An Introduction to Quantum Phenomena and their Effect

An Introduction to Quantum Phenomena and their Effect on Computing Peter Shoemaker MSCS Candidate March 7 th, 2003

Overview • How quantum computers work • How they differ from classical computers • Quantum algorithms • State of Research

What is Quantum Computing? Definition: A fundamentally new mode of information processing that can be performed only by harnessing physical phenomena unique to quantum mechanics • Highly theoretical at current state of research • Has potential for computing power far beyond classical computers

Where Would Quantum Computers Excel? • Cryptography • Searching (Grover’s Algorithm) • Factoring Large Numbers (Shor’s Algorithm) • Simulating Quantum Mechanical Systems

Classical Computers • Use bits which contain either a zero or a one • Operate on these bits using a series of binary logic gates • Components have been decreasing in size (logic gates and wires are currently less than 1 micron wide) • Classical designs are reaching theoretical limit of miniaturization (only a few atoms) • 1 micron = 10 -4 cm • 1 atom = 10 -8 cm

Classical Computers (cont. ) • On the atomic scale matter obeys the rules of quantum, not classical physics • Must develop some form of quantum technology to further computer research • Quantum technology could not only further reduce the size of components, but could allow for development of new algorithms based on quantum concepts

Physics Concepts Qubit (Quantum bit) • When an electron is placed in a magnetic field the spin of the electron is either aligned with the field (spin-up state) or opposite the field (spin-down state) • Nuclei of atoms exhibit the same property • Can set the spin state of an particle using energy pulse • Spin up-state represents a 1, spin-down a 0 • This representation of 1 and 0 using electron spins is known as a qubit • These states could also be represented by electron charge instead of spins

Physics Concepts (cont. ) Superposition • Can also set the spin state of a qubit to be a “superposition” of the two states, i. e. both 0 and 1 simultaneously • Measurement of a superposed qubit destroys the superposition and will yield either a 1 or a 0 • Operations can be performed on a superposed qubit without destroying the superposition

Superposition Passing a photon through a half-silvered mirror

Superposition (cont. ) Quantum interference

Superposition (cont. ) Quantum Interference

Physics Concepts (cont. ) Entanglement • Two Particles can also be prepared in an entangled state where performing an operation on one particle performs the same operation on another • Multiple operations can be performed on particles in an entangled state without destroying the entanglement or the superposition of either particle • Measuring the spin of either particle destroys the superposition of both states but places both particles in a distinct state • Entangled particles can be large distances apart (even billions of light years)

So What’s the Point? • While a single classical bit can store either 0 or 1, a single qubit can simultaneously store both 0 and 1 • Two qubits can store four states simultaneously while two classical bits can store one of four states • Three qubits can store eight states • In general if L is the number of qubits in a quantum register, that register can store 2 L different states simultaneously • Classical registers only store one state • More importantly, any operation on such a quantum register can be performed on all 2 L states in a single operation

Classical vs. Quantum • It takes classical computer 2 L operations to perform the same calculation as one quantum operation on L qubits • With only 500 qubits (2500 states) a quantum computer could represent numbers larger than the number of atoms in the known universe • The 2500 operations necessary to perform the same operation to 2500 would take an incredible amount of time on a classical computer • As the number of qubits increases the performance gain over classical computers grows exponentially do to the parallel ness of operations

Is It This Simple? • Measuring the superposed states in a quantum register collapses entangled particles into single binary states • Though we can operate on 2 L numbers simultaneously we can’t retrieve all 2 L results • Must have special quantum algorithms that can exploit this parallelism • Quantum computers are fragile • Needs some form of quantum error correction to ensure accuracy • Implementing a large scale quantum computer is beyond the current state of research

Quantum Algorithms • When a qubit is measured it will only return either a 0 or a 1 • In terms of a quantum register, only one of the 2 L states stored in that register will be selected • Which state is returned is governed by probability amplitudes • Quantum algorithms manipulate this probability so that a state containing a correct result will be selected

Quantum Probability • A qubit is represented as a complex linear superposition that satisfies the normalizing condition • i. e: where A and B are complex numbers and |A|2 + |B|2 = 1 • |A|2 + |B|2 represent the probability that when the qubit is measured it will measure as being in the 0 or 1 state respectively • Quantum algorithms will alter these probabilities based on which states contain the desired result

Searching (Grover’s Algorithm) • Developed by Lov K. Grover in 1996 • Provides an efficient algorithm for searching un-indexed data • Linear search takes n/2 operations on average • Grover’s algorithm takes about sqrt(n) operations on average • Performance gains vs. linear search grow larger as input size increases

Searching (cont. ) Implementation of Grover’s Algorithm: 1. Choose enough qubits so that there is one state for each data entry 2. Match each data entry to a different quantum state 3. Change sign of the probability amplitude for the target state 4. Perform “inversion about average” on all probability amplitudes 5. Repeat #4 times 6. Measure quantum state

Cryptography • Current encryption standard is RSA public key encryption • RSA encryption relies on the computational complexity of factoring large composite numbers into the product of two primes • No polynomial time algorithm is known for factoring on conventional computers • The best classical algorithm runs in O(e^(64/9)1/3(ln N)1/3(ln ln N)2/3) steps • Quantum computers can theoretically factor large composite numbers in polynomial time

Cryptography (cont. ) • In 1994 a 129 digit number was successfully factored using approximately 1600 workstations scattered around the world • The entire factorization took eight months • Using this as an estimate, it would take roughly 800, 000 years to factor a 250 digit number with the same computer power • A 1000 digit number would require 1025 years (significantly longer than the known age of the universe)

Cryptography (Shor’s Algorithm) • Developed in 1994 by Peter Shor • Shor’s quantum algorithm runs in O((log n)2 * log n) steps on a quantum computer with O(log n) steps of post-processing done on a classical computer • Could factor a 1000 digit number in only a few million steps • Overall factoring takes polynomial time

Quantum Cryptography • New algorithms for cryptography must be created if quantum computers become viable • Could used entangled particles over long distances to transmit messages securely • Can detect eavesdroppers

Obstacles • Decoherence - the tendency of a particle to decay from a given quantum state into an incoherent state as it interacts, or entangles, with the state of the environment • Decoherence can be partially prevented by shielding particles from external influences • Error Correction – Must be able to maintain coherence of quantum systems before they will be usable • Error correction is difficult since quantum systems can’t be measured without destroying the state of the system • Quantum computer hardware is in its infancy

Current State of Quantum Computing Research • Qubits based on electron charge remain coherent for a few pico seconds at best • Qubits using spin-states remain coherent for several nano seconds • Scientists have entangled two particles as of 1999 • In 1999 MIT researches developed a 2 -qubit quantum computer • In 2000 IBM built a five bit quantum-computer which solved the order-finding function (determining the period of a function) in a single step • A new method developed only a few weeks ago entangled 3 electrons using semiconductors in place of complicated lab setups

Questions?