 
Summary: Closed Timelike Curves Make Quantum
and Classical Computing Equivalent
Scott Aaronson
MIT
John Watrous
University of Waterloo
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) fixedpoint of some evolution operator. Our conclusion is then a consequence of
the following theorem: given any quantum circuit (not necessarily unitary), a fixedpoint of the
circuit can be (implicitly) computed in polynomial space. This theorem might have independent
applications in quantum information.
