## Closed Timelike Curves Make Quantum and Classical Computing Equivalent

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Aaronson, Scott

Watrous, John

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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.

Comment: 15 pages

Comment: 15 pages

##### Keywords

Quantum Physics, Computer Science - Computational Complexity