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.
Symmetries in a Hamiltonian play an important role in quantum physics because they correspond directly with conserved quantities of the related system. In this Letter, we propose quantum algorithms capable of testing ...
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Symmetries in a Hamiltonian play an important role in quantum physics because they correspond directly with conserved quantities of the related system. In this Letter, we propose quantum algorithms capable of testing whether a Hamiltonian exhibits symmetry with respect to a group. We demonstrate that familiar expressions of Hamiltonian symmetry in quantum mechanics correspond directly with the acceptance probabilities of our algorithms. We execute one of our symmetry-testing algorithms on existing quantum computers for simple examples of both symmetric and asymmetric cases.
The inherent parallelism of quantum systems determined not only the investigation of innovative applications that, can be developed using these high performance computing systems, but. also of ways to improve the perf...
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ISBN:
(纸本)9781605584133
The inherent parallelism of quantum systems determined not only the investigation of innovative applications that, can be developed using these high performance computing systems, but. also of ways to improve the performances over, the classical case. Exploiting this parallelism recently led to the emergence. of innovative ideas in the field of computer graphics, sketching the development of quantum rendering and quantum computational geometry. Following these tracks, we propose a new quantum algorithm) for the RANdom SAmple, Consensus (RANSAC) voting scheme. In this paper we. show that, by exploiting the unique features of quantum computing, generating uniform superpositions of states in the problem space and applying quantum operators to all states simultaneously, the performance of our quantum algorithm is orders of magnitude faster than the classical variant.
We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state |x⟩ that is proportional to the solution of the system of linear equation...
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We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state |x⟩ that is proportional to the solution of the system of linear equations Ax→=b→. The time complexities of our algorithms are O(κ2log(κ)/ε) and O(κlog(κ)/ε), where κ is the condition number of A and ε is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of A, the projector onto the initial state |b⟩, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of κ. Like previous methods, our techniques yield an exponential quantum speed-up under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.
In this paper, we demonstrate that the logic computation performed by the DNA-based algorithm for solving general cases of the satisfiability problem can be implemented more efficiently by our proposed quantum algorit...
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In this paper, we demonstrate that the logic computation performed by the DNA-based algorithm for solving general cases of the satisfiability problem can be implemented more efficiently by our proposed quantum algorithm on the quantum machine proposed by Deutsch. To test our theory, we carry out a three-quantum bit nuclear magnetic resonance experiment for solving the simplest satisfiability problem.
We present quantum algorithms for performing nearest-neighbor learning and k-means clustering. At the core of our algorithms are fast and coherent quantum methods for computing the Euclidean distance both directly and...
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We present quantum algorithms for performing nearest-neighbor learning and k-means clustering. At the core of our algorithms are fast and coherent quantum methods for computing the Euclidean distance both directly and via the inner product which we couple with methods for performing amplitude estimation that do not require measurement. We prove upper bounds on the number of queries to the input data required to compute such distances and find the nearest vector to a given test example. In the worst case, our quantum algorithms lead to polynomial reductions in query complexity relative to Monte Carlo algorithms. We also study the performance of our quantum nearest-neighbor algorithms on several real-world binary classification tasks and find that the classification accuracy is competitive with classical methods.
Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering and underlies the solution of time-homogeneous linear ordinary differe...
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Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering and underlies the solution of time-homogeneous linear ordinary differential equations, simulation of discrete-time Markov chains, or discovering the spectral properties of matrices with iterative methods. In this paper, we investigate the possibility of speeding up matrix powering of sparse stable Hermitian matrices on a quantum computer. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms: (i) a fast-forwarding algorithm that builds on construction of Apers and Sarlette [quantum Inf. Comput. 19, 181 (2019)] and (ii) an algorithm based on Hamiltonian simulation. Furthermore, by mapping the N-bit parity determination problem to a matrix powering problem, we provide no-go theorems that limit the quantum speedups achievable in powering non-Hermitian matrices.
The von Neumann and quantum Rényi entropies characterize fundamental properties of quantum systems and lead to many theoretical and practical applications. quantum algorithms using a purified quantum query model ...
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The von Neumann and quantum Rényi entropies characterize fundamental properties of quantum systems and lead to many theoretical and practical applications. quantum algorithms using a purified quantum query model can speed up quantum entropy estimation, while little is known about the complexity of using identical copies of the quantum state. This paper presents quantum entropy estimation algorithms with a cost of copies scaling polynomially in the rank of the state. In contrast to current methods that depend on the dimension of the system, our methods could provide exponential resource savings in the scenario of low-rank states. Furthermore, we show how to construct quantum circuits using primitive single-qubit or two-qubit gates efficiently and thus provide practical methods for estimating quantum entropies of quantum systems. We also conduct simulation experiments to show the effectiveness and noise robustness of our algorithms.
We have developed a visual syntax for representing concepts that are contingent on temporal properties (time-dependent semantics). A within-group (N=24) experiment was conducted to identify the representations that co...
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ISBN:
(纸本)0769523978
We have developed a visual syntax for representing concepts that are contingent on temporal properties (time-dependent semantics). A within-group (N=24) experiment was conducted to identify the representations that conveyed best a given semantic. We then applied our representations to the visualization of algorithms in quantum computing and carried out a second experiment (N=16) on subjects unfamiliar with the semantic concepts that were tested. The results show that our representations are intuitive and facilitate a good level of understanding of the algorithms.
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