Closed Timelike Curves Make Quantum and Classical Computing Equivalent

TL;DR

This paper demonstrates that if closed timelike curves (CTCs) existed, quantum and classical computing would be computationally equivalent, both limited to the power of PSPACE, resolving an open problem in computational complexity.

Contribution

It proves that CTCs would make quantum and classical computers equally powerful, both confined to PSPACE, and provides a new theorem on fixed-points of quantum circuits.

Findings

Quantum and classical computing are equivalent under CTCs, both in PSPACE.

A fixed-point of any quantum circuit can be computed in polynomial space.

The theorem on fixed-points may have broader applications in quantum information.

Abstract

While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to nontrivial insights in general relativity, quantum information, and other areas. In this paper we show that if CTCs existed, then quantum computers would be no more powerful than classical computers: both would have the (extremely large) power of the complexity class PSPACE, consisting of all problems solvable by a conventional computer using a polynomial amount of memory. This solves an open problem proposed by one of us in 2005, and gives an essentially complete understanding of computational complexity in the presence of CTCs. Following the work of Deutsch, we treat a CTC as simply a region of spacetime where a "causal consistency" condition is imposed, meaning that Nature has to produce a (probabilistic or quantum) fixed-point of some evolution operator. Our conclusion is then a consequence of the following theorem: given any quantum circuit (not necessarily unitary), a fixed-point of the circuit can be (implicitly) computed in polynomial space. This theorem might have independent applications in quantum information.