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 the theoretical limit of miniaturization (only a few atoms) 1 micron = cm 1 atom = 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 (2 500 states) a quantum computer could represent numbers larger than the number of atoms in the known universe The operations necessary to perform the same operation to 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 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 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?