1. 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

2. 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

3. 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

4. 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

5. Revolutions in Technology
Classical Physics
Quantum Physics
(approx. 30% of GDP of USA)

6. 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

7. 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:

8. Macroscopic Superpositions
Possible?
Or impossible?
Open question!

9. 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

10. 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: ?

11. 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

12. 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

13. 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

14. 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
( )
S := A1B1 + A1B2 + A2B1 – A2B2 ≤ 2
Experiment: Sexp = 22  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

15. Bell Experiment Across 144 km
T. Scheidl, R. Ursin, J. Kofler, et al., PNAS 107, (2010)

16. 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

17. Quantum Key Distribution (QKD)   1  1  1  1  
Basis: / / / / / / / …
Result: …
Basis: / / / / / / / …
Result: …
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

18. 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, (2018)

19. Moore‘s Law (1965) Transistor size 2000  200 nm 2010  20 nm
Graphik:
© Kurzweil Technologies
Gordon Moore

20. Classical Computer
Logical Gates
Circuits

21. 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

22. 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

23. 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

24. 2-Qubit Quantum Gates 2 qubits: 4 x 4 matrices
Basic operation: controlled NOT = CNOT
CNOT |c|t = |c|tc
A small circuit:
|0A|0B + |1A|1B
( |xA|yB)
creates entanglement
|0A|0B
|0A H
|0B
(|0A+|1A) |0B = |0A|0B + |1A|0B

25. Shor-Algorithmus 1994 due to Peter Shor
Task: Factoring of a b-bit number (important for RSA cryptography)
541 1987 = z (easy)
= 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
years < 1 second
L. M. K. Vandersypen et al., Nature 414, 883 (2001)

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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