Presentation on theme: "Quantum Computing and Qbit Cryptography"— Presentation transcript:
1 Quantum Computing and Qbit Cryptography Patrick Lii5 May 2009Physics 138
2 Outline Motivation for Quantum Computing A Review of Classical ComputersQbits and Quantum AlgorithmsQuantum CryptographyConclusionGoal of this section is to answer the question: why do we need quantum computers?
3 What is a Quantum Computer? A quantum computer (QC) is a computational device which operates on data using quantum algorithmsQC in proof-of-concept stageCurrent motivations:CryptographyFactorizationDatabase searchingyour laptop operates under all the laws of quantum mechanics, but that does not make it a quantum computerA quantum computer relies on quantum algorithmsThat is it employs purely quantum mechanical phenomena (such as superposition of states and entanglement) to process dataPhysicists at IBM have created 7-qbit computers and gotten them to perform tasks, but this is a far cry from our 32 and 64-bit computers; so it’s mostly proof of conceptMuch of the motivation for QC is based on the idea of exotic technologiesQuantum cryptography – a theoretically unbreakable cryptography methodShor’s algorithm – used in prime factorization of large-value integersGrover’s algorithm – efficient searching of a large, unsorted databaseQuantum teleportation –replicate the state of entangled particlesGuess and check – password crackers; solutions of differential equationsLarge-value problemsGovernmental interest in quantum computing was re-ignited in 1994 by Peter Shor who proposed a quantum algorithm which could be used to efficiently factor large numbers -> this has application in defeating all sorts of encryptions (specifically RSA)
4 Classical versus Quantum Computers Example: Large number factorizationQCs ->advantage of parallelismqbits are in superpositions of states‘backwards compatible’ w/ classical algorithmsIn order to compare CCs and QCs, we use the example of large-value factorizationthis is one of the major problems in computationClassical computers (CCs) factor large numbers w/ brute forceQCs technically do the same thing, however, they can do it in parallelbecause qbits are superpositions of states, as opposed to classical bits which are limited to 0, 1a single thread in a quantum processor is equivalent to hundreds of threads in a classical processorQCs are backwards compatible w/ classical algorithmsWe can use qbit gates to emulate classical gates s.t. classical algorithms can run on a QC
5 Performance Advantage of QCs classical: ~1-10 gflops of computing powerquantum: ~10 tflopsFactorization speed:for an integer N with size:the factorization time of a classical comp is:For a QCClassical 32bit computers in our homes have 1-10 gflops (manufacturers claim, gflops, but that’s not true)An eqivalent 32-qbit quantum computer has an estimated ~10 teraflops (2-3 order of mag improvement)a flop stands for FLOating Point Operations per Second – simply a measure of processing speed/capacityAside: fastest computer in the world today: 1.64 petaflops at Oak Ridge National Lab (1.64 x 10^15 floating point operations per second!!)Going back to factorization examplefor an integer N w/ size n = log2Nclassical: scales like 2^sqrt(n)this is called super-polynomial timequantum: scales like n^3polynomial time (for obvious reasons)there is also exponential timeas problem size scales linearly, computation time scales polynomially or exponentially etcas such polynomial time is the fastest, exponential time is the slowestin these eqns, A and B are just scaling factorsSo why do we need quantum computers?for certain applicationsORNL’s Jaguar Supercomputer
6 Speed ComparisonAssume CC and QC can factor a 78 digit number (n = 256) in 1 hourClassical ComputerQuantum Computern = 256 (AES)n = 512n = 1024n = 2048 (RSA)1 hr4.11 days7.47 years~73000 years1 hr8 hrs2.76 days21.3 daysSuppose we have a QC and CC that can both factor a 78 digit number (size = 256) in 1 hourcontemporary supercomputers are capable of thisThat same classical computer would take 4.11 days to factor a size = 512 number, and over 73,000 years to factor a n = 2048 numberhowever, as you can see, the quantum computer can do it in just 8hrs and 21.3 days (over 10-million-fold improvement)The RSA encryption is based on the idea that it is too computationally expensive to break the encryption; but a quantum computer can defeat it with relative easeas you can imagine, this has generated a lot of interest from the governmentQuestions so far?QC easily defeats RSA encryption!
7 Outline Motivation for Quantum Computing A Review of Classical ComputersQbits and Quantum AlgorithmsQuantum CryptographyConclusionIn this section I’m going to very quickly talk about classical computing (try to go through this quickly)
8 The Classical Computer Classical bits (cbits): 0 or 12 cbits 4 states,3 cbits 8 statesn cbits 2n statesdata is represented in binary138 q Classical bits (cbits) can be in one of two states: 0 or 1defined by bit states on the hard drive platter or voltages in the transistors of a processorpairs of cbits can have 4 states total: 00, 01, 10, or 11in general, when you have n cbits, you are limited to 2^n statesData on a classical computer is represented in binary code (strings of bits)All just review
9 Classical Operations Based on logic gates Example: 1-bit gate AND GateBit 1Bit 2Output1Based on logic gatesExample: 1-bit gateNOT gateX:0 1X:1 02-bit gates:AND/NANDOR/NORXOR/XNORAll classical operations are based on logic gates; input bits and returns a bitand we have operations that take 1-bit, 2-bit, 3-bit, and onExample of 1-bit operationNOT gate: simply flips state of bit; 0 goes to 1, 1 goes to 22-bit operationsAND: returns single bit with value of 1 iff both inputs are 1 (NAND does just the opposite, 1 as long as both inputs are not 1)OR: if either are 1NOR: no inputs are 1XOR: exclusive or; returns 1 iff one input is 0 and the other is 13-bit operationsToffoli and Fredkin gatesMore higher-order operationsXOR GateBit 1Bit 2Output1
10 Classical AlgorithmsAll 1, 2, 3-cbit gates together form universal setclassical algorithm: a complex operation that uses a sequence of classical gatesTogether all 1, 2 and 3-cbit gates form a universal setyou can create (or at least approximate) every other higher-order gate using only 1, 2, and 3 bit gatesA classical algorithm is a complex operation which utilizes a sequence of classical gates to perform a complex taskQuestions?
11 Outline Motivation for Quantum Computing A Review of Classical ComputersQbits and Quantum AlgorithmsQuantum CryptographyConclusionNow we get into quantum computing
12 The Quantum Bit (Qbit)Unlike cbits, state of a qbit is a superposition of 1 and 0:w/ normalization condition:In matrix form:n qbits are in superposition of 2n statesqbits can be any two-level quantum systemLike classical computers, quantum computers utilize bits to store datahowever, unlike a cbit which can only be in the 0 or 1 state, a quantum bit (qbit) can be in a superposition of both the 0 and 1 statedescribed by this wavefunction with the usual normalization conditionscan be described by this matrix (this is all the usual QM notation)If 1 qbit is in a superposition of 2 states (0 and 1) then n qbits are in a superposition of 2^n states (this is analogous to cbits)In principle, qbits can be any two level quantum system w/ orthogonal basesup/down spin of an electron1st and 2nd energy lvls of an atomx, y polarization of light
13 The Quantum Bit In general, for n bits: w/ normalization: In general we can represent the wavefunction of a qbit like this
14 Qbit Entanglement Purely quantum property of qbit Two qbits are entangled if wavefunction cannot be written as product of 1 qbit statesOne of the intersting aspects of qbits is that they have the unique property of quantum entanglementand this is a purely quantum property (exclusive to quantum computers)Two qbits can be said to be entangled if their combined wave function cannot be written as a product of two one particle statesyou cannot measure state of one of the entangled qbits without implicitly measuring the otherWe can design 2 qbit gates that can act on entangled pairs as well as nonentangled pairsas opposed to a 2-bit classical gate which simply takes two bits which are in no way interactingYou can also form a two qbit wavefuncton by multiplying two 1-qbit states together|Ψ⟩=α_0 β_0 |00⟩+ α_0 β_1 |01⟩+α_1 β_0 |10⟩+α_1 β_1 |11⟩But these are not entangled
15 Quantum Logic Gates All quantum operations are unitary UU† = U†U = 1Gate can be any unitary quantum operatorEx: quantum NOT gate2-bit gates can operate on entangled pairsUnlike a classical gate which takes an input and returns an output; a quantum gate manipulates the state of the qbitAll quantum gates are unitary: preserves normality of the qbitthey are all linear operators in the Hilbert space of the systemGate simply modifies amplitude of each statein principle, a quantum gate can be any unitary quantum operatorIn this example, the quantum not gate simply flips the 0 state into a 1 stateQuantum logic gate using lasers
16 Important Quantum Gates CNOT GateBit 1 InBit 2 InBit 1 OutBit 2 Out1Conditional NotHadamard Transformationπ/8 Phase GateCNOT – conditional not:classical analog is the exclusive or (XOR) gate shown earlierexclusively quantum gate because it utilizes some degree of entanglement between the two input qbitsdifference w/ classical XOR gate: input 2 AND returns 2 bitsflips the second bit iff the first bit is onHadamard Transformationhadamard matrices are a fascinating problem in math, generalized fourier transformsbut for QC, they map the qbit basis states into equally weighted superpositionsit is also self inversing (applying it to RHS of eqns returns 0 and 1 states)preserves amplitudephysical example: if 0 and 1 are represented by orthogonal x, y polarization states of lightHadamard transform would be an optical rotator which rotates the x polarized light to 45 degree polarized lightpretty easily implementedPhase Gateused to modify phase of 1 state; preserves normality because it only modifies phaseclick3 gates form a universal setany higher order n-qbit gate can be approximated to any arbitrary accuracy using only these 3 gatesThese 3 gates form a universal set
17 The Measurement Gate Born’s Law M gate Most important gate in QC Given qbit:Probability of measuring state = amplitude squaredM gate Most important gate in QCCollapses qbit wavefunctionResult based on probabilityWe may not always get “correct” answerIrreversible!From QM, we know that for a given superposition of states, probability of measuring one of those states is given by the square of the amplitude coefficientthis is just Born’s LawWe must have some way of extracting information from our qbits -> m gate fulfills this purposeAs such, the measurement gate is the most important gate in quantum computingit extracts information from the qbit by collapsing it’s wavefunction and taking a measurement of its stateWe can see the measurement gate in this quantum circuit diagraminitially, the uncollapsed state contains amplitude information for all the orthogonal basis stateshowever, once we measure it, it collapses to a single state (w/ probability proportional to the square of the amplitude) and we lose all that informationUnlike a cbit, where we just measure the state w/o changing anything, collapsing the qbit wavefunction alters italso means that measuring the qbit is an irreversible processonce we measure it, we destroy all other amplitude information it might containa bit problematic as this means that we cannot recheck our calculationExample of a measurement gate is some sort of laser or photon probe
18 Quantum AlgorithmsSimilarly to classical algorithms, quantum algorithms are sequences of quantum gatesIn general, QCs have a simple processing structure:Complex processing lies in the U GatesLike a classical algorithm, a quantum algorithm simply uses a sequence of quantum gates to perform a task or complex operationYou can break down the processing strucutre of a QC into a simple diagramqbits are prepared and sent through multiple logical gates which alter the amplitudes of each statethen they’re simply read out by an M gateHowever, this picture is misleading as all of the complex processing is done in the U gates stage
19 Shor’s Algorithm Developed by Peter Shor in 1994 Efficient factorization of large numbersRSA EncryptionBased on multiplying 2 very large prime numbers (~200 digits each)CCs cannot factor this in a reasonable timeHowever, using Shor’s algorithm, a QC canLots of interest from governmentShor’s algorithm was developed by Peter Shor in 1994—then a researcher at AT&T’s Bell Labs, now a math professor at MITI’m not going to go too into depth on this as it’s a bit too involved for a talk (more like a class)Essentially, it is a proof-of-concept algorithm which allows a quantum computer to quickly and efficiently factor large integers into their prime factorsthe speeds that I quoted at the beginning of this talk were based on processing times using this algorithmShor’s algorithm employs quantum fourier transform and a periodicity algorithmRSA encryption is based on multiplying two very large prime numbers together (like 200 digits each) to form an encryption keythe actual encryption process is a little more complicated than thata classical computer cannot factor this ‘key’ in any reasonable amount of time (at beginning we found years)however, with Shor’s algorithm, a quantum computer canthis has generated a lot of interest from the governmentIf you want to learn more, check our Mermin’s book whole chapter describing this
20 Physical Implementations of QCs In 2001, a group at IBM led by Vandersypen created a 7-qbit QCNMR implementationUsed it to demonstrate Shor’s algorithm by factoring 15 into 3 and 5Other possibilitiesOptical latticesPolarized lightDiamond basedSuperconductor (SQUIDs)Trapped ionany two level system w/ orthogonal basesI’ve already talked a little about physical implementations of QCs with photons as qbitsin 2001, a group at IBM created a 7-qbit quantum computer using an NMR implemenationqbits were just the atoms in the magnetic field; energy splitting allowed them to be orthogonalused it to demonstrate Shor’s algorithmOther possibilities:optical lattices isolate each atom and utilize energy levels as qbitsdiamond based quantum computers crystalline carbon w/ nitrogen vacancies are stable (long coherence time) and are dense (many qbits)in general, we can create a quantum computer from any two level system with orthogonal basesThe biggest problem in quantum computing is controlling the decoherence time of the qbitsif the qbits interact with anything other the our intended quantum gates, our results become uselessqbits must be shielded from external fields, other particles, photonsThe fundamental problem for quantum engineers trying to implement a QC:we need quantum bits that can be poked and prodded into performing calculations in concert with other qubitsyet robust enough to maintain their states over long periods of timeQuestions?Biggest problem in implementation of QC: controlling decoherence of qbits
21 Outline Motivation for Quantum Computing A Review of Classical ComputersQbits and Quantum AlgorithmsQuantum CryptographyConclusionNow we’ll talk a little bit about quantum cryptography schemes and using qbits to securely transfer information
22 Quantum Cryptography (BB84) Called BB84: Bennett and Brassard 1984Method of secure key distributionCreated using only 1-qbit gatesCan be implemented using current tech (transmission w/ polarized light)interception can be detectedBB84 is a quantum key distribution scheme proposed by Bennett and Brassard in 1984Bascially allows for perfectly secure message transmissionsgenerates a one time use key using 1-qbit gates and use it to encode a messagecan be implemented using polarized lightand one huge advantage of this is that any interception of the key transmission can be detected
23 Message Security Say we want to transmit the number 83 In binary: (7-bits)We securely (and randomly) generate a key w/ equal bit-lengthtake:We then use this key to encode the message“flip” message bits everywhere the key equals 1Message becomesimpossible for someone w/o a key to unencrypt thisCryptography comes down to:Random key generationSecure key distributionSay we want to transmit the number 83in binary this is a 7-bit numberWe securely and randomly generate a key which has an equal number of bits as the messageWe then use this key to encode our message by flipping the message bits everywhere the key equals 1this is equivalent to applying the CNOT or XOR gates to the two bit stringsSince this is not pattern based and we only use the key once before discarding it, you cannot unencrypt it without a keyso the problem of quantum cryptography comes down to randomly generating and securely distributing a keythe BB84 method handles both of these simultaneously
24 Key Generation I −|<−<|<− Alice sends Bob a long stream of photons (qbits)She randomly assigns each a type: circular or linear polThen, randomly assigns a polarization sub-state based on the typeLH or RH for circX or Y for linearExample: Alice sends 8 qbits−|<−<|<−Legend:— X, 0 bit| Y, 1 bit> RHCP, 0 bit< LHCP, 1 bitKey distribution is at the crux of quantum cryptography—Let’s say Alice (who is an agent at the Pentagon) wishes to send Bob at Interpol headquarters a secure message.in order to do so, she and Bob must agree on a secure key w/o revealing it on a classical channel where it can be eavesdropped onwe can assume that they have contact over the phoneAlice and Bob decide to generate/distribute the key with photons transmitted over a fiber optic cableso Alice sends Bob a long stream of photonsfor each photon she randomly assigns a type: either circularly or linearly polarizedthen for each circular or linear type, she randomly assigns a polarization state (X, Y for linear, LH RH for circ) and records the pol state for each photon before transmitting it outthey have agreed on what each polarization state represents X0, Y1, LH0, et ceteraSo in this example, Alice sends 8 randomly polarized qbits with these polarizations
25 Key Generation II O+O++OOO And he measures: <|<−−><> Bob randomly decides on a linear or circular measurement of each incoming photonFor measurement, Bob chooses:O+O++OOOLegend:+ LinearO CircularAnd he measures:<|<−−><>for reference, Alice sent: −|<−<|<−At the other end of the fiber optic cable, Bob takes each photon and randomly puts it through either a linear polarizer or a circular analyzerat this point, Bob has no idea know which of his choices agree with Alice’s—for all he knows, his measurements are completely meaninglessIn our example, Bob randomly chooses:And measures:
26 Key Generation IIIBob calls Alice and tells her his choice of measurement (circ or lin) for each photonAlice then tells Bob which of his types agree with her transmission typesNYYYNNYNThey then use the agreeing values as a keyIn example, A&B have 4 agreeing qbits: |<-<Their key is: 1101Bob then calls Alice on an insecure phone and tells her which type of measurement he used to measure each incoming photonhe does not reveal the values he measured, just what type of measurement he usedAlice looks at her notepad and compares Bob’s choices with the types that she choseshe then tells Bob which of his measurement types agree with the type she used for transmissionin the example, we see that he got 4 right (which is what we expect because he is randomly choosing the measurement type)They both throw out the ones Bob didn’t get the right type for (about half of the transmitted qbits) and use the rest as a keysince they’ve already agreed on what each polarization means, they can get a key from itSo now they both have a randomly generated key; this satisfies the first condition of cryptographynow we just have to worry about secure transmission and their key transmission being interceptedQuestions?
27 Eavesdroppingfor ‘Eve’ to eavesdrop on A&B’s transmission, she must also randomly make circ or lin measurements of each photonThis changes polarization of about half the qbits1/4th of Bob’s result will not agree w/ Alice’s prepA&B can compare some ‘check bits’ over the phone to see if anyone is eavesdroppingWe can assume Alice and Bob are in secure facilities—once they get the key, it won’t leakthe only way for someone to get the key is to eavesdrop on or intercept the transmissionthey have to do it in a way s.t. A&B don’t find outIn order for a third party (who we’ll call Eve) to eavesdrop on Alice and Bob’s key transmission, Eve must also randomly make circular or linear measurements of each photon that Alice sends through the fibersince this is random, this changes the polarization state of about half of the photonsThis means that about 1/4th of the qbits which Alice and Bob measure the same way will not agree with each otherthey can find out if anyone is eavesdropping by sacrificing some of the bits in their key and comparing them over the phone
28 Qbit Interception Suppose Eve uses a more sophisticated attack: intercepts the transmissionprocesses it in a QCrestores it to original state and sends it back off to BobThis is defeated by the no-cloning theoremForbids creation of identical copy of an arbitrary stateEve gets no useful information from her interceptionSuppose Eve uses a more sophisticated attack in which she intercepts the transmission and puts it through a quantum computer to get information on the keyand then she restores the original state and sends it back off to BobclickWe’ll this doesn’t work. It turns out that the requirement for Alice’s qbit to be returned to its initial state prevents thisthe no cloning theorem which forbids the creation of an identical copy of an arbitrary state makes it impossible for Eve to extract useful information from the qbitsSo the BB84 scheme can neither be eavesdropped on or interceptedthis takes care of our second cryptographic requirement that it is not only a random key generator, but also a secure distribution methodIn theory, cannot be defeated—this is a very powerful application of qbitsA&B’s whole setup—sending device, fiber optic cable, and receiving device is really just one large quantum computerQuestions?
29 Outline Motivation for Quantum Computing A Review of Classical ComputersQbits and Quantum AlgorithmsQuantum CryptographyConclusionFinally have come to the conclusion
30 Future Developments in QC Largely in proof-of-concept stageformidable technological obstaclesStill need to:discover more algorithmsovercome decoherence of qbitsDeeper understanding of QM may make it easier to do thisWe are decades away from truly powerful QC (~2050?)Quantum computers are still largely in the proof of concept stagePhysicists and engineers must overcome formidable technological obstacles before quantum computers can be on par with classical computershowever, once they do, we will literally be in a new era of computing we don’t yet have the understanding of what potential QCs haveWe still need to work on finding algorithms which provide performance gains over CCsand we need to design physical implementations which can prevent the decoherence of quantum bits while allowing manipulation of the qbitsIf we can attain a deeper understanding of QM phenomena (especially measurement) we may be able to more quickly achieve these goalsAs of right now, we are still decades away from a quantum computer that’s on par with my laptop
31 ConclusionQuantum computers are based on enacting quantum operations on qbitsQuantum operations are simply unitary operators in the Hilbert space of the systemQCs have the potential to vastly outperform classical computers because of the QM nature of their operationsQCs are still many years off; however, they will fundamentally change computation as we know itQbits can also be employed in generating an undefeatable cryptography scheme which may prove useful once RSA encryption is defeated by QCs
32 ReferencesQuantum Computing (General) Kaye, Phillip, Raymond Laflamme, and Michele Mosca. An Introduction to Quantum Computing. 1st ed. Oxford: Oxford University Press, Print. Lieven M.K. Vandersypen et al. (1999). "Separability of Very Noisy Mixed States and Implications for NMR Quantum Computing". Phys. Rev. Lett 83: 1054–1057. Mermin, David. Quantum Computer Science. 1st Ed. Cambridge: Cambridge University Press, Print. [great introductory resource for Quantum Computers from a professor at Cornell, not rigorous however] Classical Computing [cool logic gate simulator] Quantum Cryptography C. H. Bennet and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing”, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, p. 175 (1984) [BB84 transmission simulator] Shor’s Algorithm Shor, P. (1994) Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, Santa Fe, NM, Nov , 1994.
33 Since qbit must emerge in original state: Let:|Фμ>, μ = 0, …, 3 = four states of Alice’s qbits (X, Y, RH, LH)|ψ> = initial state of qbits on Eve’s QCSince qbit must emerge in original state:Eve must find a U that yields four distinct |ψμ>