Quantum Computing: An Introduction CS 6800: Advanced Theory of Computation Western Michigan University February 15, 2016 Lawrence Kalisz

Quantum Computing: Outline What is Quantum Computing? Bits vs. Qubits (Q1) Superposition (Q2) Decoherence (Q3) Algorithms (Q4) Applicatons (Q5) Quantum Computing. (n.d.). Retrieved February 5, 2016, from http://d.ibtimes.co.uk/en/full/1382290/quantum-computing-super-atom.jpg

Classical computing Moore’s Law: Moore's law is the observation that the number of transistors in a dense integrated circuit doubles approximately every two years.

Quantum computing Quantum Computing: studies theoretical computation systems (quantum computers) that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Digital computers require data to be encoded into binary digits (bits), each of which is always in one of two definite states (0 or 1), quantum computation uses quantum bits (qubits), which can be in superpositions of states.

Bits vs. qubits Bit: A bit (short for binary digit) is the smallest unit of data in a computer. A bit has a single binary value, either 0 or 1. The bit is the basic unit of information. It is used to represent information by computers. An analogy to this is a light switch—its off position can be thought of as 0 and its on position as 1. Quantum Computing. (n.d.). Retrieved February 5, 2016, from http://www.nature.com/nature/journal/v479/n7373/fig_tab/nature10681_F1.html

qubits Q1: What is a Qubit? Qubit: In quantum computing, a qubit or quantum bit is a unit of quantum information or the quantum analogue of the classical bit. A qubit uses the spin of an atom to represent the current value. At any one time, the qubit is both a 0 and a 1. It is only when the bit is read that it reduces to a single value of 0 or 1.

Qubit spin

Qubit Bloch sphere Bloch Sphere: The possible states for a single qubit can be visualized using a Bloch sphere. A classical bit could only be at the "North Pole" or the "South Pole“. The rest of the surface of the sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state would lie on the equator of the sphere, on the positive y axis. (2015). In Qubit. Retrieved February 5, 2016, from https://en.wikipedia.org/wiki/Qubit

Quantum superposition Q2: What is Superposition? The principle of superposition claims that while we do not know what the state of any object is, it is actually in all possible states simultaneously, as long as we don't look to check. It is the measurement itself that causes the object to be limited to a single possibility.

Quantum calculation States: Example: 2 bits vs. 2 qubits Classical: 00, 01, 10, 11 Quantum: 2^n

Quantum decoherence Q3: What is quantum decoherence? Quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. Occurs when a system interacts with its environment in a thermodynamically irreversible way. (2015). Decoherence. Retrieved February 5, 2016, from http://www.nature.com/nature/journal/v464/n7285/fig_tab/nature08812_F1.html

Algorithms Q4: What is Shor’s Algorithm: Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm (an algorithm that runs on a quantum computer) for integer factorization formulated in 1994. Informally it solves the following problem: given an integer N, find its prime factors. Grover's algorithm is a quantum algorithm that finds with high probability the unique input to a black box function that produces a particular output value, using just O(N^1/2) evaluations of the function, where N is the size of the function's domain.

Applications Q5: Name at least one application for quantum computing? Optimization Radiotherapy Protein Folding Machine Learning Object / Pattern Recognition Video Compression / Processing

Hardware D-Wave Currently the only commercial quantum computer manufacturer. Machines at NASA, Google, Wall Street 1152 qubit Public access to machine over cloud webapplication

D-Wave Quantum Computer

Examples: TSP Traveling Salesperson: The traveling salesman problem is a problem in graph theory requiring the most efficient (i.e., least total distance) Hamiltonian cycle a salesman can take through each of n cities. No general method of solution is known, and the problem is NP-hard. Given proper polynomial map, quantum computer can give highest probable solution in minutes.

Examples: RSA RSA: RSA is one of the first practical public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and differs from the decryption key which is kept secret. In RSA, this asymmetry is based on the practical difficulty of factoring the product of two large prime numbers, the factoring problem. Classical RSA factorization: NSA: 768 bit RSA factored – 3 Years 1024 RSA would take 3000 Years Quantum RSA factorization using Shor’s: 768 bit RSA factored – minutes 1024 RSA – minutes (with higher qubits)

References: [1] (2014, October 11). Quantum Computer in a Nutshell (Documentary). Retrieved February 5, 2016, from https://www.youtube.com/watch?v=0dXNmbiGPS4 [2] (2015, October 28). You don't know how Quantum Computers work! Retrieved February 5, 2016, from https://www.youtube.com/watch?v=ZoT82NDpcvQ [3] (2015). In RSA. Retrieved February 5, 2016, from https://en.wikipedia.org/wiki/RSA_%28cryptosystem%29 [4] (2015). In Qubit. Retrieved February 5, 2016, from https://en.wikipedia.org/wiki/Qubit [5] (2015). In Shor's Alogrithm. Retrieved February 5, 2016, from https://en.wikipedia.org/wiki/Shor's_algorithm [6] (2016). In D-Wave. Retrieved February 4, 2016, from http://www.dwavesys.com/tutorials/background-reading-series/quantum-computing-primer