S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for PH. Technical Report 98-008, Computer Science Department, Boston University, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....quantum setting. Two such ways are as follows. First, we may view a nondeterministic process as a probabilistic process, and consider whether the resulting process has zero or nonzero probability of success. Along these lines, Adleman, DeMarrais, and Huang [1] and Fenner, Green, Homer, and Pruim [18] have defined QNP to be Research partially supported by Canada s NSERC. the class of languages L for which there exist polynomial time quantum Turing machines that accept with nonzero probability if and only if the input is in L. This class coincides with the counting class co C= P [18, 19] ....

....Pruim [18] have defined QNP to be Research partially supported by Canada s NSERC. the class of languages L for which there exist polynomial time quantum Turing machines that accept with nonzero probability if and only if the input is in L. This class coincides with the counting class co C= P [18, 19]. This notion of quantum nondeterminism has also been investigated recently in the context of communication complexity and query complexity by de Wolf [28] Second, we way view nondeterminism as it relates to verification. A common way to view NP is that NP is the class of languages consisting of ....

S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy. Proceedings of the Royal Society, London A, 455:3953--3966, 1999.

....and PL Next, we recall some definitions and facts from counting complexity and space bounded complexity. Counting complexity is a powerful technique that has its origins in the work of Valiant [22] and has had a number of applications in complexity theory (including in quantum computing [9, 11]) For further information on counting complexity, see the survey of Fortnow [10] and the references therein. Counting complexity was applied to space bounded computation in [3, 4] to which the reader is referred to for proofs of the theorems stated in this section. For more general background ....

S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy. In Proceedings of the Sixth Italian Conference on Theoretical Computer Science, 1998.

.... a space s QTM accepting precisely those strings in the given language with nonzero probability corresponds to the counting class co C =SPACE(s) and hence contains NSPACE(s) This characterization may be viewed as the space bounded analogue of a recent result of Fenner, Green, Homer, and Pruim [11] that equates quantum NP and co C = P. Simple relationships between co C =SPACE(s) and one sided error space bounded quantum classes are examined as well. The remainder of this paper has the following organization. In Section 2 we define the quantum Turing machine model and space bounded quantum ....

S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for PH. Technical Report 98-008, Computer Science Department, Boston University, April 1998.

....[2] singularity of integer matrices is in C= L, from which it follows that testing non singularity of A can be performed in co C=SPACE(s) From Propositions 4.4 and 5.6, we have Corollary 5. 7 NQSPACE(s) co C=SPACE(s) This may be viewed as the space bounded analogue of the result QNP = co C=P [13]. 6 Conclusion and open problems Figure 1 is a diagram which summarizes the relationships between some of the quantum and classical space bounded classes we have discussed in this paper. s DSPACE(s 2 ) RevSPACE(s 2 ) s PrSPACE(s) PrQSPACE(s) PrQ H SPACE(s) s BPSPACE(s) H H H H H H H H H H ....

S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for PH. Preprint, 1997.

....and Lemma 3.10. Corollary 3.11 There exists a relativized world where BQP has no hards sets for BPP. In particular, BQP has no complete sets in this world. Lemma 3.2 shows how to compute the probability acceptance of a quantum Turing machine with a GapP function. Fenner, Green, Homer and Pruim [FGHP98] give a result in the other direction. Theorem 3.12 (FGHP) For any GapP function f there exists a polynomial time quantum Turing machine M and a polynomial p such that for all x, Pr(M (x) accepts) f(x) 2 p(jxj) From Lemma 3.2 and Theorem 3.12 we immediately get a new characterization of ....

S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for PH. Technical Report 98-008, Computer Science Department, Boston University, 1998.

....Brassard and Vazirani [BBBV97] give a relativized world where NP is not contained in BQP. We do not know any nonrelativized consequences of NP in BQP or if BQP lies in the polynomial time hierarchy. What about quantum variations of NP and interactive proof systems Fenner, Green, Homer and Pruim [FGHP99] consider the class consisting of the languages L such that for some polynomialtime quantum Turing machine, x is in L when M(x) accepts with positive probability. They show the equivalence of this class to the counting class co C=P. Watrous [Wat99] shows that every language in PSPACE has a ....

S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for PH. Proceedings of the Royal Society of London, 455:3953-3966, 1999.

....and EQACC, respectively. For example, the class NQACC corresponds to the case where x is in the language if the expectation of the observed state after applying the QACC operator is non zero. This is analogous to the de nition of the class NQP in Adleman et al. 10] and discussed in Fenner et al. [11] where it is shown that NQP is classically of high complexity, and in particular is hard for the polynomial time hierarchy. In this way we obtain natural classes of languages which correspond to those de ned classically by families of small depth circuits. In these terms, for example, we can ....

....within F. Green, S. Homer, C. Moore, and C. Pollett 39 TC 0 , those problems computed by constant depth threshold circuits. We have been unable to verify this, and in fact the only classical upper bound for these language classes that we know of is the very powerful counting class coC=P (see [11]) We do give some evidence for this proposed TC 0 upper bound here and further provide some techniques which may prove useful in solving this problem, and which certainly remove some signi cant obstacles. One obstacle is, not surprisingly for quantum computation, exponential growth in the size ....

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S. Fenner, F. Green, S. Homer, and R. Pruim, \Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy." Royal Society of London A (1999) 455, pp 3953 - 3966.

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S. Fenner, F. Green, S. Homer, and R. Pruim. Determining acceptance possibility for a quantum computation is hard for PH. Technical Report 98-008, Computer Science Department, Boston University, 1998.

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Fenner, S., Green, F., Homer, S., and Pruim, R. Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy. Proceedings of the Royal Society, London A 455 (1999), 3953--3966.

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