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1University of Strathclyde Quantum InformationStephen M. BarnettUniversity of StrathclydeThe Wolfson Foundation
2Probability and Information 2. Elements of Quantum Theory 3. Quantum Cryptography4. Generalized MeasurementsEntanglementQuantum Information Processing7. Quantum Computation8. Quantum Information Theory0 Motivation1 Digital electronics2 Quantum gates3 Principles of quantum computation4 Quantum algorthims5 Errors and decoherence6 Realizations?
3CMOS Device Performance Device performance doubles roughly every 5 years!
4P - solvable problems (computing time is polynomial in input size) ClassicalDeterministicAlgorithmFactoring Discrete logarithm Quantum simulations ...ClassicalProbabilisticAlgorithmQuantum Computing
5What happens to RSA? What happens to money?? Quantum algorithms: scaling of computing time with N~2n1. F.T. to determine periodicitiesf(x+r) mod N = f(x) mod Nfind rClassical: O(N) = O(2n)Quantum: O(log2N) = O(n2)2. Shor’s factoring algorithmN = pqfind p and q given NWhat happens to RSA? What happens to money??Exponential speed up!Naïve classical (trial): O(N1/2) = O(2n/2)Best known classical: O(2^[n1/3log2/3n])Shor’s algorithm: O(polynomial[logN]) = O(polynomial n)
6What is a computation?Generation of an output number (string of bits) based on aninput number.“Black Box” orComputer… …OutputInput… …How does the computer achieve this?
76.1 Digital electronicsPhysical bit - electrical voltage+5V = 10V = 0Single bit operationNOT gate1
9Two bit operationsNAND gate1Not all the gates are neededA small set of gates (e.g. NAND, NOT) is universal in that any logical operation can be made from them.
106.2 Quantum gatesSingle qubit operationsHHadamardSPhaseTp / 8and many more
11Two qubit operations - CNOT gate control bittarget bitCNOT gate can make entangled states
12We can break up any multi-qubit unitray transformation into a sequence of two-state transformations:
13It follows that we can realise any multi-qubit transformation as a sequence of single-qubit and two-qubit unitary transformations. This is the analogue of the universality of NAND and NOT gates in digital electronics.
14The CNOT gate, together with one qubit gates are universal control bit 1target bitcontrol bit 2Exercise:Construct the Toffoli gate using just CNOT gates and single qubitgates. Try to use as few gates as possible.
20Parallel quantum computation Can input a superposition of many possible bit strings a.Output is an entangled stated with values of f (a) computed for each a.
21Deutsch’s algorithm A f (A) Black Box A black box that computes one of four possible one-bit functions:Constant functions:Balanced functions:We wish to know if the function is constant or balanced. We can do this by performing two computations To give f (0) and f (1) .Can we do it in one step?
22A quantum computer allows solution in a single run: + for constant- for balancedare orthogonal states and so canbe identified without error.
23Exponential speed up. Exponential speed up Constant: 0 or 1 Suppose our box computes a one bit function of n bits and that this function is either constant or balanced.Constant: 0 or 1independent of inputBalanced: 0 or 1 for exactlyhalf of the possible inputsGuaranteed classical solution incomputationsQuantum?Orthogonal states for constant or balanced functions so solution in ONE computation.Exponential speed up.
246.4 Quantum algorithms1. F.T. to determine periodicitiesf(x+r) mod N = f(x) mod NClassical: O(N) = O(2n)Quantum: O(log2N) = O(n2)find r2. Shor’s factoring algorithmN = pqfind p and q given NNaïve classical (trial): O(N1/2) = O(2n/2)Best known classical: O(2^[n1/3log2/3n])Shor’s algorithm: O(polynomial[logN]) = O(polynomial n)3. Grover’s search algorithm - searching a databaseClassical: O(N)Quantum: O(N1/2)
25Factorisation algorithm Example: N = 15, m = 2=> FN(0) = 1FN(1) = 2FN(2) = 4FN(3) = 8FN(4) = 1FN(5) = 2…=> r = 4=> mr/2 – 1 = 3mr/2 + 1 = 5Both OKExample: N = 15, m = 11=> FN(0) = 1FN(1) = 11FN(2) = 1FN(3) = 11…=> r = 2=> mr/2 – 1 = 10 => GCD 5mr/2 + 1 = 12 => GCD 3Both OKN: Given big integer to be factorisedm: Small integer chosen at randomn = 0,1,2, …1. Make the series FN(n) = mn mod N2. Find the period r : FN(n+r) = FN(n)3. The greatest common divisor of N and mr/2±1 divides N
26Shor’s algorithm to factorise N 1. Find integers q and M such that: q = 2M > N2 and prepare two registers each containing M qubits.2. Set the qubits in the first register in the state (|0> + |1>)/21/2 and those in the second in the state |0>.
273. Choose an integer m at random and entangle the two registers so that This can be achieved by a unitary transformation (on a suitably programmed quantum computer) within polynomial time.4. Fourier transform for register 1:5. Measurement on register 1:=> k = multiple of q/r is obtained with high probability=> r = q/k
28Phase error Bit flip error 6.5 Errors and decoherence Interaction with the environment introduces noise and causes errorsPhase errorBit flip error
29Deutsch’s algorithmPhase errorIn this caseBit flip errorIn this case
30Scaling requires about 300 qubits. This gives Probability that a given qubit has no error in time tProbability that none of n qubits has an error in time tLet t be the time taken to perform a gate operation. For an efficient algorithm we might need n2 operations.The number of required gate operations tends to grow at least logarithmically in the nrequires about 300 qubits. This givesDecoherence is a real problem. We need efficient error correction!
31Quantum error-correction An error can make any change to a state so it is not obvious that error-correction is possible.The key idea, of course, is redundancy!This is a simultaneous eigenstate ofwith eigenvalue +1 in both cases.
33“OK, I’m convinced. Where can I buy one?” We can, in fact correct any single-qubit error using the 7-qubit Steane code:“OK, I’m convinced. Where can I buy one?”All the states differ in least four qubits – they are also common eigenstates of 6 operators with eigenvalue +1.Any single-qubit error is detectable from a unique pattern of changes to these.
34Ion-trap implementation - Cirac & Zoller, Wineland et al, Blatt et al. Single ion qubits coupled by their centre of mass motion
35Centre of mass motion acts as a ‘bus’ We can entangle theionic qubits using thecentre of mass motion.
39DiVincenzo’s criteria for implementing a quantum computer Well defined extendible qubit array - stable memoryPreparable in the “000…” stateLong decoherence time (>104 operation time)Universal set of gate operationsSingle-quantum measurementsD. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/ ; “The Physical Implementation of Quantum Computation,” quant-ph/
40SummaryQuantum information is radically different to its classical counterpart. This is because the superposition principle allows for many possible states.Our inability to measure every property we might like leads to information security, but generalised measurements allow more possibilities than the more familiar von Neumann measurements.Entanglement is the quintessential quantum property. It allows us to teleport quantum information AND it underlies the speed-up of quantum algorithms.Quantum information technology will radically change all information processing and much else besides!