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A. Gilyén, S. Lloyd, Ewin Tang (2018)
Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimensionArXiv, abs/1811.04909
Yu Tong, Dong An, N. Wiebe, Lin Lin (2020)
Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functionsArXiv, abs/2008.13295
P. Hawkes (2018)
Chapter 12 – The Finite-Element Method (FEM)
[ (2003)
Numerical Analysis in Modern Scientific ComputingSpringer New York. DOI:DOI:https://doi.org/10.1007/978-0-387-21584-6 10.5555/862457
Chen-Fu Chiang, P. Wocjan (2010)
Quantum algorithm for preparing thermal Gibbs states - detailed analysis
Almudena Vazquez, Stefan Woerner (2020)
Efficient State Preparation for Quantum Amplitude EstimationarXiv: Quantum Physics
G. Low, Vadym Kliuchnikov, N. Wiebe (2019)
Well-conditioned multiproduct Hamiltonian simulationarXiv: Quantum Physics
Branislav Nikoli (2005)
Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods
R. Feynman (1999)
Simulating physics with computersInternational Journal of Theoretical Physics, 21
A. Harrow, A. Hassidim, S. Lloyd (2008)
Quantum algorithm for linear systems of equations.Physical review letters, 103 15
D. Ventura, T. Martinez (1998)
Initializing the Amplitude Distribution of a Quantum StateFoundations of Physics Letters, 12
[ (2019)
Well-conditioned multiproduct Hamiltonian simulationarXiv e-prints (2019). arXiv:1907.11679 [quant-ph].
D. Berry, Graeme Ahokas, R. Cleve, B. Sanders (2005)
Efficient Quantum Algorithms for Simulating Sparse HamiltoniansCommunications in Mathematical Physics, 270
Julien Gacon, Christa Zoufal, Stefan Woerner (2020)
Quantum-Enhanced Simulation-Based Optimization2020 IEEE International Conference on Quantum Computing and Engineering (QCE)
we reproduce their ﬁndings to illustrate how to modify our algorithm to solve the Poisson equation in d dimensions
G. Brassard, P. Høyer, M. Mosca, A. Montreal, B. Aarhus, Cacr Waterloo (2000)
Quantum Amplitude Amplification and EstimationarXiv: Quantum Physics
J. Demmel (1992)
The Componentwise Distance to the Nearest Singular MatrixSIAM J. Matrix Anal. Appl., 13
M. Möttönen, J. Vartiainen, V. Bergholm, M. Salomaa (2004)
Transformation of quantum states using uniformly controlled rotationsQuantum Inf. Comput., 5
[ k =1 ∂x 2 k on a grid with mesh size h = 1 / ( N + 1) using divided
D. Knuth (1962)
Evaluation of polynomials by computerCommunications of the ACM, 5
Adam Holmes, Anne Matsuura (2020)
Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions2020 IEEE International Conference on Quantum Computing and Engineering (QCE)
[ (2020)
Quantum-enhanced simulation-based optimizationarXiv e-prints (2020). arXiv:2005.10780 [quant-ph].
D. Berry, Andrew Childs, Robin Kothari (2015)
Hamiltonian Simulation with Nearly Optimal Dependence on all Parameters2015 IEEE 56th Annual Symposium on Foundations of Computer Science
P. Deuflhard, A. Hohmann (2003)
Numerical Analysis in Modern Scientific Computing
G. Strang (1968)
On the Construction and Comparison of Difference SchemesSIAM Journal on Numerical Analysis, 5
Graeme Ahokas (2004)
Improved algorithms for approximate quantum fourier transforms and sparse hamiltonian simulations
Stefan Woerner, D. Egger (2018)
Quantum risk analysisnpj Quantum Information, 5
Yudong Cao, A. Papageorgiou, I. Petras, J. Traub, S. Kais (2012)
Quantum algorithm and circuit design solving the Poisson equationNew Journal of Physics, 15
A. Dewes, F. Ong, V. Schmitt, Romain Lauro, Nicolas Boulant, P. Bertet, D. Vion, D. Estève (2011)
Characterization of a two-transmon processor with individual single-shot qubit readout.Physical review letters, 108 5
N. Wiebe, D. Braun, S. Lloyd (2012)
Quantum algorithm for data fitting.Physical review letters, 109 5
Zhikuan Zhao, Jack Fitzsimons, J. Fitzsimons (2015)
Quantum assisted Gaussian process regressionArXiv, abs/1512.03929
E. Saff, V. Totik (1989)
Polynomial Approximation of Piecewise Analytic FunctionsJournal of The London Mathematical Society-second Series, 39
[ (2020)
Efficient state preparation for quantum amplitude estimationarXiv e-prints (2020). arXiv:2005.07711 [quant-ph].
D. Berry, Andrew Childs (2009)
Black-box hamiltonian simulation and unitary implementationQuantum Inf. Comput., 12
P. Hawkes, E. Kasper (1996)
The Finite-Element Method (FEM)
(2011)
“Eﬃcient algorithm for initializing the amplitude,”
[ (2003)
Quantum mechanics, path integrals and option pricing: Reducing the complexity of financeNonlinear Physics - Theory and Experiment II. World Scientific
Andrew Childs, Robin Kothari (2009)
Limitations on the simulation of non-sparse HamiltoniansQuantum Inf. Comput., 10
Wen-Chyuan Yueh (2005)
EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES, 5
[ (2015)
Quantum linear systems algorithm with exponentially improved dependence on precisionSIAM J. Comput, 46
S. Bravyi, S. Sheldon, A. Kandala, D. McKay, J. Gambetta (2020)
Mitigating measurement errors in multiqubit experimentsarXiv: Quantum Physics
Andrew Childs, N. Wiebe (2012)
Hamiltonian simulation using linear combinations of unitary operationsQuantum Inf. Comput., 12
[ (2000)
Quantum amplitude amplification and estimationarXiv e-prints (2000). arXiv:quant-ph/0005055.
Y.-C. Chen, Che-Rung Lee (2017)
Augmented Block Cimmino Distributed Algorithm for solving tridiagonal systems on GPU
Changpeng Shao, Hua Xiang (2018)
Quantum circulant preconditioner for a linear system of equationsPhysical Review A
D. Berry, Leonardo Novo (2016)
Corrected quantum walk for optimal Hamiltonian simulationQuantum Inf. Comput., 16
Andrew Childs, Robin Kothari (2010)
Simulating Sparse Hamiltonians with Star Decompositions
Andrew Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, D. Spielman (2002)
Exponential algorithmic speedup by a quantum walk
Andrew Childs (2008)
On the Relationship Between Continuous- and Discrete-Time Quantum WalkCommunications in Mathematical Physics, 294
E. Tadmor (1986)
The exponential accuracy of Fourier and Chebyshev differencing methodsSIAM Journal on Numerical Analysis, 23
[ (2012)
Hamiltonian simulation using linear combinations of unitary operationsQuantum Info. Comput. 12, 11–12 (Nov. 2012), 901–924. Retrieved from http://dl.acm.org/citation.cfm?id=2481569.2481570. 10.5555/2481569.2481570, 12
Thomas Häner, M. Rötteler, K. Svore (2018)
Optimizing Quantum Circuits for ArithmeticArXiv, abs/1805.12445
Ewin Tang (2018)
A quantum-inspired classical algorithm for recommendation systemsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
Y. Subaşı, R. Somma, Davide Orsucci (2018)
Quantum Algorithms for Systems of Linear Equations Inspired by Adiabatic Quantum Computing.Physical review letters, 122 6
K. Temme, S. Bravyi, J. Gambetta (2016)
Error Mitigation for Short-Depth Quantum Circuits.Physical review letters, 119 18
Raban Iten, R. Colbeck, I. Kukuljan, J. Home, M. Christandl (2015)
Quantum Circuits for IsometriesPhysical Review A, 93
D. Berry, Andrew Childs, R. Cleve, Robin Kothari, R. Somma (2013)
Exponential improvement in precision for simulating sparse HamiltoniansProceedings of the forty-sixth annual ACM symposium on Theory of computing
G. Long, Yang Sun (2001)
Efficient scheme for initializing a quantum register with an arbitrary superposed statePhysical Review A, 64
(2019)
“Qiskit: An open-source framework for quantum computing,”
Ewin Tang (2018)
Quantum-inspired classical algorithms for principal component analysis and supervised clusteringArXiv, abs/1811.00414
S. Chin (2008)
Multi-product splitting and Runge-Kutta-Nyström integratorsCelestial Mechanics and Dynamical Astronomy, 106
[ (2011)
Simulating sparse Hamiltonians with star decompositionsIn Proceedings of the 5th Conference on Theory of Quantum Computation, Communication, and Cryptography (TQC’10). Springer-Verlag, Berlin, 94–103. Retrieved from http://dl.acm.org/citation.cfm?id=1946127.1946135. 10.5555/1946127.1946135
H. Wang (1981)
A Parallel Method for Tridiagonal EquationsACM Trans. Math. Softw., 7
[ (2020)
Efficient quantum circuits for accurate state preparation of smooth, differentiable functionsarXiv e-prints (2020). arXiv:2005.04351 [quant-ph].
6) U1(1.6) |b = (cos(0.35), sin(0.35)) T . The angles for the Ry gates in the conditioned rotation part were calculated with the the Univer-salQCompiler software
A. Kandala, K. Temme, A. Córcoles, A. Mezzacapo, J. Chow, J. Gambetta (2018)
Error mitigation extends the computational reach of a noisy quantum processorNature, 567
A. Montanaro, S. Pallister (2015)
Quantum algorithms and the finite element methodPhysical Review A, 93
Robert Corless, G. Gonnet, D. Hare, D. Jeffrey, D. Knuth (1996)
On the LambertW functionAdvances in Computational Mathematics, 5
Christa Zoufal, Aurélien Lucchi, Stefan Woerner (2019)
Quantum Generative Adversarial Networks for learning and loading random distributionsnpj Quantum Information, 5
Lov Grover, T. Rudolph (2002)
Creating superpositions that correspond to efficiently integrable probability distributionsarXiv: Quantum Physics
V. Shende, S. Bullock, I. Markov (2005)
Synthesis of quantum logic circuitsProceedings of the ASP-DAC 2005. Asia and South Pacific Design Automation Conference, 2005., 1
Suguru Endo, Qi Zhao, Y. Li, S. Benjamin, Xiao Yuan (2018)
Mitigating algorithmic errors in a Hamiltonian simulationPhysical Review A
Martin Plesch, vCaslav Brukner (2010)
Quantum-state preparation with universal gate decompositionsPhysical Review A, 83
S. Aaronson (2015)
Read the fine printNature Physics, 11
A. Adler (2015)
Lambert-W Function
S. Lloyd, M. Mohseni, P. Rebentrost (2013)
Quantum principal component analysisNature Physics, 10
Masuo Suzuki (1991)
General theory of fractal path integrals with applications to many‐body theories and statistical physicsJournal of Mathematical Physics, 32
R. Brath (2020)
DistributionsVisualizing with Text
Daniel Egloff (2012)
Pricing Financial Derivatives with High Performance Finite Difference Solvers on GPUs
J. Butcher (2003)
Numerical methods for ordinary differential equations
L. Richardson
The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry DamPhilosophical Transactions of the Royal Society A, 210
B. Baaquie, C. Corianò, M. Srikant (2002)
Quantum Mechanics, Path Integrals and Option Pricing: Reducing the Complexity of FinancearXiv: Soft Condensed Matter
J. Arrazola, A. Delgado, Bhaskar Bardhan, S. Lloyd (2019)
Quantum-inspired algorithms in practiceQuantum, 4
D. Aharonov, A. Ta-Shma (2003)
Adiabatic quantum state generation and statistical zero knowledge
M. Schuld, I. Sinayskiy, Francesco Petruccione (2016)
Prediction by linear regression on a quantum computerPhysical Review A, 94
[ (2002)
Creating superpositions that correspond to efficiently integrable probability distributionsarXiv e-prints (2002). arXiv:quant-ph/0208112.
[
Mitigating measurement errors in multi-qubit experimentsarXiv e-prints (2020). arXiv:2006.14044 [quant-ph].
J. Shewchuk (1994)
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
N. Stamatopoulos, D. Egger, Yue Sun, Christa Zoufal, Raban Iten, Ning Shen, Stefan Woerner (2019)
Option Pricing using Quantum ComputersQuantum, 4
Leonard Wossnig, Zhikuan Zhao, A. Prakash (2017)
Quantum Linear System Algorithm for Dense Matrices.Physical review letters, 120 5
B. Clader, B. Jacobs, C. Sprouse (2013)
Preconditioned quantum linear system algorithm.Physical review letters, 110 25
Nikolay Raychev, I. Chuang (2010)
Quantum Computation and Quantum Information: Bibliography
R. Somma, Andrew Childs, Robin Kothari (2015)
Quantum linear systems algorithm with exponentially improved dependence on precisionBulletin of the American Physical Society, 2016
Anmer Daskin (2015)
Quantum Principal Component Analysis
S. Chin, J. Geiser (2010)
Multi-product operator splitting as a general method of solving autonomous and nonautonomous equationsIma Journal of Numerical Analysis, 31
A. Hoorfar, Mehdi Hassani (2008)
INEQUALITIES ON THE LAMBERTW FUNCTION AND HYPERPOWER FUNCTIONJournal of Inequalities in Pure & Applied Mathematics, 9
[ (2010)
Quantum Computation and Quantum InformationCambridge University Press
diﬀerences leads to a system of linear equations ∆ h (cid:126)v = (cid:126)f h . (A2)
[ (1986)
The exponential accuracy of Fourier and Chebyshev differencing methodsSIAM J. Numer. Anal., 23
[ (2016)
Corrected quantum walk for optimal Hamiltonian simulationQuantum Info. Comput. 16, 15-16 (Nov. 2016), 1295–1317. Retrieved from http://dl.acm.org/citation.cfm?id=3179439.3179442. 10.5555/3179439.3179442, 16
E. Dumitrescu, A. McCaskey, G. Hagen, G. Hagen, G. Jansen, G. Jansen, T. Morris, T. Morris, T. Papenbrock, T. Papenbrock, R. Pooser, R. Pooser, D. Dean, P. Lougovski (2018)
Cloud Quantum Computing of an Atomic Nucleus.Physical review letters, 120 21
Yazhen Wang (2012)
Quantum Computation and Quantum InformationStatistical Science, 27
Raban Iten, Oliver Reardon-Smith, Emanuel Malvetti, Luca Mondada, Ethan Redmond, Ravjot Kohli, R. Colbeck (2019)
Introduction to UniversalQCompilerArXiv, abs/1904.01072
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/ε2) 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.
ACM Transactions on Quantum Computing – Association for Computing Machinery
Published: Jan 14, 2022
Keywords: Quantum
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