A quantum-information-theoretic complement to a general-relativistic implementation of a beyond-Turing computer

There exists a growing literature on the so-called physical ChurchTuring thesis in a relativistic spacetime setting. The physical ChurchTuring thesis is the conjecture that no computing device that is physically realizable (even in principle) can exceed the computational barriers of a Turing machine. By suggesting a concrete implementation of a beyond-Turing computer in a spacetime setting, István Németi and Gyula Dávid (2006) have shown how an appreciation of the physical Church-Turing thesis necessitates the confluence of mathematical, computational, physical, and indeed cosmological ideas. In this essay, I will honour István’s seventieth birthday, as well as his longstanding interest in, and his seminal contributions to, this field going back to as early as 1987 by modestly proposing how the concrete implementation in Németi and Dávid (2006) might be complemented by a quantuminformation-theoretic communication protocol between the computing device and the logician who sets the beyond-Turing computer a task such as determining the consistency of Zermelo-Fraenkel set theory. This suggests that even the foundations of quantum theory and, ultimately, quantum gravity may play an important role in determining the validity of the physical Church-Turing thesis. ∗I am grateful to István Németi and Hajnal Andréka for discussions on the topic and for their willingness to share old material from their personal archive. I thank Gergely Székely for his great patience with my procrastinating, Joseph Berkovitz and Kent Peacock for correspondence, and the audience at the István-Fest, as well as John Dougherty and the referees for this journal for comments.