Basic Intro to Quantum Mechanics

• Does not obey classical laws
• Governed by Probability
• Nothing real until an observation has been made

• Newton and many other scientists view light as being a tiny particle. They drew this conclusion from the basic behaviors of light. The primary characteristics of light are that light bounces off surfaces in the same way a ball would, and light travels in a straight line as particles do. The photoelectric effect also gave evidence to the fact that light was a particle.

• Phillip Lenard discovered that electrons are emitted when light is shined upon a metallic surface. This was called the photoelectric effect. He hypothesized that if brighter light were used the electrons would fly off with a higher velocity. But he found that the electrons did not fly off any faster, just more electrons flew off. This holds true only if the frequency of the light remains constant.

• Einstein explained this fact using Plank’s equation of E=hv. He stated that light was not a wave, it was indeed a particle. It comes in packets of energy. Every time such a light quanta hits the metallic surface, it ejects an electron. Increasing the brightness merely increases the number of light quanta, not the energy of each photon.

• Physicist Chistiaan Huygens came up with the idea that light was a wave and not a particle. But at the time the main observed effects of light could only be explained it light were a particle

• This experiment was devised by Physicist Thomas Young to test whether or not light acts as a wave, similar to water. If two circular, non-concentric ripples of water meet, one will observe the process of interference. Where the crests or the troughs of the ripples meet, the magnitude will add, but where the crest of one meets the trough of another, the water will appear flat.

• Young tested this theory in regards to light by shooting light through a double slit and observing the pattern it projected on a surface. He found that light produces an interference pattern not simply a projection of the two slits. This provided evidence that light was surely a wave.

• Quantum theory largely comes from the concept that energy in terms of atoms is not emitted in any amount, but is limited to an exact piece of energy (quanta). This was the beginning of a theory to unify the wave nature and particle nature of light

• Blackbody radiation is the phenomenon of hot objects glowing with different distinct colors. In the early 1900s scientists tried to explain this by saying that as an object gets hotter, the electrons are excited and release a stream of photons. Classical physics said that as the temperature increase towards infinity, the frequency of the light emitted would also go to infinity. Instead scientists found that the light cut off after a certain point.

• Plank to the challenge of explaining this deviation. His formula fit the radiation curve nicely. He found that as the frequency of the light increases, more and more energy was required to eject photons. At incredibly high frequencies large amounts of energy is required to release more quanta so the amount released drops drastically.

• Many experiments showed that light behave as both a particle and a wave. Thus physicists decided that a wave-particle duality exists as part of nature. Many seemed worried about this possibility, but needlessly. Nature has no problem with light being both a particle and a wave. It is only confusing when we view light as a particle and a wave
• This led to deBroglie’s conclusion that all matter posses both a wave state and a matter state. This follows the principle of complementarity which says that one can not measure both the wave characteristic and the particle characteristic of matter. If one performs an experiment to test wave nature, one will find a wave. And if one performs an experiment for particle nature one will find a particle

• Based off of the wave-particle duality conclusion, think about shooting bullets through the double slit. If one tests for the probability that a bullet will go through the right slit and likewise for the other slit. The probability of the bullet hitting the detector is the sum of the pervious probabilities.
• On the other hand, if this experiment is done using electrons, the probability of the electron hitting the detector is not the sum of the probabilities. This effect comes from interference. We are now dealing with the probability function of the electron. To obtain the probability of the electron hitting the detector, one must not square the wave function

• There is one special change that can be made to the experiment to completely change the results. If a measurement is taken at the point the particles enter the double slit, the location of the electrons is exactly known and the wave function collapses. One will no longer see a diffraction pattern on the detector. This effect is due to uncertainty and the collapse of superposition.

• This principle states that we can never know a particle’s location and speed with infinite accuracy, If we know a particle’s speed with infinite precision, we have no clue where it is. The same holds true for knowing a particle’s position. This principle can be explained in physical terms. In order to see something, a photon must impart some energy to it. We now know where the particle is, but the collision with the photon affected the particle’s velocity.
• Uncertainty can be thought of as the protective principle of quantum mechanics. If one could measure position and momentum at the same time, quantum mechanics would collapse. Many have tried clever ways to defeat this principle, but no one has succeeded.

• The concept of uncertainty leads to the idea of superposition. This says that an unknown particle obeys a certain probability function. In other words a particle only has a probability of being in a certain state. This comes from an extension of the idea of the double slit experiment with electrons. The following equation describes the wave function y. Where the probability of getting either a 0 or a 1 is determined by a and b. Where a and be must satisfy the following equation.

• From this concept on can see that for the probability of getting either a 0 or a 1 to be equal a = b. Thus:

• This is one of the more amazing effects in quantum mechanics. Two particles can become entangled and become one system. Then if one of the particles is disturbed, the other one is also affected by this perturbation.
• For example: an atom releases two photons polarized in exactly different ways. Let’s say this pair is entangled to the same system. Until one measures one of the photons both of their states are unknown. But when an observer measures one of the photons, its polarization is defined. And because the two are entangled, the state of the other photon will then be defined as well. This change is completely independent of distance.

• Interaction between two entangled particles is completely independent of the distance between the particles. This action is attributed to non-locality. This eerie phenomenon allows an instantaneous change in a particle even if it is in another galaxy. This concept has been used to test teleportation, and been done successfully.

• Schrodinger’s Cat
• A basic example of quantum mechanics is that of Schrodinger’s Cat. Schrodinger proposed a thought experiment in which a cat was place in a room that contains a closed vial of poison and a radioactive particle. This radioactive atom has a 50% chance of decaying at any time. When it does decay it will open the vial of poison and the cat will die
• You have now created a superposition of a live cat and a dead cat. The cat can no longer be described by classical methods, but only by a probability function.

• Copenhagen
• The Copenhagen interpretation of quantum mechanics involves the abstract view of particles. This idea says that we cannot say what a particle does, even if it does exist while we are not looking at it. This basically says that a microscopic particle does not exist until we observe it (interact with it).

• Many Worlds
• The many worlds theory gets rid of the idea of probability and wave functions and reduces it to alternate worlds. The cat is both dead and alive, and both possibilities are equally real. When we make an observation, we are forced to decide between the two different worlds that then becomes real for us.
• On the surface this may seem like science fiction, but this is merely the consequence of taking the ideas of quantum mechanics literally.

Basics of Quantum Computers

The Qubit

• The basic unit to a computer is the bit. A digital computer uses Shannon bits. These bits can only represent a 0 or a 1 at a given moment. This only allows one number to be represented by any number of bits.
• A quantum computer uses qubits. These bits are described by some state of the system. Often a spin state or a polarization. Generally these states are denoted by  or  . What makes qubits so special is superposition. This allows a qubit to have a probability of being either a 0 or a 1. This allows a set of n qubits to represent 2ⁿ different numbers simultaneously.
• This leads to the idea of quantum parallelism. If a qubit is in superposition, it is acting as both a zero and a one. This allows computations to be performed simultaneously as if the computer were computing a given function for all possible initial states. Thought there is some limitation as to how the results can be measured.

• One cannot read off a quantum computer in the way one can read the state of a normal computer. If one reads off a qubit in superposition, the qubit will collapse into one of the possible solutions. This renders the calculation point less because it is then no more efficient than a normal computer. The parallelism of the computer will have stopped.

• Every interaction with the environment constitutes a measurement. For this reason, one must isolate the quantum computer. Otherwise the quantum states will collapse from superposition and all calculations will be wrong. This problem is commonly referred to as decoherence. If the accidental photon hits the system, the quantum states will collapse for what was hit.

• A quantum computer utilizes gates like any other computer does. But these gates are not based on transistors and electrical circuits which direct electron flow; these gates are generally based on laser manipulation, rf field manipulation, or application of NMR.
• The implementation of the following gates is very complex and their description is mostly limited to mathematical discussion, but following this section two examples of actual quantum computers are provided.
• One very important requirement of quantum gates is that they are completely reversible. This is due to the fact that no energy change can occur during calculation. Classical systems dissipate heat during computation, but a quantum computer can not if it is to maintain superposition.

• Walsh-Hadamard Gate
• This is one of the most basic types of one-qubit gates. This gate puts a state of either  or  into a superposition of both states. This gate can also bring a superposition of sates into a state of  or 

 

• CNOT
• This is a controlled-not gate. The basic gate from which basically any other gate can be built. It consists of one path that allows the first variable to pas through normally, and the second variable will XOR with the first. Thus we have a gate that will be reversible and can be expanded to form other gates. The following unitary transformations show the results of the CNOT gate based on definite inputs

• Cytosine in D₂O

• This system involves the spin of the extra protons, those not bonded to nitrogen molecules. The spin of the other three protons can be ignored because they are involved in rapid exchange to the solvent the molecule is in. One then has a two-spin system capable of quantum computation

• ¹³C-labeled Chloroform

• This molecule can be used to create a two-spin system through the interaction between the proton and the center carbon atom. One special modification that must be made to the molecule is changing the carbon. A normal carbon has no spin, so a neutron must be added to give it spin.

• Grover’s Searching Algorithm
• Lou Grover devised a revolutionary algorithm to search unsorted databases. This allows the database to be search in square root of N calculations instead of N/2 calculations. Which is a huge increase in speed.
• The mathematics behind the function are very complicated, but basically the algorithm attempts to amplify the correct answer. With each iteration of the function, the amplitude of the correct answer in increased and thus one can find the location of the target object

• Shor’s Prime Factorization Algorithm
• This algorithm was recently developed and poses a great threat to digital encryption and security systems. Many security systems such as DES and RSA use the difficulty of factoring numbers to provide security. A normal computer could take months to crack an RSA key.
• Shor’s algorithm reduces the complexity of factoring numbers to polynomial time. Now factoring a 400-digit number will not take in excess of several thousand years, but less that one year. For this reason, some people are beginning to fear what quantum computers could accomplish.
• This algorithm is far more complex than Grover’s algorithm. Simply though, it tries to find the period of a certain function. Which can be accomplished rather quickly using quantum parallelism

• NMR
• This approach to quantum computing involves the use of a Nuclear Magnetic Resonance device. In a magnetic field, particles will begin to line up with the field. A high frequency pulse will rotate certain molecules relative to the magnetic field lines. This allows the atoms to be put in superposition and then calculation to ensue based on the conditions of the system. NMR is one way to execute different quantum gates.
• Ion Entrapment
• Another possibility for quantum computing is a group of atoms in a linear trap. A laser pulse could then be directed at one of the atoms to induce it to change spin. This would be another way to execute quantum gates.

• Cryptography by non-clonability
• Teleportation by entanglement and non-locality