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Quantum Information Quantum Information Marco Chiani University of Bologna, Italy Dept. of Electrical, Electronic, and Information Engineering \Guglielmo Marconi" Aperitivo con AI: there is plenty of WORK at the bottom! Bologna, February 17, 2021 M. Chiani, Univ. of Bologna Introduction to Quantum Information Summary Introduction Quantum Mathematical Tools Quantum algorithms Secure quantum communication M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum hypes? Quantum information technologies: really promising? M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum information applications Quantum sensing: use of quantum mechanical phenomena such as entanglement to yield higher statistical precision than purely classical approaches Quantum computation: algorithms using quantum mechanical phenomena such as superposition and entanglement Quantum communications: to secure communication to move quantum information to improve classical communication M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum bits - qubits M. Chiani, Univ. of Bologna Introduction to Quantum Information Define \state" of a classical/quantum system ... \state" means whatever information is required about a specific system, in addition to physical laws, in order to predict its behavior in future experimentsy Classical (macroscopic) example: switch \open" or \closed" ! state is one classical bit Quantum example: spin of an electron. Measured in any direction (Stern-Gerlach), two possible results, \same (+1)" or \opposite (−1)", with some probabilities. 2 To predict its behavior we need a two-dimensional unit norm complex C vector [α; β]T (referred to a specific direction, let's say along z) yFano, Ugo. "Description of states in quantum mechanics by density matrix and operator techniques." Reviews of Modern Physics (1957) M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum example: spin of an electron, some experiments ) Measurements are not gentle: after we measure, the state becomes what has been observed (collapses). ) If we know the spin along z, we know nothing about the spin along x. M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum state: spin of an electron / photon polarization α 1 0 state = = α + β j i = α j0i + β j1i (qubit) β 0 1 j i = α jui + β jdi j i = α jHi + β jV i The state allows to calculate the probabilities for spin measurements in any possible direction, as well as the behavior in future experiments. P(measuring \up") = jαj2 2 2 α; β 2 C; jαj + jβj = 1 P(measuring \down") = jβj2 M. Chiani, Univ. of Bologna Introduction to Quantum Information Some examples of qubit physical implementation (incomplete list) Physical support Name Information support j0i j1i Polarization encoding Polarization of light Horizontal Vertical Photon Number of photons Fock state Vacuum Single photon state Time-bin encoding Time of arrival Early Late Amplitude-squeezed Phase-squeezed Coherent state of light Squeezed light Quadrature state state Electron Spin Spin Up Down Electrons Electron number Charge No electron One electron Nuclear spin addressed Nucleus Spin Up Down through NMR Optical lattices Atomic spin Spin Up Down Uncharged superconducting Charged superconducting island Superconducting charge qubit Charge island (Q = 0) (Q = 2e, one extra Cooper pair) Josephson junction Superconducting flux qubit Current Clockwise current Counterclockwise current Superconducting phase qubit Energy Ground state First excited state Singly charged quantum Electron localization Charge Electron on left dot Electron on right dot dot pair Quantum dot Dot spin Spin Down Up van der Waals Electron on Electron localization Charge Electron on top sheet heterostructure bottom sheet from Wikipedia M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum computing M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum computing: modifying quantum states Quantum Mechanics: it is possible to act on physical systems to change states according to linear transformations U s.t. U yU = I (unitary transformations) 0 1 Single qubit gate example: X = 1 0 j i X X j i X (α j0i + β j1i) = α j1i + β j0i \bit flip” Two-qubits gate example: controlled-not, CNOT (a; b 2 f0; 1g) jai jai jbi + ja ⊕ bi It is possible to realize an arbitrary U by using single-qubit and CNOT elementary gates. M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum computing The state of n qubits is a complex vector with dimension N = 2n . Example n = 3 α0 j000i + α1 j001i + ··· + α7 j111i Exponential compression: 40 qubits ) 240 ' 1012 2 However, when we measure we see just one configuration, with prob. jαj j . 2 Example: jα7j is the probability to measure j111i 2 2 jα0j jαN−1j j0i j1i jN − 1i M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum computing Classical vs. Quantum jxi jxi x f (x) Uf j0i jf (x)i P P Exploiting superposition: input x jxi j0i ) output x jxi jf (x)i M. Chiani, Univ. of Bologna Introduction to Quantum Information Main quantum algorithms Polynomial-time Algorithms for Prime Factorization and Discrete Logarithms (Shor's algorithm) Quantum search, spacep of N elements. Classical search operations: O(N) Quantum search O( N) (Grover's algorithm) M. Chiani, Univ. of Bologna Introduction to Quantum Information THE algorithm Polynomial-time Algorithm for Prime Factorization on a Quantum Computer (Shor, 1994) Ingredients: Factorization by order-finding Period-finding by Quantum Fourier Transform M. Chiani, Univ. of Bologna Introduction to Quantum Information Tools from number theory The integer N as a factor in common with x − 1 if x 2 mod N = 1 and x mod N 6= ±1 To find x: generate random y, then find the period of y a mod N: y 0 mod N; y 1 mod N; ::: Factoring () period-finding M. Chiani, Univ. of Bologna Introduction to Quantum Information Tools from signal analysis: Fourier (Warning: imprecise statements, just to give the idea) A function with period P in time has frequency components at frequencies k=P. P P 0 N − 1 time Fourier 0 N 2N N − 1 P P freq M. Chiani, Univ. of Bologna Introduction to Quantum Information Tools from signal analysis: Fourier (Warning: imprecise, just to give the idea) P P 0 N − 1 j0i jN − 1i Quantum Fourier Transform j0i jN=Pi j2N=Pi jN − 1i M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum order-finding, f (a) = f (a + P) (Warning: imprecise, just to give the idea) m=P j0i QFT QFT Uf j0i Generate the superposition P jxi by QFT of a Dirac's delta x P Uf = unitary, computes f (a) in a register ) output x jxi jf (x)i Measure the second register. For example it turns out f (x) = 12 Left in the first reg: superposition of all jxi with x giving f (x) = 12 ) periodic Then, Quantum Fourier Transform: the measured frequency is some random multiple of 1=P By repeating a couple of times, P can be derived with high probability M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum and AI Quantum machine learning The state of n qubits is a complex vector with dimension N = 2n N x 2 C () jxi of log2 N qubits Classical Quantum FFT O(N log N) O((log N)2) Eigenvect, eigenval of sparse matrices O(N2) O((log N)2) Matrix inversion O(N3) O((log N)3) Table: Number of operations for basic linear algebra subroutines M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum and AI ... Quantum deep learning Quantum convolutional neural networks Quantum principal component analysis TensorFlow Quantum Quantum neural networks ... M. Chiani, Univ. of Bologna Introduction to Quantum Information Some existing quantum computers Company Cloud Access Technology Quantum Computer qubits SDK/Lang. IBM Q Montreal 27 QisKit/Python IBM Yes Superconducting IBM Q Manhattan 65 QisKit/Python IBM Q Santiago 5 QisKit/Python Amazon Rigetti Through AWS Superconducting Aspen-8 32 Braket/Python Superconducting, Amazon D-Wave Through AWS Quant. D-Wave 2000Q 2048∗ Braket/Python Annealer∗ Amazon IonQ Through AWS Trapped Ion - 79 Braket/Python Bristlecone 72 Cirq/Python Google No Superconducting Sycamore 53 Cirq/Python Honeywell On-demand Trapped Ion System Model HØ 6 - Photonic Quan- Strawberry Xanadu Yes - 12 tum Computing Fields/Python OriginQ Yes Superconducting Wu Yuan 6 QPanda/C++ M. Chiani, Univ. of Bologna Introduction to Quantum Information Quantum computing and cryptography Quantum Computing: Progress and Prospects 98 TABLE 4.1 Literature-Reported Estimates of Quantum Resilience for Current Cryptosystems, under Various Assumptions of Error Rates and Error-Correcting Codes Copyright National Academy of Sciences. All rights reserved. Quantum Time Algorithm # Logical # Physical Required Quantum-Resilient Key Security Expected to Defeat Qubits Qubits to Break Replacement Cryptosystem Category Size Parameter Cryptosystem Required Requireda Systemb Strategies AES-GCMc Symmetric 128 128 Grover’s algorithm 2,953 4.61 × 106 2.61 × 1012 encryption 192 192 4,449 1.68 × 107 years 256 256 6,681 3.36 × 107 1.97 × 1022 years 2.29 × 1032 years RSAd Asymmetric 1024 80 Shor’s algorithm 2,050 8.05 × 106 3.58 hours Move to NIST- encryption 2048 112 4,098 8.56 × 106 28.63 hours selected PQC 4096 128 8,194 1.12 × 107 229 hours algorithm when available ECC Asymmetric 256 128 Shor’s algorithm 2,330 8.56 × 106 10.5 hours Move to NIST- Discrete-log encryption 384 192 3,484 9.05 × 106 37.67 hours selected PQC probleme-g 521 256 4,719 1.13 × 106 55 hours algorithm when available SHA256h Bitcoin N/A 72 Grover’s Algorithm 2,403 2.23 × 106 1.8 × 104 mining years PBKDF2 with Password N/A 66 Grover’s algorithm 2,403 2.23 × 106 2.3 × 107 Move away from 10,000 iterationsi hashing years password-based authentication Quantum Computing: Progress and Prospects. Washington, DC, USA: Nat. Acad. Press, 2019. M.
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