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An Introduction to Quantum Computing and Quantum Random Number Generators

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An Introduction to Quantum Computing and Quantum Random Number Generators An introduction to Quantum Computing and Quantum Random Number Generators Giuseppe Vallone email: [email protected] INFN and The Future of Scientific Computing - 4 May 2018 Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 2 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 3 Intro Quantum Computing QRNG Conclusions What is Quantum Information? Information Theory Quantum Mechanics , Merging two big XXth century revolutions: information theory (Shannon, Turing) and Quantum Mechanics. Pag. 4 Intro Quantum Computing QRNG Conclusions Examples of applications Quantum computer Quantum cryptography Quantum metrology Pag. 5 Intro Quantum Computing QRNG Conclusions Examples of applications Quantum sensing Quantum imaging Quantum random number Quantum simulation generation Pag. 6 Intro Quantum Computing QRNG Conclusions But... ...be aware of fake! Pag. 7 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 8 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 9 Intro Quantum Computing QRNG Conclusions One-slide review of classical computation Input Output n bits m bits Universal gates: any function f : f0; 1gn ! f0; 1gm can be built from two elementary gates: NAND and COPY. Pag. 10 I Superposition principle: if j 1i and j 2i are physical states, any linear combination is a physical state: jΨi = aj 1i + bj 2i a; b 2 C I From classical bit (two orthogonal states j0i and j1i) to quantum-bit , or qubit: 2 2 j i = αj0i + βj1i α; β 2 C ; jαj + jβj = 1 Intro Quantum Computing QRNG Conclusions Quantum computer: superposition principle I Quantum mechanics: physical states are represented as vectors j i Pag. 11 I From classical bit (two orthogonal states j0i and j1i) to quantum-bit , or qubit: 2 2 j i = αj0i + βj1i α; β 2 C ; jαj + jβj = 1 Intro Quantum Computing QRNG Conclusions Quantum computer: superposition principle I Quantum mechanics: physical states are represented as vectors j i I Superposition principle: if j 1i and j 2i are physical states, any linear combination is a physical state: jΨi = aj 1i + bj 2i a; b 2 C Pag. 11 Intro Quantum Computing QRNG Conclusions Quantum computer: superposition principle I Quantum mechanics: physical states are represented as vectors j i I Superposition principle: if j 1i and j 2i are physical states, any linear combination is a physical state: jΨi = aj 1i + bj 2i a; b 2 C I From classical bit (two orthogonal states j0i and j1i) to quantum-bit , or qubit: 2 2 j i = αj0i + βj1i α; β 2 C ; jαj + jβj = 1 Pag. 11 Intro Quantum Computing QRNG Conclusions State superposition above 10000 AMU, 810 atoms Matter–wave interference of particles selected from a molecular library with masses exceeding 10000 amu, Phys. Chem. Chem. Phys. 15, 14696 (2013) Pag. 12 I Manipulate the state into jΨouti = UjΨini. U is the algorithm I Measure each qubit (usually in the computational basis) Intro Quantum Computing QRNG Conclusions Circuit model of quantum computation 0 1 Input 0 1 Output n n bits qubits 0 1 I Prepare the system into jΨini = j 0i ⊗ j 0i ⊗ · · · ⊗ j n−1i Pag. 13 I Measure each qubit (usually in the computational basis) Intro Quantum Computing QRNG Conclusions Circuit model of quantum computation 0 1 Input 0 1 Output n n bits qubits 0 1 I Prepare the system into jΨini = j 0i ⊗ j 0i ⊗ · · · ⊗ j n−1i I Manipulate the state into jΨouti = UjΨini. U is the algorithm Pag. 13 Intro Quantum Computing QRNG Conclusions Circuit model of quantum computation 0 1 Input 0 1 Output n n bits qubits 0 1 I Prepare the system into jΨini = j 0i ⊗ j 0i ⊗ · · · ⊗ j n−1i I Manipulate the state into jΨouti = UjΨini. U is the algorithm I Measure each qubit (usually in the computational basis) Pag. 13 Intro Quantum Computing QRNG Conclusions Universal gates Any quantum algorithm U can be built by the following universal gates: Hadamard Single qubit rotation CNOT gate ' 1 1 i σz 0 1 H = p1 Rz(') = e 2 1 0 0 0 2 1 −1 i ' ! B0 1 0 0C e 2 0 B C = −i ' @0 0 0 1A 0 e 2 0 0 1 0 Pag. 14 Intro Quantum Computing QRNG Conclusions Shor algorithm Problem 1: factoring Given an integer number N, find its prime factors. n1=3(log n)2=3 The best classical algorithm requires O(e 2 ) with n = log2N Problem 2: order finding Choose a generic a such that 1 < a < N. The order r of a modulo N is the minimum positive integer r such that ar = 1 mod N. Hard (probably superpolinomial) problem. Equivalence By using O(n3) classical operations Problem 1 can be reformulated as Problem 2 Pag. 15 Intro Quantum Computing QRNG Conclusions Shor algorithm Order-finding by quantum computer x Va : jxijyi ! jxijy ⊕ a mod Ni 3 I The Va operation requires O(n ) gates 2 I DFT requires O(n ) gates Pag. 16 Intro Quantum Computing QRNG Conclusions Grover algorithm problem: find x0 given a quantum black box U! such that ( −|xi for x = x0 U!jxi = jxi for x 6= x0 p x0 is found with high probability using O( N) evaluations A classical computer cannot solve the problem with less than O(N) operations Pag. 17 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 18 Intro Quantum Computing QRNG Conclusions One-way quantum computer G. Vallone, et al., One-way quantum computation with two-photon multiqubit cluster states, Phys. Rev. A 78, 042335 (2008) Pag. 19 Intro Quantum Computing QRNG Conclusions Quantum simulation Quantum Simulation, Rev. Mod. Phys. 86, 154 (2014) Pag. 20 Intro Quantum Computing QRNG Conclusions Adiabatic Quantum Computing H(t) = (1 − t)H0 + tHP 1 T = O( 2 ) gmin Adiabatic Quantum Computing, Rev. Mod. Phys. 90, 015002 (2018) Pag. 21 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 22 Intro Quantum Computing QRNG Conclusions Photons Reprogrammable optical circuit that implements all possible linear optical protocols up to the size of the circuit (six photon input) Universal linear optics, Science 349, 711 (2017) Pag. 23 Intro Quantum Computing QRNG Conclusions Trapped ion Quantum computers,Nature 464, 45 (2010) Pag. 24 Intro Quantum Computing QRNG Conclusions Ultracol atoms in lattice Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017) Pag. 25 Intro Quantum Computing QRNG Conclusions Nuclear Magnetic Resonance (NMR) NMR techniques for quantum control and computation, Rev. Mod. Phys. 76, 1037 (2005) Pag. 26 Intro Quantum Computing QRNG Conclusions Quantum dots Quantum computers, Nature 464, 45 (2010) Pag. 27 Intro Quantum Computing QRNG Conclusions Superconducting qubits Quantum computers, Nature 464, 45 (2010) Pag. 28 Intro Quantum Computing QRNG Conclusions D-wave Quantum annealing: finding the global minimum of a given function by using quantum fluctuations. Pag. 29 Intro Quantum Computing QRNG Conclusions Google Superconducting qubit processor A blueprint for demonstrating quantum supremacy with superconducting qubits, [arXiv: 1709.06678] Pag. 30 Intro Quantum Computing QRNG Conclusions IBM IBM’s quantum processors made up of superconducting transmon qubits https://quantumexperience.ng.bluemix.net/qx/qasm Pag. 31 Intro Quantum Computing QRNG Conclusions Microsoft “Our approach focuses on topological quantum computing through Majorana fermions, which promise to yield fast, stable quantum bits, also known as qubits.” https://www.microsoft.com/en-us/quantum/technology Pag. 32 Intro Quantum Computing QRNG Conclusions Summary 1 Introduction 2 Quantum Computing Circuit model Alternatives to circuit model Implementations 3 Quantum Random Number Generators 4 Conclusions Pag. 33 Intro Quantum Computing QRNG Conclusions What is a random number A random number is a number generated by an unpredictable process Pag. 34 Intro Quantum Computing QRNG Conclusions Why random numbers? Random numbers are crucial in several applications: 1 Information technology and security (also QKD) 2 Scientific simulation (meteorology, biology, physics...) 3 Lottery/gaming Pag. 35 Head or tail? How random is it? Intro Quantum Computing QRNG Conclusions Generators based on classical physics How we generate random numbers? Pag. 36 Intro Quantum Computing QRNG Conclusions Generators based on classical physics How we generate random numbers? Head or tail? How random is it? Pag. 36 Intro Quantum Computing QRNG Conclusions Generators based on classical physics Output depends deterministically from initial conditions Pag. 37 Intro Quantum Computing QRNG Conclusions Pseudo Random Number Generators Pseudo-random numbers are generated by a deterministic algorithm that produces a sequence that “resemble” a random sequence PROS CONS I simple I period I fast I not-uniformity I correlations Pag. 38 Intro Quantum Computing QRNG Conclusions but.... Von Neumann (1903-1957) (among the father of information theory "Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin" Pag. 39 Intro Quantum Computing QRNG Conclusions Problems of PRNG V1 12589822 V2 490623226 RANDU V3 682947310 V4 1829558474 used by IBM from ’60 to ’90 V5 535857758 V6 1781505818 31 V7 1571347790 Vk+1 = 65539 · Vk mod 2 V8 1984468970 V9 2059651006 V10 940136250 Pag. 40 Intro Quantum Computing QRNG Conclusions Problems of PRNG V1 12589822 V2 490623226 RANDU V3 682947310 V4 1829558474 used by IBM from ’60 to ’90 V5 535857758 V6 1781505818 31 V7 1571347790 Vk+1 = 65539 · Vk mod 2 V8 1984468970 V9 2059651006 V10 940136250 Pag.
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