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Quantum Computer Outline

History of the theory

1920s Copenhagen interpretation formed

1936 Alan Turing - Universal Model of computation

1957 Many worlds interpretation proposed by Hugh Everett III

1960s Rolf Landaur (student at time) “computation is physics” ends up as a researcher at Thomas J. Watson Research Center.

1977-78 David Deutsch proposes quantum computer as a method to test the multiple universes theory

1979 Paul Benioff – Argonne National Laboratory in Illinois 1^st - published model for computer based on quantum mechanical components

1981 Feynman proposes a universal simulator – a simulation of any physical situation that can be run in real-time or faster.

1984 Feynman proposes a cleaner model for computer based on quantum mechanics

1985 David Deutsch’s 77-78 paper finally published

1985 Deutsch writes and has published another paper about a Universal quantum computer

Introduction (why we need one)

-         Moore’s law

o       “The number of transistors in silicon chips doubles every 1.5 years”

o       “The cost of a chip manufacturing plant doubles every 4 years.”

o       Limits on size

§         Wavelength of light is about 0.5 microns

§         People thought the limit was 0.5 to 1 microns

§         With focusing and shorter wavelength got us past this

§         Currently .11 micron chips are possible.

o       Should end in about 2010

-         Some problems (traveling salesman, factoring) can’t be solved with current computers

o       P vs NP and NP complete

o       Modeling problems

o       Encryption standards

o       Maybe mention quantum encryption

-         Allows for truly random numbers

o       Today’s “random numbers” are usually just pulled from a long list of numbers stored somewhere in the computer.

A Universal quantum computer can

1.      Act as Feynman’s universal simulator

2.      Do everything a classical computer could do

3.      Take advantage of quantum parallelism to do things a classical computer cannot

Main ideas

Interference

-         Interference arises from the fact that the wavelike aspects of quantum particles can overlap one another to cause unusual and distinctive patterns of behavior

-         Two slit experiments with photons and electrons

-         Constructive and destructive

            Superposition

-         The combination of two or more states

-         Particles with spin +½ or -½ 

-         Polarization of photons

-         Discrete energy levels in exited atoms

            Entanglement

-         Polarized photons from a prism

-         Two photons created from a reaction

-         Entanglement is the ability of quantum systems to exhibit correlations between states within a superposition. If we have two qubits each in a superposition of 0 and 1, the qubits are said to be entangled if the measurement of one qubit is always correlated with the result of the measurement of the other qubit.

Decoherence and dissipation

-         It is extremely difficult to isolate a quantum entity from its environment

-         There are many different things that can disrupt a qubit: heat, cosmic rays, the structure holding the qubit, other qubits, just about anything.

-         Dissipation is where a qubit loses energy to it’s environment (an electron naturally falling to a lower energy state

-         Decoherence is when the qubit becomes entangled with its environment and no longer possesses the proper superposition.

Copenhagen interpretation

-         There is no need to give intrinsic properties (i.e. position and velocity) to isolated quantum entities such as electrons.

-         Properties of quantum systems only make sense in the measurements made.

-         A quantum entity isn’t there until we try to measure it.

-         But what really counts as an observer

Many Worlds interpretation

-         Any time a decision is made, the world splits. In one world one path was take and in another, the other path

-         Example, Every morning I wake up and decide to wear either shoes or sandles. In one universe I wear shoes, in another I wear sandles.

-         Example, If a qubit is in a superposition of states, a measurement of the qubit will result in two universes. In one univers I will see a 0. In the other I will see a 1.

-         The many worlds interpretation removes the need to define what exactly counts as an observer

Quantum gates

Square Root of NOT sqr(NOT)

-         sqr(NOT) * sqr(NOT) = NOT

-         achived differently with different things

o       light – 45 degree polarization filter

-         electrons in energy levels – shine the light required to get the electron into the next energy level for half the time required.

Hadamard gate

-         Similar to square root of NOT gate

-         One H-gate puts qubit in superposition

-         Two H-gates leaves qubit unchanged

2-qubit XOR gate

-         Same truth table as controlled NOT

-         When combined with other single qubit gates creates a universal gate

Other gates of interest

Controlled NOT

            Controlled-controlled NOT (Toffoli gate)

What we actually have

-         Many different qubits

o       Heteropolymer

§         Linear array of atoms as memory cells.

§         Each atom can be in a grounded or excited state (0 or 1)

§         Uses laser pulses to change between excited states

o       Ion Trap

§         Uses electromagnetic fields to trap ions

§         Each atom can be in a grounded or exited state (0 or 1)

§         A Beryllium ion can be used to encode two qubits

§         Decoherence time of about a millisecond.

o       Cavity QED (Quantum Electrodynamics)

§         Uses the polarization of photons for the bits

§         Can implement an XOR gate

§         Only currently works on small quantum systems

o       NMR (Nuclear Magnetic Resonance)

§         Uses a test-tube-sized sample of some liquid

§         Each atom in the liquid is a qubit

§         Uses the spin of one of the nuclei of one atom of the molecules

o        “If we shine light right on anything it can be a qubit

o       Coffee mug example

-         Entangled qubits

-         The Square Root of NOT operators

-         Some XOR operations

Problems to overcome

-         Decoherence and dissipation

-         Enough qubits in a system to do effective calculations

Testing the Many Universes Theory

1. Assume we have a fully functional quantum computer

2. Take one qubit and set it to 0

3. Apply sqr(not)

4. Have the computer look at the value of the qubit

                        - In one Universe there will be a 0 and in another a 1

5. The computer announces that it has made the measurement.

6. The computer then “forgets” all knowledge of the value observed

                        - This allows the Universes to still produce interference (2-slit experiment)

7. Perform another sqr(not) operation on the qubit

8. If the Many Universes Theory is correct the measured value will always be 1

9. If the Many Universes Theory is incorrect the measured value can be either 0 or 1 at random.

Factoring Numbers using Quantum Computers (Shor’s algorithm)

1.      Set one of the registers into a superposition of all possible numbers

2.      Compute x^a mod n in the second register where x is a random number that doesn’t share any factors with n, a is the number (or superposition of numbers) in the first register and n is the number to be factored

3.      Carry out a Fourier transform on the first register and then observe its content.

4.      From the result calculate the order of x because it has a good chance of predicting one of the factors of n.

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page last updated December 2003