C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM Journal on Computing, 26:1510--1523, 1997. 15  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... with work of Deutsch [3] Deutsch and Jozsa [4] and Berthiaume and Brassard [5] A seminal paper is the in depth study of quantum complexity which was conducted by Bernstein and Vazirani [6] Research in this area continued with such papers as that of Bennett, Bernstein, Brassard and Vazirani [7], Simon [8] Berthiaume and Brassard [9] and Adleman, Demarrais and Huang [10] which broadened the study of quantum complexity classes in analogy with their classical counterparts. In this paper we continue in this examination and consider quantum circuit classes de ned in analogy with classical ....

C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM Journal on Computing, 26:1510-1523, 1997.

....is, HD E jwihwj where HD has no special dependence on w can find jwi any faster, even if HD is allowed to depend on time. Our Hamiltonian is clearly not of this form, so their lower bounds aren t directly applicable here. Indeed, it is only by the lower bounds shown for digital quantum search [1] that we know that our Hamiltonian is optimal to simulate a small digital quantum circuit. It is an interesting question whether one can deduce the same lower bound by more direct means. 5 Further Research and Open Problems We have seen how Grover s algorithm can be described much more simply ....

C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM J. Comp., 26(5):1510--1523, 1997, quant-ph/9701001.

....bounds for non Boolean non symmetric functions are known. We prove our lower bound of 1 (ln(N) 1) by utilizing what we refer to as a weighted all pairs inner product argument. We hope that this proof technique, which extends previous work by especially Bennett, Bernstein, Brassard and Vazirani [4] and Ambainis [2] will be of use elsewhere. We rst give a description of our proof technique in Section 2, by utilizing which we prove a general lower bound in Section 3. We then apply the technique to ordered searching in Section 4. In Section 5, we give a new quantum algorithm for ordered ....

....have (x; y) 0. Hereby, we may ignore f and just consider the scenario in which we are given weight function . 3 General lower bound The rst general technique for proving lower bounds for quantum computing was introduced by Bennett, Bernstein, Brassard and Vazirani in their in uential paper [4]. Their beautiful technique is nicely described in Vazirani s exposition [14] Our technique is a natural generalization of theirs, but it can also be viewed as a generalization of Ambainis powerful entanglement lower bound approach recently proposed in [2] provided one casts his technique using ....

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Bennett, C. H., E. Bernstein, G. Brassard and U. Vazirani, \Strengths and weaknesses of quantum computation", SIAM J. Comp., 26:1510-1523, 1997.

....the problem of determining the exact accepting probability of a quantum computation, and possibly even to locate QAP within the polynomial hierarchy. Our work shows that this is probably not the case as if QAP is in the polynomial hierarchy then this hierarchy collapses. Work of Bennett et al. [3] and recently of Fortnow and Rogers [13] has suggested that quantum computation with bounded error probability (BQP) is most likely unable to solve NP hard problems. Combined with our result, this implies that BQP is even less likely than PH to contain QAP . We take this as evidence that quantum ....

....studied (see [15, 12] for example) It has been found [20, 21] that there are counting problems at least as difficult as any problem in PH, and thus (likely) much more difficult than any NP problem. The relationship between quantum computing and counting problems has been previously observed [18, 13, 3]. Our result further strengthens the connections between quantum computation and counting complexity and strengthens previous results in this area by providing the first example of a quantum computation problem whose complexity can be precisely characterized in terms of a counting class. The ....

C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM Journal on Computing, 26:1510--1523, 1997.

....and T , and which outputs j 2 i such that T (j 2 i) SUCCESS. Then R can be used to invert any one way permutation. The existence of one way permutations, easy to compute in quantum polynomial time but hard to invert, is likely because of the results of Bennett, Bernstein, Brassard and Vazirani [BBBV97]. They show that, relative to a length preserving permutation oracle chosen at random, with probability 1 the class NP co NP cannot be solved in time 0(2 n=3 ) on a QTM. It must be realized that if the class NP co NP can be solved in polynomial time, then one way permutations cannot exist: for ....

BENNETT, C. H., E. BERNSTEIN, G. BRASSARD AND U. VAZI- RANI, "Strengths and weaknesses of quantum computation", SIAM J. Computing 26, 5 (1997), pp. 1510--1523.

....hard for #P. By analogy, one might have hoped QAP would be significantly easier than the problem of determining the exact accepting probability of a quantum computation, and possibly even to locate QAP within the polynomial hierarchy. Our work shows that this is not the case. Work of Bennet et al. [BBBV] and recently of Fortnow and Rogers [FR97] has suggested that quantum computation with bounded error probability (BQP) is most likely unable to solve NP hard problems. Combined with our result, this implies that BQP is even less likely than PH to contain QAP . We take this as evidence that quantum ....

....probability exists classically as well; in the classical case, bounded error computation corresponds to BPP and determining non zero acceptence probability, as mentioned before, corresponds to NP. The relationship between quantum computing and counting problems has been previously observed ([Sim94, FR97, BBBV]) Our result further strengthens the connections between quantum computation and counting complexity and strengthens previous results in this area by providing the first example of a quantum computation problem whose complexity can be precisely characterized in terms of a counting class. The ....

C.H. Bennet, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. Manuscript.

....of Shor [Sh] showing that factoring and computing discrete logs are computable in polynomial time on a quantum computer. In view of these results it is natural to ask whether quantum computers can solve all problems in the class NP (or NP co Gamma NP ) in polynomial time. Bennett, et al. [BBBV] gave evidence that this question is unlikely to be resolved without a major breakthrough in complexity theory, by showing that relative to a random oracle, NP 6 BQT IME(o(2 n=2 ) This result is best possible, up to constant factors, since there is a matching upper bound that follows from ....

....IME(o(2 n=2 ) This result is best possible, up to constant factors, since there is a matching upper bound that follows from Grover s [Gr] quantum search algorithm. Boyer, et al. BBHT] obtain the exact constants in the upper and lower bounds, thus exhibiting a tight bound on quantum search. [BBBV] also showed that relative to a random permutation oracle NP co Gamma NP 6 BQT IME(o(2 n=3 ) As in all oracle lower bounds, the results of [BBBV] are proved by establishing a lower bound on the number of oracle queries that the algorithm must make. More generally, in the black box model, ....

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Bennett, C., Bernstein, E., Brassard, G. and Vazirani, U., "Strengths and Weaknesses of Quantum Computation", Siam Journal of Computing, 26, October, 1997 (special issue on quantum computation) .

....of Shor [Sh] showing that factoring and computing discrete logs are computable in polynomial time on a quantum computer. In view of these results it is natural to ask whether quantum computers can solve all problems in the class NP (or NP co Gamma NP ) in polynomial time. Bennett, et al. [BBBV] gave evidence that this question is unlikely to be resolved without a major breakthrough in complexity theory, by showing that relative to a random oracle, NP 6 BQT IME(o(2 n=2 ) This result is best possible, up to constant factors, since there is a matching upper bound that follows from ....

....factors, since there is a matching upper bound that follows from Grover s [Gr] Supported by a JSEP grant. Email: vazirani cs.berkeley.edu. quantum search algorithm. Boyer, et al. BBHT] obtain the exact constants in the upper and lower bounds, thus exhibiting a tight bound on quantum search. [BBBV] also showed that relative to a random permutation oracle NP co Gamma NP 6 BQT IME(o(2 n=3 ) As in all oracle lower bounds, the results of [BBBV] are proved by establishing a lower bound on the number of oracle queries that the algorithm must make. More generally, in the black box model, ....

[Article contains additional citation context not shown here]

Bennett, C., Bernstein, E., Brassard, G. and Vazirani, U., "Strengths and Weaknesses of Quantum Computation", Special issue of Siam J. Comp., October, 1997.

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C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM Journal on Computing, 26:1510--1523, 1997. 15

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C.H. Bennet, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. Manuscript.

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C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM Journal on Computing, 26:1510--1523, 1997. 15

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C. H. Bennett, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. SIAM Journal on Computing, 26(5):1510-1523, 1997, quantph /9701001. 40

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C.H. Bennet, E. Bernstein, G. Brassard, and U. Vazirani. Strengths and weaknesses of quantum computation. Manuscript.

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