A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory.InProceedings of the 7th IEEE Structure in Complexity Theory Conference, pages 132--137. IEEE, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(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 ....

A. Berthiaume and G. Brassard, The Quantum Challenge to Structural Complexity Theory, Proceedings of the 7th IEEE Conference on Structure in Complexity Theory, 132-137, 1992.

....models of quantum computing, the quantum Turing machine and quantum circuits were defined by Deutsch[Deu85, Deu89] Yao has shown[Yao93] that these two models have polynomially equivalent computational power when the circuits are uniform. In a sequence of papers oracles have been exhibited [DJ92, BB92, BV97, Sim97] relative to which quantum Turing machines are superpolynomially more powerful than probabilistic ones. These results culminated in the seminal paper of Shor[Sho97] where he gave polynomial time quantum algorithms for the factoring and the discrete logarithm problems. A quantum ....

A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proc. 7th Structure in Complexity Theory, pp. 132--137, 1992.

....The first formal models of quantum computing, the quantum Turing machine and quantum circuits were defined by Deutsch[13, 14] Yao has shown[35] that these two models have polynomially equivalent computational power when the circuits are uniform. In a sequence of papers oracles have been exhibited [16, 7, 9, 34], relative to which quantum Turing machines are more powerful than classical (probabilistic or non deterministic) ones. These results culminated in the seminal paper of Shor[33] where he gave polynomial time quantum algorithms for the factoring and the discrete logarithm problems. A quantum ....

A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proc. 7th Structure in Complexity Theory, pp. 132--137, 1992.

....[24, 45] Finally, several researchers have explored the computational power of quantum Turing Machines. Early work by Deutsch and Jozsa [22] showed how to exploit some inherently quantum mechanical features of QTMs. Their results, in conjunction with subsequent results by Berthiaume and Brassard [12, 13], established the existence of oracles under which there are computational problems that QTMs can solve in polynomial time with certainty, whereas if we require a classical probabilistic Turing machine to produce the correct answer with certainty, then it must take exponential time on some inputs. ....

Berthiaume, A., and Brassard, G., The quantum challenge to structural complexity theory, Proceedings of 7th IEEE Conference on Structure in Complexity Theory, 1992, pt. 132--137.

....# 1 = 0 # 1 = # 0 # 1 6= 0 # 1 6= # 0 SPP # 1 = # 0 # 1 = # 0 2 # 1 6= # 0 # 1 6= # 0 2 BPP # 1 # 0 # 0 # 1 # 1 # 0 # 0 # 1 In this table, is a xed constant with 0 1 and p(n) is a value polynomial in the length n of the input. Berthiaume and Brassard [5] introduced the class HalfP using the name C= P[half] A central notion of this paper is the notion of a Boolean function which is de ned to be a mapping from f0; 1g n to f0; 1g for some integer n 0 which is called its arity. Circuits are a standard way of representing Boolean functions. We ....

....4.6 Any nontrivial absolute (gap, relative) counting property of circuits is UP8 hard with respect to polynomial time Turing reducibility. In particular, a nontrivial absolute (gap, relative) counting problem on circuits is not solvable in polynomial time unless P=UP. Berthiaume and Brassard [5] considered the following class HalfP consisting of all sets L which have a nondeterministic machine M such that x 2 L i exactly the half of the total number of paths are accepting and x = 2 L i no path is accepting. The class HalfP lies between P and the Quantum Computation Class QP [5] The ....

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Andre Berthiaume, Gilles Brassard, The Quantum Challenge to Structural Complexity Theory, Procceedings 7th Annual Conference on Structure in Complexity Theory, 1992, 132-137.

....a possibility was given by Deutsch and Jozsa ( DJ] who presented a simple promise problem that can be solved efficiently without error on Deutsch s quantum computer, but that requires exhaustive search to solve deterministically without error in a classical setting. Brassard and Berthiaume ([BB1]) recast this problem in complexity theoretic terms, and thus constructed an oracle relative to which the quantum computer is exponentially more efficient than the classical (zero error) Turing Machine. In [BB2] they exhibited a similar separation for non deterministic (zero error) Turing ....

....and thus constructed an oracle relative to which the quantum computer is exponentially more efficient than the classical (zero error) Turing Machine. In [BB2] they exhibited a similar separation for non deterministic (zero error) Turing Machines. Unfortunately, the problems explored in [DJ] [BB1] and [BB2] are all efficiently solved by a (classical) probabilistic Turing Machine with exponentially small error probability. However, Bernstein and Vazirani ( BV] constructed an oracle which produces a superpolynomial relativized separation between the quantum and (classical) probabilistic ....

A. Berthiaume and G. Brassard, The Quantum Challenge to Structural Complexity Theory, Proc. 7th IEEE Conference on Structure in Complexity Theory (1992).

....The rst formal models of quantum computing, the quantum Turing machine and quantum circuits were de ned by Deutsch[13, 14] Yao has shown[35] that these two models have polynomially equivalent computational power when the circuits are uniform. In a sequence of papers oracles have been exhibited [16, 7, 9, 34], relative to which quantum Turing machines are more powerful than classical (probabilistic or non deterministic) ones. These results culminated in the seminal paper of Shor[33] where he gave polynomial time quantum algorithms for the factoring and the discrete logarithm problems. A quantum ....

A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proc. 7th Structure in Complexity Theory, pp. 132-137, 1992.

....Church s Thesis. Such a development could lead to eventual practical applications, if and when such quantum computers become buildable. Shor s Factoring Algorithm. However, this idea remained pure speculation until the last several years, when a series of papers on the power of quantum computers [19, 8, 7, 6, 39] culminated in Peter Shor s 1994 proof [37, 38] that a (somewhat idealized) quantum computer could factor large integers in polynomial time. This was an astounding discovery, since mathematicians throughout history have searched for an efficient way to factor numbers without success, since at ....

....function is either constant or unbiased, and we cannot tolerate any non zero probability of failure in determining which one it is. This seems like an unnatural problem. But in any case, following the Deutsch Jozsa paper, analysis of the power of quantum computers developed rapidly with papers [8, 7, 6] that defined various quantum complexity classes and compared them with various classical complexity classes in relativized oracle settings similar to Deutsch and Jozsa s. Quantum operations were also found to have uses in implementing various cryptographic operations; see the end of [8] for a ....

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A. Berthiaume and Gilles Brassard. The quantum challenge to structural complexity theory. In Proceedings of the Seventh Annual Structure in Complexity Theory Conference, pages 132--137, Los Alamitos, CA, 1992. Institute of Electrical and Electronic Engineers Computer Society Press.

....yet[97] One other direction is oracle results in quantum complexity. This direction compares quantum complexity power and classical complexity power when the two models are allowed to have access to an oracle, i.e. a black box which can compute a certain (possibly difficult) function in one step [38, 36, 40, 41]. In fact, the result of Bernstein and Vazirani[38] from 1993 demonstrating a superpolynomial gap between quantum and classical computational comlexity with an access to a certain oracle initialized the sequence of results leading to the Shor s algorithm. An important recent result[23] in quantum ....

Berthiaume A and Brassard G, The quantum challenge to structural complexity theory, in Proc. of the Seventh Annual Structure in Complexity Theory Conference 1992 (IEEE Computer Society Press, Los Alamitos, CA) 132--137

....large integers and compromise a good deal of modern cryptography. While the main research focus has been on finding efficient quantum algorithms for hard problems, attention has also been paid to determining the strength of quantum computation vis a vis its classical (probabilistic) counterpart [7, 5]. In this paper we take a further step in this direction by proving that testing for non zero acceptance probability of a quantum machine is classically an extremely hard problem. In fact, we prove that this problem which we call QAP ( quantum acceptance possibility ) and which is complete for ....

A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proceedings of the 7th IEEE Structure in Complexity Theory Conference, pages 132--137. IEEE, 1992.

....classical counterparts. Undoubtedly the most celebrated of these results are Peter Shor s factoring and discrete logarithm algorithms for quantum computers [23, 24] Other results include Grover s quantum searching algorithm [12] and various oracle results regarding the power of quantum computers [1, 3, 4, 7, 25]. The above examples regard the power of universal quantum machines (e.g. quantum Turing machines [1, 5] quantum circuits [6, 27] quantum cellular automata [8, 16, 17, 26] In this paper, we define two new, much more restricted quantum computational models: 1 way and 2 way quantum finite state ....

A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proceedings of the 7th Annual IEEE Conference on Structure in Complexity, pages 132--137, 1992.

....until now. All main quantum algorithms, including Shor s celebrated factoring algorithm [29] and Grover s searching algorithm [21] use it as a subroutine. All known relativized separation results for quantum computation are based on quantum algorithms that use the discrete Fourier transform [6, 4, 30, 3]. Fundamental Supported in part by the esprit Long Term Research Programme of the EU under project number 20244 (alcom it) Current address: D ept. IRO, Universit e de Montr eal. Email: hoyer IRO.UMontreal.CA. y Department of Mathematics and Computer Science, Odense University, Campusvej 55, ....

Andr'e Berthiaume and Gilles Brassard. The quantum challenge to structural complexity theory. Journal of Modern Optics, 41:2521 -- 2535, 1994.

....by EQP which stands for exact or error free quantum polynomial time (see, for example, 12, 15] 2.2. Oracle Quantum Turing Machines 15 abilistic computation. This problem is called Deutsch Jozsa problem with which we deal in Section 3.2. According to the notation of Berthiaume and Brassard [14, 15], we say that B(X) holds if for each nonnegative integer n either X## n = # or there are exactly 2 n 1 strings in X # # n . Let SX = 1 n : X # # n = # . Then the result of Deutsch and Jozsa [26] implies that SX can be recognized in linear time on a quantum computer if oracle X is ....

....n = # . Then the result of Deutsch and Jozsa [26] implies that SX can be recognized in linear time on a quantum computer if oracle X is available, provided B(X) holds. That is, SX # QP X for all X # # # for which B(X) holds. Building on work of Deutsch and Jozsa [26] Berthiaume and Brassard [14, 15] obtained a relativized separation between P and QP by the construction of an oracle X such that B(X) holds but SX ## P X . Theorem 2.2.1. There exists an oracles relative to which there is a set that can be recognized in worst case linear time by the quantum computer, yet any deterministic ....

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A. Berthiaume and G. Brassard, The quantum challenge to structural complexity theory, Proceedings of the 7th Annual IEEE Conference on Structure in Complexity Theory (Piscataway, NJ), IEEE Computer Society Press, 1992, pp. 132137.

....(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 ....

A. Berthiaume and G. Brassard, The Quantum Challenge to Structural Complexity Theory, Proceedings of the 7th IEEE Conference on Structure in Complexity Theory, 132--137, 1992.

....quantum mechanics describes atoms, quantum chemistry, etc. Modern engineering research into quantum dots as computer units and modern theoretical research into quantum computing, with its exciting potential ability of solving such hard to solve problems as factoring large integers (see, e.g. [2,3,6,7,29,30]) is at this quantization level. ffl From the practical viewpoint, quantum dots will have a huge potential of further miniaturizing computers, so, if this project is successful, we will not need to worry about it for at least a few decades. ffl However, from the fundamental viewpoint of a more ....

Berthiaume, A., and Brassard, G. "The quantum challenge to structural complexity theory", in Proceedings of the 7th IEEE Conference on Structure in Complexity Theory, 1992, pp. 132--137.

....the QTM may be capable of dissipating very small amount of energy per step. ffl Several researchers pointed out the possibilities of faster computations via QTMs as follows. D. Deutsch and R. Jozsa found a problem such that QTM can solve faster than any other classical models of computation [5, 6, 8]. P. W. Shor showed that a QTM can factor integers and find discrete logarithms in polynomial time with a bounded probability of error [16] What are major characteristics of computations performed by QTMs Let us consider a simple example. Let f : f0; 1g n f0; 1g be a Boolean function ....

A. Berthiaume and G. Brassard, "The Quantum Challenge to Structural Complexity Theory", in Proc. 7th IEEE Conference on Structure in Complexity Theory(1992), pp. 132-137.

....framework, we generalize it, and we give an exact quantum polynomial time algorithm to solve it. This provides the first evidence of an exponential gap between the power of exact quantum computation and that of classical 1 The Deutsch Jozsa problem gives rise to an oracle decision problem [7, 8]. Also, in the soon to be published journal version of their paper, Bernstein and Vazirani extend their result to a decision problem [5] bounded error probabilistic computation, even for decision problems. Of independent interest are the techniques developed to obtain our results. Two of the ....

A. Berthiaume and G. Brassard, "The quantum challenge to structural complexity theory", Proceedings of the 7th Annual IEEE Structure in Complexity Theory Conference, 1992, pp. 132 -- 137.

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A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory.InProceedings of the 7th IEEE Structure in Complexity Theory Conference, pages 132--137. IEEE, 1992.

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Andr`e Berthiaume and Gilles Brassard. The quantum challenge to structural complexity theory. In Proceedings of the 7th IEEE Structurein Complexity Theory Conference, pages 132--137. IEEE, 1992.

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A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proceedings of the 7th IEEE Structure in Complexity Theory Conference, pages 132--137. IEEE, 1992.

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A. Berthiaume and G. Brassard, "The quantum challenge to structural complexity theory," Proceedings of the Seventh IEEE Conference on Structure in Complexity Theory, IEEE, pp. 132--137, 1992.

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Berthiaume A and Brassard G, The quantum challenge to structural complexity theory, in Proc. of the Seventh Annual Structure in Complexity Theory Conference 1992 (IEEE Computer Society Press, Los Alamitos, CA) 132--137

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A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Osamu Barrington, David Mix; Brassard, Gilles; Hemachandra, Lane; Leivant, Daniel; Chair, Tim Long; Nisan, Noam; Royer, James; Watanabe, editor, Proceedings of the 7th Annual Conference on Structure in Complexity Theory (SCTC '92), pages 132-137, Boston, MA, USA, June 1992. IEEE Computer Society Press.

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A. Berthiaume and G. Brassard. The quantum challenge to structural complexity theory. In Proc. 7th Structure in Complexity Theory, pp. 132--137, 1992.

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A. Berthiaume and G. Brassard (1992a) "The quantum challenge to structural complexity theory," in Proceedings of the Seventh Annual Structure in Complexity Theory Conference, IEEE Computer Society Press, Los Alamitos, CA, pp. 132--137.

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