Closed timelike curves make quantum and classical computing equivalent
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.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 February 2009
 DOI:
 10.1098/rspa.2008.0350
 arXiv:
 arXiv:0808.2669
 Bibcode:
 2009RSPSA.465..631A
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity
 EPrint:
 15 pages