Quantum and classical computing Dalibor HRG EECS FER
How to think?
Review / Classical computing Classical computing: Turing machine (A.Turing,1937.), computability (functions and predicates), Computational Complexity – theory of classical computation. Bool’s algebra and circuits, today computers, (logic). Algorithms and complexity classes (P, P/poly, PSPACE, NP, NP- complete, BPP,…) – measuring how efficient is algorithm, can it be useful?
Famous mathematical questions today: P – predicates which are decidable in polynomial time (head moves of Turing machine) PSPACE – predicates decidable in polynomial space (cells on Turing machine’s track) Review / Classical computing
NP – we can check some solution in polynomial time, but finding it, is a difficult problem. Predicate: SAT, HC (hamiltonian cycle),TSP (travelling salesman problem), 3- SAT,… Karp’s reducebility: NP – complete: each predicate from NP is reducible to 3– SAT predicate.
Review / Classical computing
NANOTECHNOLOGY
Review / Quantum computing (R. Feynman,Caltech,1982.) – impossibility to simulate quantum system! (D. Deutsch, Oxford, CQC, 1985.) – definition of Quantum Turing machine, quantum class (BQP) and first quantum algorithm (Deutsch-Jozsa). Postulates of quantum mechanics, superposition of states, interference, unitary operators on Hilbert space, tensorial calculation,…
Quantum mechanics Fundamentals: dual picture of wave and particle. Electron: wave or particle?
Quantum mechanics
Waves!
Secret of the electron Does electron interfere with itself?
Quantum mechanics Discrete values of energy and momentum. State represent object (electron’s spin, foton’s polarization, electron’s path,…) and its square amplitude is probability for outcome when measured. Superposition of states, nothing similar in our life. Interference of states.
Qubit and classical bit Bit: in a discrete moment is either “0” (0V) or “1” (5V). Qubit: vector in two dimensional complex space, infinite possibilities and values. Physically, what is the qubit?
Qubit
System of N qubits Unitary operators: legal operations on qubit. Unitary operators: holding the lengths of the states. Important!!
Tensors For representing the state in a quantum register. Example, system with two qubits: State in this systems is:
Quantum gates Quantum circuits (one qubit): Pauli-X (UNOT), Hadamard (USRN). (two qubits): CNOT (UCN).
Quantum parallelism All possible values of the n bits argument is encoded in the same time in the n qubits! This is a reason why the quantum algorithms have efficiency!
Quantum algorithms (1) Time Quantum operators Initial state Measurement
Quantum algorithms (2) Idea: 1. Make superposition of initial state, all values of argument are in n qubits. 2. Calculate the function in these arguments so we have all results in n qubits. 3. Interference ( Walsh-Hadamard operator on the state of n qubits or register) of all values in the register. We obtain a result. f
(No-cloning theorem) Wooters & Zurek 1982 Unknown quantum state can not be cloned. Basis for quantum cryptology (or quantum key distribution).
Quantum cryptology (1) Quantum bits Alice Bob Eve
Quantum cryptology (2) Public channel for authentication
Quantum teleportation Bennett 1982 It is possible to send qubit without sending it, with two classical bits as a help. AliceBob EPR Alice & Bob share EPR (Einstein,Podolsky,Rosen) pair. Classical bits.
Present algorithms? Deutsch-Josza Shor - Factoring 1994., Kitaev - Factoring Grover - Database searching 1996., Grover - Estimating median
Who is trying? Aarhus Berkeley Caltech Cambridge College Park Delft DERA (U.K.) École normale supérieure Geneva HP Labs (Palo Alto and Bristol) Hitachi IBM Research (Yorktown Heights and Palo Alto) Innsbruck Los Alamos National Labs McMaster Max Planck Institute-Munich Melbourne MIT NEC New South Wales NIST NRC Orsay Oxford Paris Queensland Santa Barbara Stanford Toronto Vienna Waterloo Yale many others…
Corporations?