Richard Jozsa. Characterizing classes of functions computable by quantum parallelism. Proceedings of the Royal Society of London A, 435:563--574, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... The algorithms of Deutsch [38] Deutsch Jozsa [39] Berthiaume Brassard [23] Bernstein Vazirani [22] Simon [92] Grover [48] and Buhrman van Dam [28] give examples of problems for which we have a quantum reduction in the query complexity of a problem, whereas the lower bounds of Jozsa [59], Bennett et al. 19] and Beals et al. 10] show that there are also limits to the advantage that quantum computation can give us. The general picture that has emerged from these results is that we can only expect a superpolynomial difference between classical and quantum computation if we can ....

Richard Jozsa. Characterizing classes of functions computable by quantum parallelism. Proceedings of the Royal Society of London A, 435:563--574, 1991.

....(QTM) He also described a universal simulator for QTMs with an exponential overhead. More recently, Bernstein and Vazirani constructed a universal QTM with only a polynomial simulation overhead [7] The power of QTMs was compared to that of classical probabilistic TMs in a sequence of papers [17, 11, 2, 7]. The most striking evidence that QTMs can indeed be more powerful than probabilistic TMs was obtained by Shor[22] who built his work on an earlier result of Simon [21] Shor has shown that the problems of computing the discrete logarithm and factoring can be This research was supported by ....

R. Jozsa, Characterizing classes of functions computable by quantum parallelism, Proceedings of the Royal Society of London, A435:563-574, 1991.

....Since this is one sided, it is below the classical Saks WigdersonSantha lower bound of 1 3 for one query routines. With # = sin 1 ( 1 # 10 ) it has error probability 1 10 for all inputs, below the two sided classical bound of 1 6. No exact onequery quantum algorithm for OR is possible [24], 20] The MEASURE 1 [0] gate may be replaced by a CHADAMARD [0 1] gate, which effects a Hadamard on qubit 1 if qubit 0 is in the state 1# and otherwise does nothing, yielding an algorithm with the same output probabilities but different characteristics when used as a building block in larger ....

R. Jozsa, "Characterizing classes of functions computable by quantum parallelism," in Proceedings of the Royal Society of London A 435, 1991, pp. 563--574.

....(QTM) He also described a universal simulator for QTMs with an exponential overhead. More recently, Bernstein and Vazirani constructed a universal QTM with only a polynomial simulation overhead [7] The power of QTMs was compared to that of classical probabilistic TMs in a sequence of papers [17, 11, 2, 7]. The most striking evidence that QTMs can indeed be more powerful than probabilistic TMs was obtained by Shor[22] who built his work on an earlier result of Simon [21] Shor has shown that the problems of computing the discrete logarithm and factoring can be efficiently solved on a QTM, whereas ....

R. Jozsa, Characterizing classes of functions computable by quantum parallelism, Proceedings of the Royal Society of London, A435:563--574, 1991.

....induces a linear transformation corresponding to the program. Let U f be such a linear transformation corresponding to the program for f . Then, by the linearity of U f , we have U f ( X Y ) U f ( X) U f ( Y ) This type of computation is called quantum parallel computation [7, 10]. We can use the same procedure even when we have to evaluate the values of f for exponentially many assignments. The above equation means that a QTM can compute a superposition of the values of f quickly. But, according to the current quantum physics, it is not certain whether we can read each ....

R. Jozsa, "Characterizing Classes of Functions Computable by Quantum Parallelism", Proc. R. Soc. Lond., A 435(1991), pp. 563-574.

....given probabilistically according to the laws of quantum mechanics, and it may well not reoccur if you start the whole process all over again. Unfortunately, the probability of success is never better than 2 Gamman in the worst case when there are 2 n outputs in equally weighted superposition [20], unless you wish to compute a constant function of the outputs. The simplest example of such behaviour is useless but instructive: any single one of the 2 n outputs can be obtained with probability 2 Gamman of success. This is useless because the same can be achieved classically: secretly ....

....the travelling salesperson problem [10, 21] This would be a major breakthrough. Unfortunately, even the or of two bits in quantum superposition is not computable by quantum parallelism: no measurement can give the or with nonzero probability of learning that the correct answer has been obtained [20]. Nevertheless, Deutsch and Jozsa have discovered that the or of exponentially many bits in quantum superposition can be computed efficiently under the promise [19] that either all the outputs are 0 or that exactly half of them are 1 [13] Moreover, this can be done with certainty in the sense ....

Jozsa, R., "Characterizing classes of functions computable by quantum parallelism", Proceedings of the Royal Society, London, Vol. A435, 1991, pp. 563 -- 574.

....an open question raised in [BV93] We also develop a theory of quantum communication complexity, and use it as a tool to prove that the majority function does not have a linear size quantum formula. For other developments on quantum complexity, see Berthiaume and Brassard [BB92] and Jozsa [Jo91]. Qquantum effects have also been studied in the context of cryptographic protocols by Wiesner, Bennett, Brassard, Cr epeau, and others; for more information on this subject, see [Br93] for an up to date survey and the references in the recent paper [BCJL93] For work in quantum systems from the ....

R. Jozsa, "Characterizing classes of functions computable by quantum parallelism, " Proceedings of the Royal Society of London A435 (1991), 563-574.

....every other day, while the classical computer never does. More to the point for this paper, the logical or function of the outputs is not computable by quantum parallelism in the sense of the above template: no measurement can give the or with nonzero probability of being certain of the answer [14] Nevertheless, Deutsch and Jozsa have discovered that it can be computed efficiently in the context of some promise problems 4 [9] where the promise is that either all the outputs are 0 or that exactly half of them are 1. Moreover, this can be done with certainty in the sense that the result ....

Jozsa, R., "Characterizing classes of functions computable by quantum parallelism", Proceedings of the Royal Society, London, Vol. A435, 1991, pp. 563 -- 574.

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