From the Foundations of Quantum Mechanics to Modern Quantum Information Technologies Johannes Kofler AKKT Forschungsgesellschaft Johannes Kepler University Linz, Austria Institute of Bioinformatics Johannes Kepler University Linz, Austria 5 July 2018
Disclaimer “I think I can safely say that nobody understands quantum mechanics.” Richard Feynman Nobel prize in physics (1965) for one of the formulations of quantum mechanics
Outlook Introduction Basic Concepts of Quantum Mechanics Quantum States & Schrödinger Equation Superposition Entanglement Schrödinger Cat States Bell experiments on Local Realism Quantum Cryptography Quantum Computer
The Beginnings of Quantum Mechanics 1900: Max Planck Radiation law: quantization of energy 1905: Albert Einstein Explanation of the photoelectric effect: quanta of light = photons 1913: Niels Bohr Atomic model: stable orbits and quantum jumps 1925/26: Werner Heisenberg & Erwin Schrödinger: Quantum mechanics
Revolutions in Technology Classical Physics Quantum Physics (approx. 30% of GDP of USA)
The Schrödinger Equation Quantum state (t) describes probabilities for measurement results of physical observables (position, momentum, spin, etc.) Schrödinger equation determines time evolution of state Hamiltonian (matrix describing interactions) Quantum state (vector of complex numbers) One of the fundamental equations in physics: allows to compute an infinity of physical phenomena Low energy limit of relativistic quantum mechanics
The Double Slit Experiment Classical Physics Quantum Physics Particles (e.g. marbles) Waves (e.g. sound, water) Quanta (photons, electrons, atoms, molecules, …) wave particle duality superposition: |left slit + |right slit Pictures: http://www.blacklightpower.com/theory/DoubleSlit.shtml
Macroscopic Superpositions Possible? Or impossible? Open question!
Randomness Classical Randomness Quantum Randomness (e.g. roulette, weather) Quantum Randomness (e.g. radioactive decay, photon on a beam splitter) Randomness is subjective In principle everything can be computed (deterministic chaos) Individual events apparently cannot be predicted Randomness is objective
Completeness of Quantum Mechanics EPR 1935 Can the probability character of quantum mechanics be reduced to an underlying „mechanism“ Are there hidden variables beneath the quantum state? Albert Einstein Boris Podolsky Nathan Rosen Statistical mechanics: Quantum mechanics: ?
Quantenzustände Superposition: | = | + | = | Entanglement polarization: horizontal vertical Entanglement (at least two particles) |AB = |AB + |AB vertically polarized non-linear crystal = |AB + |AB B Exp. Alice Bob uv laser A basis: result basis: result 1 2 3 4 5 6 7 8 /: /: /: /: /: /: /: /: horizontally polarized local: random results global: perfect correlations
Local Realism Classical world view: Realism: Objects have definite properties independent of measurement (existence of hidden variables) Locality: Any physical influences cannot be faster than the speed of light (backbone of Einstein’s relativity theory) external world passive observer
Classical Correlations Alice und Bob are in different laboratories A referee prepares pairs of dice and sends one die to Alice and one die to Bob Alice and Bob can measure color or parity Measurement 1: color Result: A1 (Alice), B1 (Bob) Measurement 2: parity Result: A2 (Alice), B2 (Bob) Possible values: +1 (even or black) –1 (odd or red) Bob Alice A1 (B1 + B2) + A2 (B1 – B2) = ±2 for all local realistic (= classical) theories A1B1 + A1B2 + A2B1 – A2B2 = ±2 local realism restricts possible correlations –2 ≤ A1B1 + A1B2 + A2B1 – A2B2 ≤ +2
one of the most striking results in physics Violation of Bell‘s inequality Pairs of dice entangled photons Color, parity different photon polarizations |AB = |AB + |AB Bell‘s inequality (1964) John S. Bell (1928-1990) S := A1B1 + A1B2 + A2B1 – A2B2 ≤ 2 Experiment: Sexp = 22 2,83 Bell inequality is violated (in hundreds of experiments, also with electrons, atoms, etc.); one of the most striking results in physics A2 B2 A1 B1
Bell Experiment Across 144 km T. Scheidl, R. Ursin, J. Kofler, et al., PNAS 107, 19708 (2010)
Cryptography One-Time-Pad Symmetric Encryption Idea due to Gilbert Vernam (1917) Security proof by Claude Shannon (1949) [no other provably secure method] Criteria for one-time-pad: random and secret key key at least as long as message key must be used only once Quantum mechanics can achieve this Quantum Key Distribution (QKD) Gilbert Vernam Claude Shannon
Quantum Key Distribution (QKD) 1 1 1 1 Basis: / / / / / / / … Result: 0 1 1 0 1 0 1 … Basis: / / / / / / / … Result: 0 0 1 0 1 0 0 … Alice and Bob tell each other measurement basis choice If basis is the same: they use the (locally random) results for the key The rest is discarded Perfect correlations yield the key: 0110… Sometimes they change measurement basis to check Bell’s inequality Eavesdropping would be detected (Bell inequality could not be violated) Security guaranteed by laws of quantum physics
Implementations of Quantum Key Distribution Fiber-Based Networks Via Satellite Xinglong, Nanshan, Graz (7600 km) Satellite-to-ground secure keys with kbit/s rate per passage of the satellite Micius Tokyo network (2010): 300 kbit/s over 45 km Cambridge: record: 200 Gbit/s quantum key over a 100 km long cable S.-K. Liao, PRL 120, 030501 (2018)
Moore‘s Law (1965) Transistor size 2000 200 nm 2010 20 nm Graphik: http://cdn.overclock.net/4/40/40a21d1b_Moores_Law.png © Kurzweil Technologies Gordon Moore
Classical Computer Logical Gates Circuits
preparation into qubits Quantum Computer Bit Qubit |Q = |0 + |1 1 „0“ or „1“ „0“ and„1“ classical input bits 01101… classical output bits 00110… preparation into qubits measurement of qubits quantum circuit
Qubits General state of a qubits: Bloch sphere: P(„0“) = cos2/2 P(„1“) = sin2/2 … phase (interference) Physical implementations: Photon polarization |0 = | |1 = | Spin of electron, atom, etc. |0 = |up |1 = |down Energy levels in atom: |0 = |ground |1 = |excited Superconducting flux: |0 = |left |1 = |right etc… | = |0 + |1 |R = |0 + i |1
Quantum Gates Quantum gates are operations on qubits Represented as unitary n x n matrices, where n = 2number of qubits on qubit state vectors |0 = (1,0)T, |1 = (0,1)T 1-qubit operations: H |0 (|0 + |1) H |1 (|0 – |1) creates superposition X (a|0 + b|1) = a|1 + b|0 NOT operation general for 1 qubit: rotations on the Bloch sphere
2-Qubit Quantum Gates 2 qubits: 4 x 4 matrices Basic operation: controlled NOT = CNOT CNOT |c|t = |c|tc A small circuit: |0A|0B + |1A|1B ( |xA|yB) creates entanglement |0A|0B |0A H |0B (|0A+|1A) |0B = |0A|0B + |1A|0B
Shor-Algorithmus 1994 due to Peter Shor Task: Factoring of a b-bit number (important for RSA cryptography) 541 1987 = z (easy) 1074967 = x y (hard) Classical: super-polynomial: , known optimum Quantum: polynomial: O(b3), probabilistic for b = 1000 (301 digits) at THz speed: classical quantum mechanical 1024 steps 1010 steps 100000 years < 1 second L. M. K. Vandersypen et al., Nature 414, 883 (2001)
Further Quantum Algorithms Search 1996: Lov Grover Task: database seach in an unsorted database of N elements (e.g. find marked page in a book) Classical: O(N) Quantum: O(N), quadratic speed-up Linear Equations 2009: Aram Harrow, Avinatan Hassidim, and Seth Lloyd (HHL) Task: solve N×N system of linear equations (with N variables) that is sparse and has low condition number Classical: O( N) Quantum: O(2 log N), exponential speed-up
Computational Complexity Classes Traveling salesman Sorting Factoring PSPACE: set of all decision problems that can be solved by a Turing machine using a polynomial amount of space NP: solutions verifiable in polynomial time NP complete: “hard” problems in NP P: problems solvable in polynomial time BQP “bounded-error quantum polynomial time”: solvable by quantum computer in polynomial time, with error probability ≤1/3
Possible Implementations NMR Trapped Ions Photons Qubits: nuclear spins IBM (2001): 15 = 3∙5 Qubits: electron energy levels NIST (2016): 219 entangled Beryllium ions Qubits: paths or polarizations Up to 10 photons, fast gates NV Centers Quantum Dots SQUIDS Qubits: energy levels MIT/Harvard: 50 nm accuracy Qubits: energy levels Long life times Qubits: currents, fluxes 72 50 49
D-Wave Canadian company „First commercial quantum computer“ Specific problem: „quantum annealing“ (finding global minimum, i.e. ground state) Currently 2048 qubits (not fully connected) Uses quantum tunneling Quantum speedup not yet demonstrated
Road to Quantum Computation Google’s Bristlecone: quantum computing chip with 72 qubits Close to „quantum supremacy“, i.e. performing calculation that is not possible on fastest supercomputers Noise: strong limitations on performance Error correction: many physical qubits per logical qubit Full quantum computer: still years to decades away Applications: Factoring Quantum simulation Quantum machine learning
Quantum Machine Learning data/algorithm CC: Classical machine learning QC: Classical machine learning applied to solve problems in quantum mechanics Finding unknown states or state transformations Computing molecular energies Solving many-body Schrödinger equation CQ: Quantum-enhanced machine learning Linear equations (HHL algorithm) Quantum-enhanced reinforcement learning (environment in superposition → speedup) Quantum sampling techniques (quantum annealing for training Boltzmann machines) QQ: Quantum computer performs machine learning on quantum data Results encoded in quantum states
Summary and Outlook Quantum mechanics is radically different from classical physics Quantum entangled states violate Bell’s inequality → the quantum world is inconsistent with local realism Quantum cryptography: quantum key distribution implemented via fibers and satellites Quantum computer: many physical implementations possible a few powerful algorithms exist, hard to find new ones useful machine years to decades away