L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26:1524--1540,1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... secure quantum key distribution [4, 27] and exponentially more efficient quantum than classical communication complexity protocols [24] Equally important for understanding the power of quantum models are upper bounds and impossibility proofs, such as the containment of BQP in PP [1, 8], the impossibility of quantum bit commitment [20] and the existence of oracles relative to which quantum computers have limited power [3, 8] In this paper we consider the potential advantages of quantum variants of zero knowledge proof systems. Zeroknowledge proof systems were first defined ....

L. Adleman, J. Demarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524--1540, 1997.

....yield inequivalent notions in the 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 ....

....in the nullspace of A that assign values (1; 0) and (0; 1) to the labels 1 and 2, respectively, is a hyperplane of dimension d 2, and each vector in this hyperplane can be extended to yield at most (p l) distinct g 2 F 1 with g y;z f . To prove (6) let us define H a = fh 2 F j Ah=0; h[1]= 1; 0) and h[2] a; 0)g for each a 2 f2; p 1g, and define T = fh 2 F j h[i] 6= h[j] for i 6= jg: We will prove that there are at least p 1 values of a for which H a T contains at least p 2d 4 l elements. As each h 2 H a T may be extended to yield (p l) distinct g ....

L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524--1540, 1997.

....of BQP which remains unscathed under variations of the model such as restricting to a small set of rational amplitudes, allowing quantum subroutines and a single measurement at the end of the computation. Bernstein and Vazirani show that BQP is contained in PSPACE. Adleman, DeMarrais and Huang [ADH97] show that BQP is contained in the counting class PP. Bennett, Bernstein, 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 ....

L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, 1997.

....yield inequivalent notions in the 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 [21] have de ned QNP to be 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 [21, 22] This notion ....

L. Adleman, J. Demarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, 1997.

....no. The physics de nition allows arbitrary real and complex entries as long as the matrices are unitary. This by itself limits the values to have 6 Fortnow absolute value at most one. But one can do much better. If we allow all possible reals, BQP can accept arbitrarily complicated languages [1]. However this result feels like cheating basically you encode hard languages directly into the entries of the matrix. Thus one requires knowing the language ahead of time to create the machine. A similar trick can also be played with probabilistic machines using noncomputable probabilities. For ....

....time to create the machine. A similar trick can also be played with probabilistic machines using noncomputable probabilities. For fairness we should only allow eciently computable matrix entries, where we can compute the ith bit in time polynomial in i. Independently Adleman, DeMarrais and Huang [1] and Solovay and Yao [26] show that we can simulate a BQP machine using eciently computable entries from the set f1; 4 5 ; 3 5 ; 0; 3 5 ; 4 5 ; 1g. Or you can get away with fewer numbers if you don t mind an irrational: f1; p 2; 0; p 2; 1g. 3.4 Don t we have to require the ....

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L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, 1997.

....the Turing machine with a quantum Turing machine yields the class BQP. We show that BQP is contained in the counting class AWPP. Based on previous results about the class AWPP [Li93] we can show that BQP is low for PP and so improve the upper bound given by Adleman, DeMarrais and Huang [ADH97] We can also use oracle results about AWPP to get a relativized world where P = BQP but the polynomial time hierarchy is innite. We also use these techniques to give a relativized world where BQP does not have complete sets. We know that BPP BQP. An important open question is whether or not ....

....to a distribution on outputs with nonnegative probabilities that add up to one. A quantum Turing machine s transition function maps to a superposition of the outputs where each output gets an amplitude which may be a complex value. In the case of BQP as dened below, Adleman, DeMarrais and Huang [ADH97] and Solovay and Yao [SY96] show that we can assume these amplitudes take one of the values in f Gamma1; Gamma 4 5 ; Gamma 3 5 ; 0; 3 5 ; 4 5 ; 1g. Bennett, Bernstein, Brassard and Vazirani [BBBV97] show that we can assume the quantum Turing machine has a single accepting ....

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L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):15241540, 1997.

....are computed within amplitude precision bounded by an inverse polynomial in the input size. Most of the algorithms we will mention (such as Shor s) are in the class BQP. BV93, BV97] showed that BQP computations can be done using unitary operations with a fixed irrational rotation. Adleman et al. [ADH97] improved this to show that BQP can be computed using only unitary operations with rational rotations, and that BQP is in the class PSPACE of polynomial space computations of (classical) TMs. ffl Quantum Gates. A set of Boolean gates are universal if any Boolean operation on arbitrarily many ....

L. M. Adleman and J. Demarrais and M.-D. A. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, (October 1997).

....probability iff f(x) 1. An example is given by two competing definitions of quantum NP : Kitaev [Kit99] defines this class as the set of languages which are accepted by polynomial time quantum algorithms that are given a polynomial size quantum certificate. On the other hand, Adleman et.al. ADH97] and Fenner et.al. FGHP98] define quantum NP as the set of languages L for which there is a polynomial time quantum algorithm whose acceptance probability is positive iff x 2 L. This quantum class was shown equal to the classical counting class co C =P in [FGHP98] using tools from [FR99] We ....

L. M. Adleman, J. Demarrais, and M. A. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524--1540, 1997.

....mapping V : Sigma Theta f1; kg Sigma (for k = bm=2 1c) where each V (x; j) is an encoding of a quantum circuit composed of quantum gates from some appropriately chosen universal set of gates. Universal sets of gates transformations have been investigated in a number of papers [1, 7, 8, 14, 15]; for the purposes of this paper, we will assume only that this set includes the Hadamard gate and any universal gate for reversible computation such as the Fredkin gate or Toffoli gate. Each encoding V (x; j) is identified with the quantum circuit it encodes. It is assumed that this encoding is ....

L. Adleman, J. Demarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524--1540, 1997.

....in general whether a given quantum basis state appears with nonzero amplitude in a superposition, or whether a given quantum bit has positive expectation value at the end of a quantum computation. This result is achieved by showing that the complexity class NQP of Adleman, Demarrais, and Huang [1], a quantum analog of NP, is equal to the counting class coC=P. 1 Introduction This decade has seen renewed interest and great activity in quantum computing. This interest has been spurred by the clear formal definition of Computer Science Department, University of South Carolina, Columbia, SC ....

....the set of transition amplitudes we allow in our model of quantum computation. The equation holds whether we allow arbitrary algebraic numbers as transition amplitudes (Theorem 4. 1) or we restrict transition amplitudes to be in a small finite set of rational numbers as described by Adleman, et al. [1] (Theorem A.1) We will assume throughout the paper that transcendental amplitudes are not allowed. The class NQP was originally defined by Adleman, Demarrais, and Huang [1] who showed that NQP PP. The sharper upper bound NQP coC= P is implicit in their proof and a recent result of Fortnow ....

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L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26:1524--1540, 1997.

....function maps to a distribution on outputs with nonnegative probabilities that add up to one. A quantum Turing machine s transition function maps to a superposition of the outputs where each output gets an amplitude which may be a complex value. To simplify matters, Adleman, DeMarrais and Huang [ADH97] and Solovay and Yao [SY96] show that we can assume these amplitudes take one of the values in the set f Gamma1; Gamma 4 5 ; Gamma 3 5 ; 0; 3 5 ; 4 5 ; 1g. The quantum Turing machines we consider here all run in polynomial time and thus have an exponential number of possible ....

....path of M A (x) is not enough to eoeect the sign of the dioeerence of the number of accepting and rejecting paths of M 0 . From Theorem 3.2 we get the same result for BQP. Corollary 3.3 BQP is low for PP. This improves and simpli es the bound given by Adleman, DeMarrais and Huang [ADH97]. Corollary 3.4 (Adleman DeMarrais Huang) BQP PP P #P PSPACE We also have a class containing BQP that is not known to contain NP as Beigel [Bei94] has a relativized world where NP is not low for PP. Fenner, Fortnow, Kurtz and Li [FFKL93] give an interesting collapse for AWPP relative ....

L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):15241540, 1997.

....insure that the machine is well formed (see below) It is known that the power of QTMs depends greatly upon the values which the transition function may take; in the absence of any restrictions, it is possible to encode a great deal of information in these values. For example, it is shown in [1] that QTMs can recognize non recursive sets in polynomial time, logarithmic space and with bounded probability of error if allowed to have arbitrary transcendental transition amplitudes. Thus we must place some restriction on these values in order to avoid this problem, and so we will insist that ....

....transcendental transition amplitudes. Thus we must place some restriction on these values in order to avoid this problem, and so we will insist that all transition functions of QTMs take only rational values. Although some quantum algorithms use algebraic transition amplitudes, it is shown in [1] that, for the case of bounded error polynomial time, machines with algebraic amplitudes are equivalent in power to ones with rational amplitudes. It is an open question not addressed in this paper whether QTMs with algebraic amplitudes are equivalent in power to ones with rational amplitudes in ....

L. Adleman, J. Demarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5): 1524--1540, 1997.

....use this relationship to obtain new limitations on the complexity of quantum computing. We show that BQP is contained in the counting class AWPP. Based on a previous result about AWPP [Li93] we can show that BQP is low for PP and so improve the upper bound given by Adleman, DeMarrais and Huang [ADH97] We can also use oracle results about AWPP to get a relativized world where P = BQP but the polynomial time hierarchy is infinite. We also use these techniques to give a relativized world where BQP does not have complete sets. We know that BPP BQP. An important open question is whether or not ....

....function maps to a distribution on outputs with nonnegative probabilities that add up to one. A quantum Turing machine s transition function maps to a superposition of the outputs where each output gets an amplitude which may be a complex value. To simplify matters, Adleman, DeMarrais and Huang [ADH97] and Solovay and Yao [SY96] show that we can assume these amplitudes take one of the values in the set f Gamma1; Gamma 4 5 ; Gamma 3 5 ; 0; 3 5 ; 4 5 ; 1g. Bennet, Bernstein, Brassard and Vazirani [BBBV97] show that we can assume the quantum Turing machine has a single accepting ....

[Article contains additional citation context not shown here]

L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524--1540, 1997.

....are computed within amplitude precision bounded by an inverse polynomial in the input size. Most of the algorithms we will mention (such as Shor s) are in the class BQP. BV93, BV97] showed that BQP computations can be done using unitary operations with a fixed irrational rotation. Adleman et al. [ADH97] improved this to show that BQP can be computed using only unitary operations with rational rotations, and that BQP is in the class PSPACE of polynomial space computations of (classical) TMs. 7.3 Quantum Parallel Complexity Classes Let NC (QNC, respectively) be the class of (quantum, ....

L. M. Adleman and J. Demarrais and M.-D. A. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, (October 1997).

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L. M. Adleman, J. Demarrais, and M.-D. Huang. Quantum computability. SIAM Journal on Computing, 26:1524--1540, 1997.

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L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26:1524--1540,1997.

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L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26:1524--1540, 1997.

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L. Adleman, J. DeMarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, 1997.

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Adleman, L. M., Demarrais, J., and Huang, M.-D. A. Quantum computability. SIAM Journal on Computing 26, 5 (Oct. 1997), 1524--1540.

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L. Adleman, J. Demarrais, and M. Huang. Quantum computability. SIAM Journal on Computing, 26(5):1524-1540, 1997.

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