The Parallel Quantum Algorithm for the Class of Optimization

2020 ◽  
Vol 30 (04) ◽  
pp. 2050014
Author(s):  
Guanlei Xu ◽  
Xiaogang Xu
Keyword(s):  
Prior Information ◽  
Quantum Algorithm ◽  
Quantum Measurements ◽  
Classical Solutions ◽  
Minimum Norm ◽  
Quantum Operations ◽  
Parallel Circuits ◽  
Successful Probability ◽  
The Given ◽  
Np Problem

For the given n numbers without any other prior information, how to obtain the minimum norm of them only by assigning their signs before them? Moreover, how to know one number is the multiplication of which ones in the given n numbers? In classical solutions, enumeration is the only way via trying one by one, whose complexity is about [Formula: see text] and this is a NP problem. In this paper, the parallel quantum algorithm is proposed to solve the two questions shown in above. Through the quantum design of linear expressions of angles in parallel circuits, only [Formula: see text] time’s quantum operations and about [Formula: see text] times’ quantum measurements in the average will give the correct answer in the successful probability of 0.97 instead of the traditional [Formula: see text] times. The example and theoretical analysis demonstrate the efficiency of the proposed method.

Functions of Minimal Norm with the Given Set of Fourier Coefficients

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 651
Author(s):  
Pyotr Ivanshin
Keyword(s):  
Fourier Series ◽  
Existence And Uniqueness ◽  
Fourier Coefficients ◽  
Nonnegative Function ◽  
Trigonometric Fourier Series ◽  
Minimum Norm ◽  
Minimal Norm ◽  
Uniqueness Of The Solution ◽  
The Given

We prove the existence and uniqueness of the solution of the problem of the minimum norm function ∥ · ∥ ∞ with a given set of initial coefficients of the trigonometric Fourier series c j , j = 0 , 1 , … , 2 n . Then, we prove the existence and uniqueness of the solution of the nonnegative function problem with a given set of coefficients of the trigonometric Fourier series c j , j = 1 , … , 2 n for the norm ∥ · ∥ 1 .

scholarly journals NP Problem in Quantum Algorithm

2008 ◽  
pp. 201-207 ◽  
Author(s):  
Masanori Ohya ◽  
Natsuki Masuda
Keyword(s):  
Quantum Algorithm ◽  
Np Problem

scholarly journals On the Brukner–Zeilinger approach to information in quantum measurements

2015 ◽  
Vol 471 (2183) ◽  
pp. 20150435 ◽  
Author(s):  
A. E. Rastegin
Keyword(s):  
Quantum Theory ◽  
Invariant Measure ◽  
Critical Points ◽  
Natural Extension ◽  
Quantum Measurements ◽  
Quantum Operations ◽  
Total Information ◽  
Complete Set

We address the problem of properly quantifying information in quantum theory. Brukner and Zeilinger proposed the concept of an operationally invariant measure based on measurement statistics. Their measure of information is calculated with probabilities generated in a complete set of mutually complementary observations. This approach was later criticized for several reasons. We show that some critical points can be overcome by means of natural extension or reformulation of the Brukner–Zeilinger approach. In particular, this approach is connected with symmetric informationally complete measurements. The ‘total information’ of Brukner and Zeilinger can further be treated in the context of mutually unbiased measurements as well as general symmetric informationally complete measurements. The Brukner–Zeilinger measure of information is also examined in the case of detection inefficiencies. It is shown to be decreasing under the action of bistochastic maps. The Brukner–Zeilinger total information can be used for estimating the map norm of quantum operations.

scholarly journals Efficiency at optimal work from finite source and sink: A probabilistic perspective

2015 ◽  
Vol 40 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Ramandeep S. Johal
Keyword(s):  
Prior Information ◽  
Present Formulation ◽  
Weighted Mean ◽  
Initial State ◽  
Standard Analysis ◽  
Valid Inference ◽  
Maximum Work ◽  
The Mean ◽  
Expected Values ◽  
The Given

AbstractWe revisit the classic thermodynamic problem of maximum work extraction from arbitrary-sized source of heat and sink, modelled as perfect gases. For a given initial state of the process, we assume ignorance of the final temperatures. We quantify the prior information about the process and assign a prior distribution to the unknown temperature(s). This requires that we also take into account the temperature values which are not regarded in standard analysis. In the present formulation, however, such values appear to be consistent with the given prior information and hence are included here in the inference. We derive estimates of the efficiency at optimal work from the expected values of the final temperatures, and show that these match with the exact expressions in the limit when any one of the systems is very large compared to the other. For other relative sizes of the source and the sink, a weighted mean is defined over the estimates from two valid inference procedures, that generalizes the procedure suggested earlier in [J. Phys. A: Math. Theor. 46 (2013), 365002]. The mean estimate for efficiency obtained in this way agrees with the results of the optimal performance quite accurately.

scholarly journals REVERSIBILITY CONDITIONS FOR QUANTUM OPERATIONS

2012 ◽  
Vol 24 (07) ◽  
pp. 1250016 ◽  
Author(s):  
ANNA JENČOVÁ
Keyword(s):  
Large Class ◽  
Quantum States ◽  
Quantum Operations ◽  
The Given ◽  
Equivalent Conditions ◽  
Nikodym Derivative

We give a list of equivalent conditions for reversibility of the adjoint of a unital Schwarz map, with respect to a set of quantum states. A large class of such conditions is given by preservation of distinguishability measures: F-divergences, L1-distance, quantum Chernoff and Hoeffding distances. Here we summarize and extend the known results. Moreover, we prove a number of conditions in terms of the properties of a quantum Radon–Nikodym derivative and factorization of states in the given set. Finally, we show that reversibility is equivalent to preservation of a large class of quantum Fisher informations and χ2-divergences.

scholarly journals AN OBSERVER-BASED DE-QUANTISATION OF DEUTSCH'S ALGORITHM

2011 ◽  
Vol 22 (01) ◽  
pp. 191-201
Author(s):  
CRISTIAN S. CALUDE ◽  
MATTEO CAVALIERE ◽  
RADU MARDARE
Keyword(s):  
Quantum Computing ◽  
Quantum Measurement ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Classical Solutions ◽  
Finite State Automata ◽  
Computational Problem ◽  
Classical Simulation ◽  
Finite State ◽  
Classical Computing

Deutsch's problem is the simplest and most frequently examined example of computational problem used to demonstrate the superiority of quantum computing over classical computing. Deutsch's quantum algorithm has been claimed to be faster than any classical ones solving the same problem, only to be discovered later that this was not the case. Various de-quantised solutions for Deutsch's quantum algorithm—classical solutions which are as efficient as the quantum one—have been proposed in the literature. These solutions are based on the possibility of classically simulating "superpositions", a key ingredient of quantum algorithms, in particular, Deutsch's algorithm. The de-quantisation proposed in this note is based on a classical simulation of the quantum measurement achieved with a model of observed system. As in some previous de-quantisations of Deutsch's quantum algorithm, the resulting de-quantised algorithm is deterministic. Finally, we classify observers (as finite state automata) that can solve the problem in terms of their "observational power".

scholarly journals Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 481
Author(s):  
Dong An ◽  
Noah Linden ◽  
Jin-Peng Liu ◽  
Ashley Montanaro ◽  
Changpeng Shao ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Monte Carlo Methods ◽  
Quantum Algorithms ◽  
General Setting ◽  
Quantum Algorithm ◽  
Mathematical Finance ◽  
Classical Solutions ◽  
Multilevel Monte Carlo

Inspired by recent progress in quantum algorithms for ordinary and partial differential equations, we study quantum algorithms for stochastic differential equations (SDEs). Firstly we provide a quantum algorithm that gives a quadratic speed-up for multilevel Monte Carlo methods in a general setting. As applications, we apply it to compute expectation values determined by classical solutions of SDEs, with improved dependence on precision. We demonstrate the use of this algorithm in a variety of applications arising in mathematical finance, such as the Black-Scholes and Local Volatility models, and Greeks. We also provide a quantum algorithm based on sublinear binomial sampling for the binomial option pricing model with the same improvement.

A numerical method of bicharacteristics For quasi-linear partial functional Differential equations

2008 ◽  
Vol 8 (1) ◽  
pp. 21-38
Author(s):  
P. ARLUKOWICZ ◽  
W. CZERNOUS
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Approximate Solutions ◽  
Classical Solutions ◽  
Difference Problem ◽  
Partial Functional Differential Equations ◽  
Solution Of Equations ◽  
Integral Systems ◽  
Functional Differential ◽  
The Given

Abstract Classical solutions of mixed problems for first order partial functional differential equations in several independent variables are approximated by solutions of an Euler-type difference problem. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. The convergence of explicit difference schemes is proved by means of consistency and stability arguments. It is assumed that the given functions satisfy the nonlinear estimates of the Perron type. Differential systems with deviated variables and differential integral systems can be obtained from the general model by specializing the given operators.

Multiscale Image Segmentation Using Energy Minimization

2013 ◽  
Vol 662 ◽  
pp. 940-943
Author(s):  
Yin Hui Zhang ◽  
Sen Wang ◽  
Zhong Hai Shi ◽  
Zi Fen He
Keyword(s):  
Image Segmentation ◽  
Energy Minimization ◽  
Prior Information ◽  
Hierarchical Modeling ◽  
Image Data ◽  
Inference Problem ◽  
Posterior Inference ◽  
The Given ◽  
Multiscale Image Segmentation

We consider the role of multiscale prior information of the object in the form of a Bayesian framework to address the posterior inference problem. The multiscale prior is implicitly estimated from the given image. We show how the multiscale prior effectively exploits the available image data for hierarchical modeling and exploiting posterior inference scheme to determine the posterior likelihood at each iteration with definite number of iteration steps. Extensive experiments show that this method achieves robust multiscale image segmentation results in the presence of dynamic Gaussian noises.

scholarly journals Algorithmic Manifold and Application to P versus NP Problem

2016 ◽  
Author(s):  
Takuya Yabu
Keyword(s):  
Time Complexity ◽  
Reduction Method ◽  
Recent Literature ◽  
Circuit Complexity ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Geometric Method ◽  
Diagonal Argument ◽  
Long Time ◽  
Np Problem

About P versus NP problem, it has been studied for long time. Recent literature has shown that the existing proof method using the diagonal argument or the circuit complexity is not effective. On the other hand, as another approach, calculation of time complexity based on the geometric method is also performed, but it is limited to the quantum algorithm, and it is an application example to the existing method of lower band derivation of quantum circuit complexity, it is essentially unchanged.　In this paper, I introduce algorithmic manifolds that explain algorithms by geometric method and show that they are topologically homogeneous with respect to P versus NP problem. And I will also discuss polynomial-time reduction method of NP problem for class P.

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