Physicist (and Catholic) Stephen M. Barr takes on the question, He writes:
Not in any direct way. That is, it doesn’t provide an argument for the existence of God. But it does so indirectly, by providing an argument against the philosophy called materialism (or “physicalism”), which is the main intellectual opponent of belief in God in today’s world.
Materialism is an atheistic philosophy that says that all of reality is reducible to matter and its interactions. It has gained ground because many people think that it’s supported by science. They think that physics has shown the material world to be a closed system of cause and effect, sealed off from the influence of any non-physical realities — if any there be. Since our minds and thoughts obviously do affect the physical world, it would follow that they are themselves merely physical phenomena. No room for a spiritual soul or free will: for materialists we are just “machines made of meat.”
Quantum mechanics, however, throws a monkey wrench into this simple mechanical view of things. No less a figure than Eugene Wigner, a Nobel Prize winner in physics, claimed that materialism — at least with regard to the human mind — is not “logically consistent with present quantum mechanics.” And on the basis of quantum mechanics, Sir Rudolf Peierls, another great 20th-century physicist, said, “the premise that you can describe in terms of physics the whole function of a human being … including [his] knowledge, and [his] consciousness, is untenable. There is still something missing.”
How, one might ask, can quantum mechanics have anything to say about the human mind? Isn’t it about things that can be physically measured, such as particles and forces? It is; but while minds cannot be measured, it is ultimately minds that do the measuring. And that, as we shall see, is a fact that cannot be ignored in trying to make sense of quantum mechanics. If one claims that it is possible (in principle) to give a complete physical description of what goes on during a measurement — including the mind of the person who is doing the measuring — one is led into severe difficulties. This was pointed out in the 1930s by the great mathematician John von Neumann. Though I cannot go into technicalities in an essay such as this, I will try to sketch the argument.