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Unformatted text preview: arXiv:0808.2669v1 [quant-ph] 19 Aug 2008 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) 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. 1 Introduction The possibility of closed timelike curves (CTCs) within general relativity and quantum gravity theories has been studied for almost a century [11, 15, 13]. A different line of research has sought to understand the implications of CTCs, supposing they existed, for quantum mechanics, computation, and information [9, 8, 5]. In this paper we contribute to the latter topic, by giving the first complete characterization of the computational power of CTCs. We show that if CTCs existed, then both classical and quantum computers would have exactly the power of the complexity class PSPACE , which consists of all problems solvable on a classical computer with a polynomial amount of memory. To put it differently, CTCs would make polynomial time equivalent to polynomial space as computational resources, and would also make quantum and classical computers equivalent to each other in their computational power. Our results treat CTCs using the causal consistency framework of Deutsch . It will not be hard to show that classical computers with CTCs can simulate PSPACE and be simulated in it (though as far as we know, this result is new). The main difficulty will be to show that quantum computers with CTCs can be simulated in PSPACE . To prove this, we need to give Email: firstname.lastname@example.org. Email: email@example.com. 1 an algorithm for (implicitly) computing fixed-points of superoperators in polynomial space. Our algorithm relies on fast parallel algorithms for linear algebra due to Borodin, Cook, and Pippenger , and might be of independent interest....
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