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Light and Matter Tim Freegarde School of Physics & Astronomy University of Southampton Quantum computation

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2 Binary computing elements e.g. half-adder circuit any computer can be built from 2-bit logic gates ABC 001 011 101 110 A B C NAND A B D C ABCD 0000 0101 1001 1110 ABC 000 011 101 110 XOR A B C carry sum gates are not reversible: output does not define input HALF-ADDER

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3 Reversible binary computing elements e.g. half-adder circuit any computer can be built from 2-bit logic gates ABC 001 011 101 110 A B C NAND A B D C ABCD 0000 0101 1001 1110 ABC 000 011 101 110 XOR A B C A 0 0 1 1 A A 0 0 1 1 carry sum gates are not reversible: output does not define input HALF-ADDER for reversible gates, additional outputs needed

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4 Reversible binary computing elements e.g. half-adder circuit any computer can be built from 2-bit logic gates ABC 001 011 101 110 A B C NAND ABCD 0000 0101 1001 1110 ABC 000 011 101 110 XOR A B C A 0 0 1 1 A A 0 0 1 1 carry sum gates are not reversible: output does not define input A B D A C 0 B A 0 D A C HALF-ADDER for reversible gates, additional outputs needed CNOT CCNOT (Toffoli)

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5 Thermodynamics of computation e.g. entropy thermodynamic quantities are associated with any physical storage of information 01 setting a binary bit reduces entropy by hence energy consumption reversible logic does not change ; no energy consumed if change is slow note that conventional logic gates consume

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6 Quantum computing electronic or nuclear spin of atom or molecule each data bit corresponds to a single quantum property electronic state of atom or molecule polarization state of single photon vibrational or rotational quantum number e.g. electron spins in magnetic field gradient electromagnetic interactions between trapped ions lift degeneracies in radiative transitions 11 10 01 00 CNOT 0 0 1 1 1 E DCBA B A evolution described by Schrödinger’s equation operations carried out as Rabi -pulses

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7 Quantum computing tiny, reversible quantum bits (qubits) for small, fast, low power computers complex wavefunctions may be superposed: 11 10 01 00 CNOT 0 0 1 1 1 E DCBA B A parallel processing: result is classical read-out: probabilistic results limited algorithms: factorization (encryption security) parallel searches (data processing)

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8 Quantum computing extension of computing from real, binary numbers to complex, continuous values extension of error-correction algorithms from digital computers to analogue computers 11 10 01 00 CNOT 0 0 1 1 1 E DCBA B A link between numerical and physical manipulation extension of quantum mechanics to increasingly complex ensembles is quantum mechanics part of computation, or computation part of quantum mechanics? statistical properties (the measurement problem)

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9 observedescribeunderstandpredictexploit quantum optics quantum mechanics Quantum information processing classical mechanics Kepler 1571 Newton 1642 Galileo 1564 H G Wells 1866 A C Clarke 1917 Planck 1858 Einstein 1879 Townes 1915 Schawlow 1921 Fraunhofer 1787 Balmer 1825 Compton 1892 Hertz 1887 De Broglie 1892 Schrödinger 1887Heisenberg 1901Feynman 1918

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10 Further reading R P Feynman, Feynman Lectures on Computation, Addison-Wesley (1996) A Turing, Proc Lond Math Soc ser 2 442 230 (1936) C H Bennett, P A Benioff, T J Toffoli, C E Shannon D Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proc Roy Soc Lond A 400 97 (1985) www.qubit.org D P DiVicenzo, “Two-bit gates are universal for quantum computation,” Phys Rev A 51 1015 (1995)

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Logic Gates & Circuits. AND Gate Input AInput BOutput X 000 010 100 111 AND Logic Gate AND Truth Table X = A. B AND Boolean Expression.

Logic Gates & Circuits. AND Gate Input AInput BOutput X 000 010 100 111 AND Logic Gate AND Truth Table X = A. B AND Boolean Expression.

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