An HHL-based algorithm for computing hitting probabilities of quantum walks

We present a novel application of the HHL (Harrow-Hassidim-Lloyd) algorithm --- a quantum algorithm solving systems of linear equations --- in solving an open problem about quantum walks, namely computing hitting (or absorption) probabilities of a general (not only Hadamard) one-dimensional quantum walks with two absorbing boundaries. This is achieved by a simple observation that the problem of computing hitting probabilities of quantum walks can be reduced to inverting a matrix. Then a quantum algorithm with the HHL algorithm as a subroutine is developed for solving the problem, which is faster than the known classical algorithms by numerical experiments.

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Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation

ACM Transactions on Quantum Computing ◽

10.1145/3490631 ◽

2022 ◽

Vol 3 (1) ◽

pp. 1-37

Author(s):

Almudena Carrera Vazquez ◽

Ralf Hiptmair ◽

Stefan Woerner

Keyword(s):

Necessary Conditions ◽

Linear Equations ◽

Circuit Complexity ◽

Quantum Algorithm ◽

Richardson Extrapolation ◽

Classical Simulation ◽

Hamiltonian Simulation ◽

Systems Of Linear Equations ◽

Amplitude Amplification ◽

Theoretical Results

We present a quantum algorithm to solve systems of linear equations of the form Ax = b , where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O (1/√ε, poly (log κ, log N )), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O (κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O (1/ε 2 ) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.

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On a Quantum Algorithm for the Resolution of Systems of Linear Equations

Recent Advances in Computational Optimization - Studies in Computational Intelligence ◽

10.1007/978-3-319-21133-6_3 ◽

2015 ◽

pp. 37-53

Author(s):

J. M. Sellier ◽

I. Dimov

Keyword(s):

Linear Equations ◽

Quantum Algorithm ◽

Systems Of Linear Equations

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Accelerated GPMHSS Method for Solving Complex Systems of Linear Equations

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.260816.051216a ◽

2017 ◽

Vol 7 (1) ◽

pp. 143-155 ◽

Cited By ~ 3

Author(s):

Jing Wang ◽

Xue-Ping Guo ◽

Hong-Xiu Zhong

Keyword(s):

Complex Systems ◽

Linear Systems ◽

Iteration Method ◽

Numerical Experiments ◽

Linear Equations ◽

Splitting Method ◽

Complex Symmetric Linear Systems ◽

Systems Of Linear Equations ◽

Complex Symmetric ◽

Symmetric Systems

AbstractPreconditioned modified Hermitian and skew-Hermitian splitting method (PMHSS) is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equations, and uses one parameter α. Adding another parameter β, the generalized PMHSS method (GPMHSS) is essentially a twoparameter iteration method. In order to accelerate the GPMHSS method, using an unexpected way, we propose an accelerated GPMHSS method (AGPMHSS) for large complex symmetric linear systems. Numerical experiments show the numerical behavior of our new method.

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A two-qubit photonic quantum processor and its application to solving systems of linear equations

Scientific Reports ◽

10.1038/srep06115 ◽

2014 ◽

Vol 4 (1) ◽

Cited By ~ 40

Author(s):

Stefanie Barz ◽

Ivan Kassal ◽

Martin Ringbauer ◽

Yannick Ole Lipp ◽

Borivoje Dakić ◽

...

Keyword(s):

Large Scale ◽

Linear Equations ◽

Quantum Algorithm ◽

Quantum Computers ◽

Single Qubit ◽

Quantum Processor ◽

Systems Of Linear Equations ◽

Significant Obstacle

Abstract Large-scale quantum computers will require the ability to apply long sequences of entangling gates to many qubits. In a photonic architecture, where single-qubit gates can be performed easily and precisely, the application of consecutive two-qubit entangling gates has been a significant obstacle. Here, we demonstrate a two-qubit photonic quantum processor that implements two consecutive CNOT gates on the same pair of polarisation-encoded qubits. To demonstrate the flexibility of our system, we implement various instances of the quantum algorithm for solving of systems of linear equations.

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The Higher Accuracy Fourth-Order IADE Algorithm

Journal of Applied Mathematics ◽

10.1155/2013/236548 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 3

Author(s):

N. Abu Mansor ◽

A. K. Zulkifle ◽

N. Alias ◽

M. K. Hasan ◽

M. J. N. Boyce

Keyword(s):

Finite Difference ◽

Heat Equation ◽

Fourth Order ◽

Linear Equations ◽

Dirichlet Boundary Conditions ◽

The Novel ◽

One Dimensional ◽

Systems Of Linear Equations ◽

The One ◽

Successive Over Relaxation

This study develops the novel fourth-order iterative alternating decomposition explicit (IADE) method of Mitchell and Fairweather (IADEMF4) algorithm for the solution of the one-dimensional linear heat equation with Dirichlet boundary conditions. The higher-order finite difference scheme is developed by representing the spatial derivative in the heat equation with the fourth-order finite difference Crank-Nicolson approximation. This leads to the formation of pentadiagonal matrices in the systems of linear equations. The algorithm also employs the higher accuracy of the Mitchell and Fairweather variant. Despite the scheme’s higher computational complexity, experimental results show that it is not only capable of enhancing the accuracy of the original corresponding method of second-order (IADEMF2), but its solutions are also in very much agreement with the exact solutions. Besides, it is unconditionally stable and has proven to be convergent. The IADEMF4 is also found to be more accurate, more efficient, and has better rate of convergence than the benchmarked fourth-order classical iterative methods, namely, the Jacobi (JAC4), the Gauss-Seidel (GS4), and the successive over-relaxation (SOR4) methods.

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New Research Results of Algorithms for Analysis of Changing Electric Circuits

Известия высших учебных заведений Электромеханика ◽

10.17213/0136-3360-2020-6-29-36 ◽

2020 ◽

Vol 63 (6) ◽

pp. 29-36

Author(s):

Ivan Lebedev ◽

◽

Nikolai Savelov ◽

Keyword(s):

Numerical Experiments ◽

Linear Equations ◽

Iterative Refinement ◽

Floating Point ◽

Electrical Circuits ◽

Electric Circuits ◽

Research Results ◽

Systems Of Linear Equations ◽

New Research ◽

Refinement Algorithm

New research results in the field of accelerated analysis of linear and linearizable electrical circuits with variable elements are presented. The considered algorithms are based on a new modification of Gaussian elim-ination for solution of systems of linear equations. The researches was directed on a systematic thorough study of the phenomenon of quasi-stabilization of the error discovered by the authors during multiple repeated cir-cuit analyzes. The Sherman-Morrison algorithm and the iterative refinement algorithm are considered as alter-natives. To control the results of numerical experiments, the representation of floating-point numbers in the bi-nary128 format of the IEEE 754 standard is used. A method for the machine formation of mathematical models of circuits with an unlimited number of elements is proposed.

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Quantum Algorithm for Systems of Linear Equations with Exponentially Improved Dependence on Precision

SIAM Journal on Computing ◽

10.1137/16m1087072 ◽

2017 ◽

Vol 46 (6) ◽

pp. 1920-1950 ◽

Cited By ~ 60

Author(s):

Andrew M. Childs ◽

Robin Kothari ◽

Rolando D. Somma

Keyword(s):

Linear Equations ◽

Quantum Algorithm ◽

Systems Of Linear Equations

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Method of Infinite System of Equations for Problems in Unbounded Domains

Journal of Applied Mathematics ◽

10.1155/2012/584704 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Dang Quang A ◽

Tran Dinh Hung

Keyword(s):

Numerical Experiments ◽

Infinite System ◽

Linear Equations ◽

Unbounded Domains ◽

Uniform Grid ◽

System Of Equations ◽

System Of Linear Equations ◽

One Dimensional ◽

Infinite Domains ◽

Artificial Boundaries

Many problems of mechanics and physics are posed in unbounded (or infinite) domains. For solving these problems one typically limits them to bounded domains and find ways to set appropriate conditions on artificial boundaries or use quasi-uniform grid that maps unbounded domains to bounded ones. Differently from the above methods we approach to problems in unbounded domains by infinite system of equations. In this paper we present starting results in this approach for some one-dimensional problems. The problems are reduced to infinite system of linear equations. A method for obtaining approximate solution with a given accuracy is proposed. Numerical experiments for several examples show the effectiveness of the offered method.

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