Quantum Learning Algorithms and Post-Quantum Cryptography∗
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Quantum Learning Algorithms and Post-Quantum Cryptography∗ Alexander M. Poremba 1 QMATH, Department of Mathematical Sciences, University of Copenhagen, 1165 Copenhagen, Denmark. 2 Department of Physics and Astronomy, University of Heidelberg, 69047 Heidelberg, Germany. Abstract. Quantum algorithms have demonstrated promising speed-ups over classical algorithms in the context of computational learning theory - despite the presence of noise. In this work, we give an overview of recent quantum speed-ups, revisit the Bernstein-Vazirani algorithm in a new learning problem extension over an arbitrary cyclic group and discuss applications in cryptography, such as the Learning with Errors problem. We turn to post-quantum cryptography and investigate attacks in which an ad- versary is given quantum access to a classical encryption scheme. In particular, we consider new notions of security under non-adaptive quantum chosen-ciphertext attacks and propose symmetric-key encryption schemes based on quantum-secure pseudorandom functions that fulfil our definitions. In order to prove security, we introduce novel relabeling techniques and show that, in an oracle model with an arbitrary advice state, no quantum algorithm making superposition queries can reliably distinguish between the class of functions that are randomly relabeled at a small subset of the domain. Finally, we discuss current progress in quantum computing technology, partic- ularly with a focus on implementations of quantum algorithms on the ion-trap architecture, and shed light on the relevance and effectiveness of common noise arXiv:1712.09289v3 [quant-ph] 17 Jun 2018 models adopted in computational learning theory. ∗This work was carried out as part of my Master's thesis at the University of Heidelberg. Contact: [email protected] Principal advisor: Gorjan Alagic, Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD Co-advisor: Thomas Gasenzer, Kirchhoff-Institute for Physics, University of Heidelberg, Germany. Table of Contents 1 List of Abbreviations . .5 2 Introduction . .6 3 Technical Summary of Results . 11 4 Cryptography . 17 4.1 Preliminaries . 17 4.2 Symmetric-Key Cryptography . 18 4.3 Security Notions . 19 Computational Security . 19 Computational Indistinguishability. 19 Semantic Security. 21 4.4 Pseudorandom Functions . 22 4.5 Learning with Errors . 23 Decision Learning with Errors. 24 Symmetric-Key Constructions and Security. 24 Separation Result. 25 5 Quantum Computation . 27 5.1 Formalism . 27 5.2 Unitary Evolution . 29 5.3 Quantum Measurement . 30 5.4 Universal Quantum Gates . 31 5.5 The Quantum Circuit Model . 33 5.6 Quantum Parallelism . 34 5.7 Decoherence . 36 Quantum Noise Models. 36 Independent Noise Models. 38 5.8 Error Correcting Codes . 39 5.9 Quantum Oracles . 41 Membership Oracles. 42 Example Oracles. 42 6 Quantum Algorithms . 44 6.1 Deutsch-Josza Algorithm . 45 6.2 Bernstein-Vazirani Algorithm . 46 7 The Quantum Fourier Transform . 49 7.1 The Quantum Fourier Transform over Finite Abelian Groups . 50 7.2 Efficient Circuit Implementations . 53 8 Quantum Learning Algorithms . 55 8.1 Computational Learning Theory . 55 Exact Learning. 56 PAC Learning. 56 8.2 Learning Parity With Noise. 58 8.3 Extended Bernstein-Vazirani Algorithm. 61 8.4 Learning with Errors with Quantum Examples. 63 9 Relabeling Games . 65 9.1 Classical Relabeling. 65 9.2 Relabeling in Quantum Algorithms . 66 Non-adaptive Relabeling. 66 Adaptive Relabeling. 69 10 Post-Quantum Cryptography . 72 10.1 Security Under Non-adaptive Quantum Chosen-Ciphertext Attacks . 72 Indistinguishability. 73 Semantic Security. 73 Equivalence of Indistinguishability and Semantic Security. 74 10.2 Quantum-secure Pseudorandom Functions . 74 10.3 Secure Constructions . 75 11 The Physical Realization of Quantum Computation . 80 11.1 DiVincenzo Criteria . 80 11.2 Ion-Trap Implementation . 82 Hyperfine Structure. 82 Experimental Setup. 84 The Hamiltonian. 87 Single-Qubit Gates. 90 Two-Qubit Gates. 92 Quantum Algorithms with Trapped Ions. 94 Decoherence and Sources of Error. ..