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osama-quantum-computing and its uses and applications
This document provides an overview of quantum computing. It begins with introductions to quantum mechanics and the basic concept of a quantum computer. Qubits can represent superpositions of states allowing quantum computers to perform massive parallelism. Data is represented using qubit states and operations involve entanglement. Measurement causes superpositions to collapse probabilistically. While quantum mechanics is strange, quantum computing may enable solving problems like factoring exponentially faster than classical computers. The document questions the Church-Turing thesis in light of quantum computing's ability.
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osama-quantum-computing and its uses and applications
- 1.QuantumComputingOsama AwwadDepartment ofComputer ScienceWestern Michigan UniversityFebruary 20, 2024
- 2.Overview Introduction DataRepresentationComputational ComplexityImplementation TechnologiesQuantum Computer Languages
- 3.Introduction to quantummechanics Quantum mechanics is a fundamental branch oftheoretical physics with wide applications inexperimental physics that replaces classicalmechanics and classical electromagnetism at theatomic and subatomic levels.
- 4.Introduction to quantummechanics Quantum mechanics is a more fundamental theorythan Newtonian mechanics and classicalelectromagnetism It provides accurate and precise descriptions formany phenomena that these "classical" theoriessimply cannot explain on the atomic andsubatomic level
- 5.What is aquantum computer? A quantum computer is a machine that performscalculations based on the laws of quantum mechanics,which is the behavior of particles at the sub-atomiclevel.
- 6.•Moore’s Law: Wehit the quantum level 2010~2020.Why bother with quantumcomputation?
- 7.Computer technology ismakingdevices smaller and smaller……reaching a point where classicalphysics is no longer a suitablemodel for the laws of physics.
- 8.Physics and Computation•Information is stored in a physical medium,and manipulated by physical processes.• The laws of physics dictate the capabilities ofany information processing device.• Designs of “classical” computers are implicitlybased in the classical framework for physics• Classical physics is known to be wrong orincomplete… and has been replaced by a morepowerful framework: quantum mechanics.
- 9.The design ofdevices on such a small scale willrequire engineers to control quantum mechanicaleffects.Allowing computers to take advantage ofquantum mechanical behaviour allows us to domore than cram increasingly many microscopiccomponents onto a silicon chip…… it gives us a whole new framework in whichinformation can be processed in fundamentallynew ways.The nineteenth century was known as the machine age, the twentiethcentury will go down in history as the information age. I believe the twenty-first century will be the quantum age. Paul Davies, Professor NaturalPhilosophy – Australian Centre for Astrobiology
- 10.“No, you’re notgoing to be able to understand it. . .. You see, my physics students don’t understandit either. That is because I don’t understand it.Nobody does. ... The theory of quantumelectrodynamics describes Nature as absurdfrom the point of view of common sense. And itagrees fully with an experiment. So I hope thatyou can accept Nature as She is -- absurd.Richard FeynmanNobody understands quantummechanics
- 11.…consider a setupinvolving a photon source,a half-silvered mirror (beamsplitter),and a pair of photon detectors.photonsourcebeamsplitterdetectorsA simple experiment in optics
- 12.50%50%Simplest explanation: beam-splitteractsas a classical coin-flip, randomly sendingeach photon one way or the other.Now consider what happens when we fire asingle photon into the device…
- 13.… consider amodification of the experiment…100%The simplest explanation is wrong!The simplest explanation forthe modified setup would stillpredict a 50-50 distribution…full mirrorThe “weirdness” of quantum mechanics…
- 14.Classical probabilities…Consider acomputation tree for a simple two-step (classical) probabilisticalgorithm, which makes a coin-flip at each step, and whose output is 0 or 1:21212121210101The probability of the computation followinga given path is obtained by multiplying theprobabilities along all branches of thatpath… in the example the probability thecomputation follows the red path is412121The probability of the computation giving theanswer 0 is obtained by adding theprobabilities of all paths resulting in 0:214141
- 15.21|021212121|1|0|121…vs quantum probabilities…In quantum physics, we have probability amplitudes, whichcan have complex phase factors associated with them.The probability amplitude associated with a pathin the computation tree is obtained by multiplyingthe probability amplitudes on that path. In theexample, the red path has amplitude 1/2, and thegreen path has amplitude –1/2.The probability amplitude for getting the answer |0is obtained by adding the probability amplitudes…notice that the phase factors can lead tocancellations! The probability of obtaining |0 isobtained by squaring the total probabilityamplitude. In the example the probability ofgetting |0 is021212
- 16.… consider amodification of the experiment…The simplest explanation forthe modified setup would stillpredict a 50-50 distribution…full mirrorExplanation of experiment0 021121100%0102102110121121
- 17.Representation of DataQuantum computers, which have not been built yet, would be based onthe strange principles of quantum mechanics, in which the smallestparticles of light and matter can be in different places at the same time. In a quantum computer, one "qubit" - quantum bit - could be both 0 and1 at the same time. So with three qubits of data, a quantum computercould store all eight combinations of 0 and 1 simultaneously. Thatmeans a three-qubit quantum computer could calculate eight timesfaster than a three-bit digital computer. Typical personal computers today calculate 64 bits of data at a time. Aquantum computer with 64 qubits would be 2 to the 64th power faster,or about 18 billion billion times faster. (Note: billion billion is correct.)
- 18.A bit ofdata is represented by a single atom that is in one oftwo states denoted by |0> and |1>. A single bit of this form isknown as a qubit
- 19.Representation of Data- QubitsA physical implementation of a qubit could use the two energylevels of an atom. An excited state representing |1> and aground state representing |0>.ExcitedStateGroundStateNucleusLight pulse offrequency fortime interval tElectronState |0> State |1>
- 20.Representation of Data- SuperpositionA single qubit can be forced into a superposition of the two statesdenoted by the addition of the state vectors:|> = |0> + |1>Where and are complex numbers and | | + | | = 11 21 2 1 22 2A qubit in superposition is in both of thestates |1> and |0 at the same time
- 21.Representation of Data- SuperpositionLight pulse offrequency for timeinterval t/2State |0> State |0> + |1>Consider a 3 bit qubit register. An equally weightedsuperposition of all possible states would be denoted by:|> = |000> + |001> + . . . + |111>1√81√81√8
- 22.Data Retrieval Ingeneral, an n qubit register can represent the numbers 0through 2^n-1 simultaneously.Sound too good to be true?…It is! If we attempt to retrieve the values represented within asuperposition, the superposition randomly collapses torepresent just one of the original values.In our equation: |> = 1 |0> + 2 |1> , represents theprobability of the superposition collapsing to |0>. The ’sare called probability amplitudes. In a balancedsuperposition, = 1/√2n where n is the number of qubits.1 2 1n
- 23.Relationships among data- EntanglementEntanglement is the ability of quantum systems to exhibitcorrelations between states within a superposition.Imagine two qubits, each in the state |0> + |1> (a superpositionof the 0 and 1.) We can entangle the two qubits such that themeasurement of one qubit is always correlated to themeasurement of the other qubit.
- 24.Measuring multi-qubit systemsIfwe measure both bits ofwe get with probability11011000 11100100 yx2xy
- 25.Measurement ||2, foramplitudes of all states matching an output bit-pattern,gives the probability that it will be read. Example:0.316|00› + 0.447|01› + 0.548|10› + 0.632|11›The probability to read the rightmost bit as 0 is |0.316|2 + |0.548|2= 0.4 Measurement during a computation changes the state of the systembut can be used in some cases to increase efficiency (measure and haltor continue).
- 26.Quantum mechanics andinformationHow does this affect communication complexity?How does this affect information security?How does this affect computational complexity?10 10 Any physical medium capable ofrepresenting 0 and 1 is in principle capableof storing any linear combination
- 27.A “Probabilistic TuringMachine” (PTM) is an abstractmodel of the modern (classical) computer.Strong Church-Turing Thesis:A PTM can efficiently simulate any realistic model ofcomputing.Widespread belief in the Strong Church-Turingthesis has been one of the underpinnings oftheoretical computer science.The Classical Computing Model
- 28.What do wemean by “efficient”?The complexity of an algorithmmeasures how much of some resource(e.g. time, space, energy) the algorithmuses as a function of the input size.e.g. the best known algorithms forfactoring an n bit number uses time in33231)(log)))(1(92.1( nnnokeO (number field sieve algorithm)
- 29.Factoring is believedto be hard on a Turingmachine (or any equivalent model), but howdo we know that there isn’t some novelarchitecture on which it is easy?
- 30.The Strong ChurchTuring thesis tells usthat all reasonable models can be efficientlysimulated by a PTM, which implies that if it’shard for a PTM it must be hard for any otherreasonable computer.i.e. we believe computational problems, likefactoring, have an intrinsic difficulty,independent of how hard we try to find anefficient algorithm.
- 31.In the early1980s, Richard Feynman observed thatit seems implausible for a PTM to efficientlysimulate quantum mechanical systems……quantum computers arequantum mechanical systems…… so quantum computing is a modelwhich seems to violate the StrongChurch-Turing thesis!
- 32.Are quantum computersrealistic?Are quantum computers realistic?The answer seems to be YES!If the quantum computers are a reasonable modelof computation, and classical devices cannotefficiently simulate them, then the Strong Church-Turing thesis needs to be modified to state:A quantum computer can efficiently simulateany realistic model of computation.
- 33.Applications• Efficient simulationsof quantum systems• Phase estimation; improved time-frequency andother measurement standards (e.g. GPS)• Factoring and Discrete Logarithms• Hidden subgroup problems• Amplitude amplification• and much more…
- 34.Quantum Algorithmsa,b G , ak = b , find kInteger Factorization (basis of RSA cryptography):Discrete logarithms (basis of DH crypto, including ECC):Given N=pq, find p and q.
- 35.Computational Complexity ComparisonClassicalQuantumFactoringElliptic CurveDiscreteLogarithms nnOe3/23/1log nOenO log nOe nOenO log(in terms of number of group multiplications for n-bit inputs)
- 36.The following cryptosystemsare insecure against suchquantum attacks:Which cryptosystems are threatenedby Quantum Computers??• RSA (factoring)• Rabin (factoring)• ElGamal (discrete log, including ECC – see Proos and Zalka)•Buchmann-Williams (principal ideal distance problem)•and others… (see MMath thesis, Michael Brown, IQC)Information security protocols must be studied in the contextof quantum information processing.http://arxiv.org/abs/quant-ph/0301141We need to worry NOW about information that needs toremain private for long periods of time.It takes a long time to change an infrastructure.
- 37.Quantum Information Security•Quantumkey establishment (available now/soon)•Quantum random number generation (available now/soon)•Quantum money (require stable quantum memory)•Quantum digital signatures (requires quantum computer)•Quantum secret sharing (requires quantum computer)•Multi-party quantum computations•and more…We can exploit the eavesdropper detection that isintrinsic to quantum systems in order to derive new“unconditionally secure” information security protocols.The security depends only on the laws of physics, andnot on computational assumptions.
- 38.Quantum computing incomputationalcomplexity theory The class of problems that can be efficiently solved by quantumcomputers is called BQP, for "bounded error, quantum, polynomialtime". Quantum computers only run randomized algorithms, so BQP onquantum computers is the counterpart of BPP on classical computers In complexity theory, BPP is the class of decision problems solvable bya probabilistic Turing machine in polynomial time, with an errorprobability of at most 1/3 for all instances. The abbreviation BPP refersto Bounded-error, Probabilistic, Polynomial time.
- 39.Quantum computing incomputationalcomplexity theory BQP is suspected to be disjoint from NP-complete and astrict superset of P, but that is not known. Both integer factorization and discrete log are in BQP.Both of these problems are NP problems suspected to beoutside BPP, and hence outside P Both are suspected to not be NP-complete There is a common misconception that quantumcomputers can solve NP-complete problems inpolynomial time (generally suspected to be false )
- 40.Quantum computing incomputationalcomplexity theory
- 41.Implementation requirements Qubitimplementation itself Control of unitary evolution Initial state preparation (qubits) Measurement of the final state(s)
- 42.Implementation Ion TrapsNuclear magnetic resonance (NMR) Optical photon computer Solid-state
- 43.Optical photon computerOne method of this type uses the interactionbetween an atom and photon in a resonator, andanother uses optical devices such as a beamsplitter, mirror, etc.
- 44.NMR NMR usesthe spin of an atomic nucleus to represent aqubit. Chemical bonds between spins are manipulated by amagnetic field to simulate gates. Spins are prepared by magnetising, and induced voltagesare used for measurement. Currently it is thought that NMR will not scale to more than about twenty qubits. In 2006, the researchers reached a 12-coherence state anddecoded it using liquid state nuclear magnetic resonancequantum information processors.
- 45.Ion Traps Thismethod uses two electron orbits of an ion(charged atom) trapped within an electromagneticfield in a vacuum to form a qubit (ion trapmethod).
- 46.Solid-state deviceThere aretwo well-known qubits of this type.1. A qubit achieved by a superconducting circuitusing a Josephson junction that creates a weakbond between two superconductors.2. A qubit achieved by a semiconductor quantumdot, which is a structure from 10 to severalhundred nanometers in size for confining anelectron.
- 47.Quantum Computer LanguagesEventhough no quantum computer has been built that hasn’t stoppedthe proliferation of papers on various aspects of the subject. Many suchpapers have been written defining language specifications. QCL - (Bernhard ¨ Omer) C like syntax and very complete.http://tph.tuwien.ac.at/»oemer/qcl.html . qGCL - (Paolo Zuliani and others)http://web.comlab.ox.ac.uk/oucl/work/paolo.zuliani/ Quantum C - (Stephen Blaha) Currently just a specification,
- 48.References “A surveyof quantum computing and automata”. E. de Doncker andL. Cucos, In Fourth World Multiconference on Systemics, Cybernetics,and Informatics (SCI'00), (2000). “The Temple of Quantum Computing”, Riley T. Perry.2004 “Quantum Computation:A Computer Science Perspective”, AndersK.H. Bengtsson. 2005 http://en.wikipedia.org/wiki/Quantum_computing http://www.nec.co.jp/rd/Eng/innovative/E3/top.html http://www.sciencedaily.com/
- 49.Q & AThankYou