The usual representation of quantum algorithms, limited to the process of solving the problem, is physically incomplete. We complete it in three steps: (i) extending the representation to the process of setting the pr...
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The usual representation of quantum algorithms, limited to the process of solving the problem, is physically incomplete. We complete it in three steps: (i) extending the representation to the process of setting the problem, (ii) relativizing the extended representation to the problem solver to whom the problem setting must be concealed, and (iii) symmetrizing the relativized representation for time reversal to represent the reversibility of the underlying physical process. The third steps projects the input state of the representation, where the problem solver is completely ignorant of the setting and thus the solution of the problem, on one where she knows half solution (half of the information specifying it when the solution is an unstructured bit string). Completing the physical representation shows that the number of computation steps (oracle queries) required to solve any oracle problem in an optimal quantum way should be that of a classical algorithm endowed with the advanced knowledge of half solution.
Two quantum algorithms of finding the roots of a polynomial function f(x) = x (m) + a (m- 1) x (m- 1) + ... + a (1) x + a (0) are discussed by using the Bernstein-Vazirani algorithm. One algorithm is presented in the ...
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Two quantum algorithms of finding the roots of a polynomial function f(x) = x (m) + a (m- 1) x (m- 1) + ... + a (1) x + a (0) are discussed by using the Bernstein-Vazirani algorithm. One algorithm is presented in the modulo 2. The other algorithm is presented in the modulo d. Here all the roots are in the integers Z. The speed of solving the problem is shown to outperform the best classical case by a factor of m in both cases.
Energy-efficient routing in wireless sensor networks has attracted attention from researchers in both academia and industry, most recently motivated by the opportunity to use software-defined network-inspired approach...
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Energy-efficient routing in wireless sensor networks has attracted attention from researchers in both academia and industry, most recently motivated by the opportunity to use software-defined network-inspired approaches. These problems are NP-hard, with algorithms needing computation time that scales faster than polynomials in the problem size. Consequently, heuristic algorithms are used in practice, which are unable to guarantee optimally. In this article, we show proof-of-principle for the use of a quantum annealing processor instead of a classical processor, to find optimal or nearly optimal solutions very quickly. Our preliminary results for small networks show that this approach using quantum computing has great promise and may open the door for other significant improvements in the efficacy of network algorithms.
We use the Clifford algebra technique (J. Math. Phys. 43: 5782, 2002;J. Math. Phys. 44: 4817, 2003), that is nilpotents and projectors which are binomials of the Clifford algebra objects gamma(a) with the property {ga...
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We use the Clifford algebra technique (J. Math. Phys. 43: 5782, 2002;J. Math. Phys. 44: 4817, 2003), that is nilpotents and projectors which are binomials of the Clifford algebra objects gamma(a) with the property {gamma(a), gamma(b)}(+) = 2 eta(ab), for representing quantum gates and quantum algorithms needed in quantum computers in a simple and an elegant way. We identify n-qubits with the spinor representations of the group SO(1, 3) for a system of n spinors. Representations are expressed in terms of products of projectors and nilpotents;we pay attention also on the nonrelativistic limit. An algorithm for extracting a particular information out of a general superposition of 2(n) qubit states is presented. It reproduces for a particular choice of the initial state the Grover's algorithm (Proc. 28th Annual ACM Symp. Theory Comput. 212, 1996).
It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity O( ro...
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It has recently been shown that starting with a classical query algorithm (decision tree) and a guessing algorithm that tries to predict the query answers, we can design a quantum algorithm with query complexity O( root GT) where T is the query complexity of the classical algorithm (depth of the decision tree) and G is the maximum number of wrong answers by the guessing algorithm [3, 14]. In this article, we show that, given some constraints on the classical algorithms, this quantum algorithm can be implemented in time O-similar to (root GT). Our algorithm is based on non-binary span programs and their efficient implementation. We conclude that various graph-theoretic problems including bipartiteness, cycle detection, and topological sort can be solved in time O(n(3/2) log(2) n) and with O(n(3/2) ) quantum queries. Moreover, finding a maximal matching can be solved with O(n(3/2)) quantum queries in time O(n(3/2) log(2) n), and maximum bipartite matching can be solved in time O(n(2) log(2) n).
An interpretation of quantum mechanics is discussed. It is assumed that quantum is energy. An algorithm by means of the energy interpretation is discussed. An algorithm, based on the energy interpretation, for fast de...
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An interpretation of quantum mechanics is discussed. It is assumed that quantum is energy. An algorithm by means of the energy interpretation is discussed. An algorithm, based on the energy interpretation, for fast determining a homogeneous linear function f(x) := s.x = s (1) x (1) + s (2) x (2) + ai + s (N) x (N) is proposed. Here x = (x (1), aEuro broken vertical bar , x (N) ), x (j) a R and the coefficients s = (s (1), aEuro broken vertical bar , s (N) ), s (j) a N. Given the interpolation values , the unknown coefficients of the linear function shall be determined, simultaneously. The speed of determining the values is shown to outperform the classical case by a factor of N. Our method is based on the generalized Bernstein-Vazirani algorithm to qudit systems. Next, by using M parallel quantum systems, M homogeneous linear functions are determined, simultaneously. The speed of obtaining the set of M homogeneous linear functions is shown to outperform the classical case by a factor of N x M.
A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is, how big is the advantage of exact quantum algorithms over their classical counterpar...
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A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is, how big is the advantage of exact quantum algorithms over their classical counterparts: deterministic algorithms? We present the first example of a total Boolean function f(x(1),..., x(N)) for which exact quantum algorithms have superlinear advantage over deterministic algorithms. Any deterministic algorithm that computes our function must use N queries but an exact quantum algorithm can compute it with O(N-0.8675...) queries. A modification of our function gives a similar result for communication complexity: there is a function f which can be computed by an exact quantum protocol that communicates O(N-0.8675... log N) quantum bits but requires Omega(N) bits of communication for classical protocols.
Certain aspects of some unitary quantum systems are well described by evolution via a non-Hermitian effective Hamiltonian, as in the Wigner-Weisskopf theory for spontaneous decay. Conversely, any non-Hermitian Hamilto...
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Certain aspects of some unitary quantum systems are well described by evolution via a non-Hermitian effective Hamiltonian, as in the Wigner-Weisskopf theory for spontaneous decay. Conversely, any non-Hermitian Hamiltonian evolution can be accommodated in a corresponding unitary system + environment model via a generalization of Wigner-Weisskopf theory. This demonstrates the physical relevance of novel features such as exceptional points in quantum dynamics, and opens up avenues for studying many-body systems in the complex plane of coupling constants. In the case of lattice field theory, sparsity lends these channels the promise of efficient simulation on standardized quantum hardware. We thus consider quantum operations that correspond to Suzuki-Lie-Trotter approximation of lattice field theories undergoing nonunitary time evolution, with potential applicability to studies of spin or gauge models at finite chemical potential, with topological terms, to quantum phase transitions—a range of models with sign problems. We develop non-Hermitian quantum circuits and explore their promise on a benchmark, the quantum one-dimensional Ising model with complex longitudinal magnetic field, showing that observables can probe the Lee-Yang edge singularity. The development of attractors past critical points in the space of complex couplings indicates a potential for study on near-term noisy hardware.
We present two general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Both methods are based on the adversary method of Ambainis. We show that they yield optimal lower boun...
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We present two general methods for proving lower bounds on the query complexity of nonadaptive quantum algorithms. Both methods are based on the adversary method of Ambainis. We show that they yield optimal lower bounds for several natural problems, and we challenge the reader to determine the nonadaptive quantum query complexity of the "1-to-1 versus 2-to-1" problem and of Hidden Translation. In addition to the results presented at Wollic 2008 in the conference version of this paper, we show that the lower bound given by the second method is always at least as good (and sometimes better) as the lower bound given by the first method. We also compare these two quantum lower bounds to probabilistic lower bounds. (C) 2009 Elsevier Inc. All rights reserved.
The computation of classical Ising partition functions, coming from statistical physics, is a natural generalization of binary optimization. This is a notoriously hard problem in general, which makes it an especially ...
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ISBN:
(纸本)9798400717987
The computation of classical Ising partition functions, coming from statistical physics, is a natural generalization of binary optimization. This is a notoriously hard problem in general, which makes it an especially interesting task to consider in the search for practical quantum advantage in near term quantum computers. In this work we view classical Ising models (on certain graphs) as quantum imaginary time evolution, which is enabled by the use of the transfer matrix mapping. We study this mapping from two points of view: (1) following Onsager and Kaufman's original solution of the 2D Ising model, which serves as a starting point, we consider more general models and the possibility of a similar Lie-theoretic solution;(2) we consider quantum algorithms for the computation of partition functions and thermal averages via transfer matrices, which can be implemented either with block encodings inside larger unitaries or by approximating the state trajectories with unitary operators.
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