Light and Matter

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

2. A B C 0 0 1 A C 0 1 1 B 1 0 1 1 1 0 A B C 0 0 0 A C 0 1 1 B 1 0 1 1 1 0 A B C D A 0 0 0 0 D B 0 1 0 1 1 0 0 1 1 1 1 0 C Binary computing elements • any computer can be built from 2-bit logic gates • e.g. half-adder circuit • gates are not reversible: output does not define input NAND XOR carry sum HALF-ADDER

3. A B C 0 0 1 A C 0 1 1 B 1 0 1 1 1 0 A B C A A 0 0 0 0 A C 0 1 1 0 B 1 0 1 1 1 1 0 1 A A B C D A 0 0 0 0 0 D B 0 0 1 0 1 1 1 0 0 1 1 1 1 1 0 C Reversible binary computing elements • any computer can be built from 2-bit logic gates • e.g. half-adder circuit • gates are not reversible: output does not define input NAND • for reversible gates, additional outputs needed XOR carry sum HALF-ADDER

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

5. reversible logic does not change ; no energy consumed if change is slow • note that conventional logic gates consume Thermodynamics of computation • thermodynamic quantities are associated with any physical storage of information 0 1 • e.g. entropy • setting a binary bit reduces entropy by • hence energy consumption

6. 1 1 1 0 0 E D C B A 11 B • operations carried out as Rabi -pulses 10 01 A 00 Quantum computing • each data bit corresponds to a single quantum property • electronic or nuclear spin of atom or molecule • 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 • evolution described by Schrödinger’s equation CNOT

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

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

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

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