Quantum Computing Keith Kelley CS 6800, Theory of Computation

Quantum Computing  Computers governed by the laws of Quantum Mechanics  Moore's Law and chip thickness: macroscopic vs microscopic  Quantum Mechanics on purpose or by accident  Mostly theoretical and a little bit experimental

Quantum Mechanics  Otherwise known as Quantum Physics  The physics of the very small: atoms, molecules and particles  As opposed to the Newtonian Mechanics we understand

Quantum Mechanics  “I think it is safe to say that no one understands quantum mechanics.”  Richard P. Feynman, The Character of Physical Law (1965)  “You see my physics students don't understand it.... That is because I don't understand it. Nobody does.”  Richard P. Feynman, Nobel Lecture, 1966

Quantum Computers: quantum bits  Quantum Computers are made of qubits  Generally, particles trapped in magnetic fields, electrical fields, crystal lattices or otherwise  Quantum Computers set and read the quantum mechanical properties of particles  Bit: traditional data storage 0s and 1s  Qubit: quantum data storage 0 and 1 at once

superposition  more than one position  both positions simultaneously

Amplitudes vs Probabilities  Macroscopic objects are governed by probabilities, a number between 0 and 1  Microscopic objects are governed by amplitudes, complex numbers  Amplitudes are treated mathematically the same as probabilities, but an amplitude has a complex (imaginary) component, denoted by a fraction of I

Particle/Wave Duality and Interference  Two waves in phase and their combined waveform

Interference

Superposition->Exponential Everything  Quantum Information: Exponential Data Storage  4 bits: 4 pieces of data  4 qubits: 2^4=16 pieces of data  Quantum Parallelism: Exponential Processing Power  4 qubits=2^4 operations

entanglement  Nonlocal correlation between qubits  Can be used to create registers  “spooky action at a distance”  [1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev., 47, 777, (1935).

Logic Gates Classical  NOT  AND  OR  NAND  NOR  XOR  XNOR Quantum  Deutsch  Hadamard  CNOT  Phase shifter gates

Reversible Gates Reversible  NOT  Toffoli  Quantum Gates Not Reversible  OR  NOR  XOR  AND  NAND

Controlled Gates  Gates with an extra bit, the control bit  Any gate U with a control bit  CNOT  Used to disentangle EPR states

Universal Gates Quantum  The Hadamard gate, the controlled-not gate, a certain phase- shift gate  Deutsch Gate(Pi/2)  (Same as a Toffoli Gate) Classical  Toffoli  AND and NOT  NAND

Toffoli Gate  Classical Gate  Universal reversible logic gate for classical operations (but not for quantum operations)  If 1 st 2 bits are set, the third is flipped

Hadamard Gate  Quantum Gate  Represented by the Hadamard Matrix

Deutsch Gate  Universal for a quantum computer as well as a classical computer  3 inputs and 3 outputs  Reversible

Models of Representation Quantum Assembly  QRAM  Quantum Turing Machine  Quantum Finite Automata

Quantum Turing Machine  Deterministic (Classical): a 6-tuple  M=(Q,Sigma,qstart,qaccept,qreject,Transition)  Nondeterministic Turing Machine  can perform one of several tasks at each step  Probabilistic Turing Machine  Same as deterministic, except the transition function accounts for probabilities of all moves  Quantum Turing Machine  Same as probabilistic, except the probabilities are complex number amplitudes

Languages  Quantum Bits and Quantum Bytes->Quantum Gates, Registers and Circuits  Quantum Machine Language  Quantum Assembly Programming  QCL  Quantum Pseudocode  QGCL  QPL and CQPL  Quantum Lambda Calculus  Q  QML  Quantum C, etc...

Applications  Encryption  Compression  Physics Modeling  Math Problems  Parallel Computing  Anything

Encryption  The bad  Breaking RSA through polynomial factoring  The good  Quantum Authentication with entanglement

Algorithms  Deutsch  Deutsch-Jozsa  Shor's  Grover's  Simon's

Deutsch-Jozsa Algorithm  A generalized form of Deutsch's algorithm  Demonstrates exponential speedup from the classical solution  No practical application

Simon's Algorithm  Finds periodicity in n function evaluations  Classical algorithm needs 2^(n-1)+1

Grover's Search  Searching unordered arrays  Quadratic speedup  Classical solution averages n/2 queries or up to n queries  Grover's does it in sqrt(n) queries

Shor's Factorization  Factors an integer  Exponential speedup  Classical: O(e^cn 1/3 log 2/3 n)  Shor's: O(n 2 log n log log n)

Hardware Implementations  Nuclear Magnetic Resonance (NMR)  Ion Traps  Linear Optics  Cavity QED  Optical Lattice  Kane Quantum Computer  Quantum Dot

Observation/Measurement  Causes: wave function collapse  Aka collapse of the state vector  Aka reduction of the wave packet  When not desired: called decoherence

Schroedinger's Cat

References  Quantum Computing for Computer Scientists, Noson S. Yanofsky and Mirco A. Manucci  quantiki.org 

Exam Question  Q. Name two algorithms for quantum computers and their approximate speedups  A1. Shor's exponential speedup of integer factorization  A2. Grover's quadratic speedup searching an unordered array  A3. Deutsch's or Deutsch-Josza's exponential speedup of algorithms that exist merely for illustration of quantum computing potential  A4. Simon's exponential speedup of periodicity count of a function

“Quantum mechanics is certainly imposing... I, at any rate, am convinced that [God] does not throw dice.” Albert Einstein letter to Max Born (December 4, 1926)