Towards efficient quantum algorithms for optimization and sampling ; Vers des algorithmes quantiques efficaces pour l'optimisation et l'échantillonnage
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| Title: | Towards efficient quantum algorithms for optimization and sampling ; Vers des algorithmes quantiques efficaces pour l'optimisation et l'échantillonnage |
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| Authors: | Szilágyi, Dániel |
| Contributors: | Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Université Paris Cité, Iordanis Kerenidis |
| Source: | https://theses.hal.science/tel-04381693 ; Other [cs.OH]. Université Paris Cité, 2022. English. ⟨NNT : 2022UNIP7185⟩. |
| Publisher Information: | CCSD |
| Publication Year: | 2022 |
| Subject Terms: | Sampling, Hamiltonian Monte Carlo, Portfolio optimization, Support-Vector machines, Second-Order optimization, Linear systems, Quantum algorithms, Monte Carlo hamiltonien, Échantillonnage, Optimisation de portefeuille, Machines à vecteurs de support, Optimisation conique, Systèmes linéaires, Algorithmes quantiques, [INFO.INFO-OH]Computer Science [cs]/Other [cs.OH] |
| Description: | Recently, with the advent of big data and large-scale machine learning, there has been an increasing demand for quantum algorithms that would be more directly applicable to practically-relevant problems. In this thesis we present three algorithms that aim to take us closer to this goal. Firstly, we develop a quantum algorithm for second-order cone programming, a class of optimization problems that is between linear and semidefinite programs in terms of expressivity and ease of solving. These problems are most commonly solved using interior-point methods, the complexity of which is mainly dictated by the cost of solving a series of linear systems. In our algorithm we solve these linear systems approximately using a quantum linear system solver, and prove that the resulting interior-point method converges to the correct solution in the same number of iterations. We give numerical evidence that the algorithm provides end-to-end speedups in certain low-precision applications such as support-vector machines and portfolio optimization. The quantum linear system solver that we use is the result of a long line of research spawned by the quantum linear system algorithm of Harrow, Hassidim and Lloyd. While it has been proven to be asymptotically optimal, its corresponding circuit is nontrivial and requires some classical preprocessing. The second contribution of this thesis is an improved quantum algorithm for linear systems, based on the optimal classical method of Chebyshev iteration. Finally, we observe that the aforementioned algorithms have the common property of approximating the unique true solution of the input problem. In general, designing such quantum algorithms is nontrivial, as often the only way of recovering (an approximation to) the true solution is by running the algorithm many times (either naively or via amplitude amplification) and computing some statistics (e.g. the mean) of the measured outputs. If quantum computers intrinsically provide sampling access to their outputs, could we exploit them to ... |
| Document Type: | doctoral or postdoctoral thesis |
| Language: | English |
| Relation: | NNT: 2022UNIP7185 |
| Availability: | https://theses.hal.science/tel-04381693 https://theses.hal.science/tel-04381693v1/document https://theses.hal.science/tel-04381693v1/file/va_Szilagyi_Daniel.pdf |
| Rights: | info:eu-repo/semantics/OpenAccess |
| Accession Number: | edsbas.AB7EAC1A |
| Database: | BASE |
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