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Quantum Computation: Introduction, Computation, Decoherence, and Applications

Explore the realm of Quantum Computation in this comprehensive guide covering qubits, gates, error correction, algorithms like Deutsch's and Grover's, and the potential impact on RSA encryption.

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Quantum Computation: Introduction, Computation, Decoherence, and Applications

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  1. Quantum Computation Jonathan Coslovsky March 2008

  2. Outline • Introduction • The qubit • Calculation (gates) • Decoherence and error correction • Applications • Experimental implementations • Experimental progress timescale • Summery

  3. Introduction:The Classical Computer • Silicon microprocessor chip – Since 1960s. The IBM System 360/20 1966

  4. Introduction:The Classical Computer • Moore’s law – The number of transistors placed on an integrated circuit is increasing exponentially.

  5. Introduction:The Problem-NP tasks • Polynomial computing tasks – P. • Non-Polynomial computing tasks-NP. • Example: Factorization: 15 3x5 91 7x13 703 ?x? 19x37 8876044532898802067 ???????x??????? Answer: 5915587277x 1500450271 Why is this important? The principle of RSA.

  6. Introduction:Solutions • Solution #1: wait for a few years for a better (classical) computer. • Problem: Today: L< 1μm. (L-Transistor size) Quantum effects become important: L~λ~nm. (λ-de Broglie wavelength of electrons). Individual atom size: L~a0~Ǻ. Moore’s law will eventually break down. • Solution #2: Find a new computing technique.

  7. Introduction:What is a Quantum Computer? • The idea: Take advantage of QM phenomena, such as superposition and entanglement. • Is it possible? 1994-Shor, P algorithm for factoring numbers.

  8. Quantum bits (qubits) • Classical bit: 0 or 1. • voltage on a transistor, magnetization of a ferromagnetic material, or intensity of a pulse of light. • Qubit: Superposition in a two state system Where . E |1> |0>

  9. Qubits:Quantum register • 3 qubits register: • Vector representation: • An N-qubit register is described by 2N (complex) amplitudes.

  10. Qubits:Candidates

  11. Qubit:Bloch sphere • A single qubit can be represented on a Bloch sphere:

  12. Calculation:Classical gates Single bit-NOT gate Two bits-C-NOT gate But what is the meaning in a qubit?

  13. Calculation:Single-qubit gates

  14. Calculation:Two-qubits gate • The controlled-NOT gate: • i.e.

  15. CalculationQuantum circuit-example Example: Step 1: Step 2:

  16. CalculationQuantum circuit-example • Any qubit gate can be formed by combining C-NOT gates with single-qubit operations. • Example:

  17. CalculationQuantum circuit-example • Toffoli gate:

  18. Calculation:Single-qubit manipulation • Every unitary transformation can be represented as a rotation over the Bloch Sphere. • Every single-qubit operator can be decomposed to: • For Example:

  19. Calculation:Single-qubit manipulation • Lets consider a 2-level atom: • Rotation is achieved by a pulse. • is set by the optical phase of the pulse. • Pulse area:

  20. Calculation:Two-qubit manipulation • Assume π-pulse at ωB’

  21. Calculation:Two-qubit manipulation • Control qubit – Excitation level of a cooled 9Be+ ion in a harmonic trap. • Target qubit – 2 hyperfine levels of the ion.

  22. Decoherence and error correction • Interactions with environment decoherence loss of information. • In order to increase decoherence time, typically low temperatures are used. • Quantum systems are fragile. We need quantum error correction algorithms. • Error rates are proportional to the ratio of operating time to decoherence time.

  23. Decoherence and error correction • Dephasing time: T2. • Number of operations:

  24. Applications:Deutsch’s algorithm • The idea: determine if a single bit function f(x) is constant or balanced. • Classical computer-2 calls. • Quantum computer-1 call.

  25. Applications:Deutsch’s algorithm

  26. Applications:Deutsch’s algorithm If f is constant: f(0)=f(1)

  27. Applications:Deutsch’s algorithm If f is balanced: f(1)=1⊕f(0)

  28. Applications:Grover’s algorithm • Quantum search algorithm. • For example: The London telephone directory: • Holmes, Sherlock 221b Baker Street 123 456 • It is easy to find Holmes’ telephone number, but, difficult to find the telephone 123 456. • Classical computer ~ NData/2 op. • Quantum computer ~ Sqrt(NData).

  29. Applications:Grover’s algorithm

  30. Applications:Grover’s algorithm

  31. Applications:Grover’s algorithm • Grover operator: • Oracle. • Hadamard gate. • CPS-conditional phase shift. • Hadamard. • Oracle: a gate that checks if we have the desired solution:

  32. Applications:Grover’s algorithm • The three other steps (H, CPS,H) perform an ‘inversion about the mean’. • Example: Ndata=4, N=2 qubits. • Suppose: target=10. • Applying oracle:

  33. Applications:Grover’s algorithm • Inversion about the mean: Mean This Algorithm can be used as a password cracker!

  34. Applications:Shor’s algorithm • Fourier transform operation: Classical computer (FFT) ~ n2n op. for N=2n numbers. Quantum Computer ~ n2 op. • But the result of QFT is stored as amplitudes, it can not be read. • But QC can find periodicity. • 1994-Peter Shor – can be used to factorize large numbers. • Is RSA encryption in danger?

  35. Applications:Is RSA encryption in immediate threat? • Largest prime number today (March 2008): 232582657-1. • In order to factorize a 1000 bit number, between 1012 and 1018 qubits are required. • Until today, systems of only a few qubits have been demonstrated. • RSA is probably safe for the next few decades.

  36. Applications:Simulation of quantum system • C.C. -> Memory required is exponential in system size. • N two-level system -> 2N amplitudes. • A relatively small molecule of 53 atoms requires 253~9x1015 bits~1 Petabyte=106 Gigabytes. • Q.C. -> N qubits required. • Unfortunately, on making the measurements, we would only obtain N bits of information.

  37. Applications:Quantum repeaters ALICE BOB L

  38. Experimental implementations • D. DiVincenzo: requirements for the system: • Scalable physically to increase the number of qubits . • Possible to prepare an initial state. • Decoherence time >> operation time. • Single- and two-qubit quantum gates must be demonstrated. • Possible to measure the state of each individual qubit.

  39. Experimental implementations:Scaling up by networks • An optional solution for the scaling up problem is using a network of small quantum computers, thus creating a larger one. • This uses “flying qubits”. Flying qubits B A

  40. Experimental progress timescale • 1995: Quantum C-NOT gate. • 1998: 2 and 3 qubit NMR QC. Execution of Grover’s algorithm. • 2000: 5 and 7 qubit NMR QC. Execution of order finding. • 2001: Execution of Shor’s algorithm (15 was factored). • 2005: Qubyte. • 2006: 12 qubit QC.

  41. Experimental progress timescale Is Moore’s law valid on a QC? QC of 1000 qubits expected on 2032. QC of 1 GigaQubyte only on 2127

  42. Outlook • Quantum computation is in some cases more efficient than classical computation. • QC is possible, at least in principle. • Small QC systems have already been demonstrated. • The main goal now is to scale up the systems.

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