the beginnings of nucleus-celled life. So basically anything on the surface of that time would have been subjected quite far below the surface, as you thought.
So what about a civilization at our level? Well, any buildings at that level would have been churned down into the crust, monuments weathered away, etc... Fossils of plant life have survived that far back, and it's possible plant fossils would show evidence of agriculture. If the civilization was global, then the spread of plants across the entire globe would be something noticed in the fossil record.
The advanced civ would have to have been on land (since anything over ocean would be buried deep, deep in the ocean, very difficult to find now). We have designed things to last an extremely long time, such as the Clock of the Long Now. But any kind of advanced machinery would have the metals fuse together over such a long time, and would not be distinguishable in the fossil record. Your best bet would be evidence of life; not say, tools.
You could look for evidence of intelligent design in the genomes of organisms.
Intelligent design refers to a scientific research program as well as a community of scientists, philosophers and other scholars who seek evidence of design in nature. The theory of intelligent design holds that certain features of the universe and of living things are best explained by an intelligent cause, not an undirected process such as natural selection. Through the study and analysis of a system's components, a design theorist is able to determine whether various natural structures are the product of chance, natural law, intelligent design, or some combination thereof. Such research is conducted by observing the types of information produced when intelligent agents act. Scientists then seek to find objects which have those same types of informational properties which we commonly know come from intelligence. Intelligent design has applied these scientific methods to detect design in irreducibly complex biological structures, the complex and specified information content in DNA, the life-sustaining physical architecture of the universe, and the geologically rapid origin of biological diversity in the fossil record during the Cambrian explosion approximately 530 million years ago.<
The website takes some pains to distance themselves from creationism, which seeks to validate a set of religious beliefs. The site states (and to me with some justification) that conflation of creationism with investigations of possible intelligent design serves to discredit the latter with rational thinkers.
I probably do not need to add that evidence of intelligent design is like the Face on Mars; a person who is ready to see it will see it. I am not aware of formal criteria that, if found, all interested parties agree is evidence of intelligence.
If someone found within the dna coding for histones a stretch of degraded "trash dna" which when translated turned out to be a fragment of a treatise on geometry that would probably count.
I would say space debris would be your best shot.
I would imagine that any matter put into orbit 2 billion would still be present in some format, whether it be dust or debris. I'm not too sure on the rate of decay for metals/plastics in space, but I'd assume most things would get smashed up beyond recognition by various collisions over the years. However, some chunks of various satellites or rockets may remain.
If you're talking super-advanced Civ though, you could always develop a super-alloy that can stand up to just about anything?
EDIT: It seems Joe has directly contradicted my ideas - he's probably right, I hadn't thought of orbital decay