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Closed Timelike Curves Make Quantum and Classical Computing Equivalent
"... 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: b ..."
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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) 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. 1
QMA/qpoly ⊆ PSPACE/poly: DeMerlinizing quantum protocols
 In TwentyFirst Annual IEEE Conference on Computational Complexity
, 2006
"... This paper introduces a new technique for removing existential quantifiers over quantum states. Using this technique, we show that there is no way to pack an exponential number of bits into a polynomialsize quantum state, in such a way that the value of any one of those bits can later be proven wit ..."
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This paper introduces a new technique for removing existential quantifiers over quantum states. Using this technique, we show that there is no way to pack an exponential number of bits into a polynomialsize quantum state, in such a way that the value of any one of those bits can later be proven with the help of a polynomialsize quantum witness. We also show that any problem in QMA with polynomialsize quantum advice, is also in PSPACE with polynomialsize classical advice. This builds on our earlier result that BQP/qpoly ⊆ PP/poly, and offers an intriguing counterpoint to the recent discovery of Raz that QIP/qpoly = ALL. Finally, we show that QCMA/qpoly ⊆ PP/poly and that QMA/rpoly = QMA/poly. 1
Interactive PCP
 In Proceedings of the 35th International Colloquium on Automata, Languages and Programming, ICALP ’08
, 2008
"... A central line of research in the area of PCPs is devoted to constructing short PCPs. In this paper, we show that if we allow an additional interactive verification phase, with very low communication complexity, then for some NP languages, one can construct PCPs that are significantly shorter than t ..."
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A central line of research in the area of PCPs is devoted to constructing short PCPs. In this paper, we show that if we allow an additional interactive verification phase, with very low communication complexity, then for some NP languages, one can construct PCPs that are significantly shorter than the known PCPs (without the additional interactive phase) for these languages. We give many cryptographical applications and motivations for our results and for the study of the new model in general. More specifically, we study a new model of proofs: interactivePCP. Roughly speaking, an interactivePCP (say, for the membership x ∈ L) is a proofstring that can be verified by reading only one of its bits, with the help of an interactiveproof with very small communication complexity. We show that for membership in some NP languages L, there are interactivePCPs that are significantly shorter than the known (noninteractive) PCPs for these languages. Our main result is that for any constant depth Boolean formula Φ(z1,...,zk) of size n (over the gates ∧, ∨, ⊕, ¬), a prover, Alice, can publish a proofstring for the satisfiability of Φ, where the size of the proofstring is poly(k). Later on, any user who wishes to verify the published
On the power of entangled quantum provers
, 2006
"... We show that the value of a general twoprover quantum game cannot be computed by a semidefinite program of polynomial size (unless P=NP), a method that has been successful in more restricted quantum games. More precisely, we show that proof of membership in the NPcomplete problem GAP3DMATCHING c ..."
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We show that the value of a general twoprover quantum game cannot be computed by a semidefinite program of polynomial size (unless P=NP), a method that has been successful in more restricted quantum games. More precisely, we show that proof of membership in the NPcomplete problem GAP3DMATCHING can be obtained by a 2prover, 1round quantum interactive proof system where the provers share entanglement, with perfect completeness and soundness s = 1 − 2−O(n), and such that the space of the verifier and the size of the messages are O(log n). This implies that QMIP ∗ log n,1,1−2−O(n) � P unless P = NP and provides the first nontrivial lower bound on the power of entangled quantum provers, albeit with an exponentially small gap. The gap achievable by our proof system might in fact be larger, provided a certain conjecture on almost commuting versus nearly commuting projector matrices is true.
Are quantum states exponentially long vectors?
, 2005
"... In this extended abstract, which is based on a talk that I gave there, I demonstrate that gratitude by explaining why Goldreich’s views about quantum computing are wrong. Why should anyone care? Because in my opinion, Goldreich, along with Leonid Levin [10] and other “extreme ” quantum computing ske ..."
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In this extended abstract, which is based on a talk that I gave there, I demonstrate that gratitude by explaining why Goldreich’s views about quantum computing are wrong. Why should anyone care? Because in my opinion, Goldreich, along with Leonid Levin [10] and other “extreme ” quantum computing skeptics, deserves credit for focusing attention on the key issues, the ones that ought to motivate quantum computing research in the first place. Personally, I have never lain awake at night yearning for the factors of a 1024bit RSA integer, let alone the class group of a number field. The real reason to study quantum computing is not to learn other people’s secrets, but to unravel the ultimate Secret of Secrets: is our universe a polynomial or an exponential place? Last year Goldreich [7] came down firmly on the “polynomial ” side, in a short essay expressing his belief that quantum computing is impossible not only in practice but also in principle: As far as I am concern[ed], the QC model consists of exponentiallylong vectors (possible configurations) and some “uniform ” (or “simple”) operations (computation steps) on such vectors... The key point is that the associated complexity measure postulates that each such operation can be effected at unit cost (or unit time). My main concern is with this postulate. My own intuition is that the cost of such an operation or of maintaining such vectors should be linearly related to the amount of “nondegeneracy ” of these vectors, where the “nondegeneracy ” may
Oneway reversible and quantum finite automata with advice
 In: Proceedings of the 6th international conference on Language and Automata Theory and Applications. LATA’12
, 2012
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MIT Abstract
, 808
"... 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: b ..."
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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) 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. 1