Quantum Mechanics 103

1. Quantum Mechanics 103 Quantum Implications for Computing

2. Schrödinger and Uncertainty • Going back to Taylor’s experiment, we see that the wavefunction of the photon extends through both slits • Therefore the photon has “traveled” through both openings simultaneously • The wavefunction of a “particle” will contain every possible path the particle could take until the particle is “detected” by scattering or being absorbed • These paths can interfere with each other to produce diffraction-like probability patterns • BUT, Schrödinger took this explanation to an extreme

3. During that time, there is a 50% chance that one of the nuclei will decay and trigger a Geiger Counter Schrödinger’s Famous Cat • Suppose a radioactive substance is put in a box with a cat for a period of time • If the Geiger Counter triggers, a gun is discharged and the cat is killed

4. Schrödinger’s Famous Cat • Until an observer opens the box to make a “measurement” of the system, • The nucleus remains both decayed and undecayed • The Geiger counter remains both triggered and untriggered • The gun has both fired and not fired • The cat is both dead and alive Disclaimer: To be truly indeterministic, this experiment must be performed in a sound-proof room with no window

5. Paradox? • Paradoxical as it may seem, the concept of “superposition of states” is borne out well in experiment • Like superposition of waves producing interference effects • Quantum Mechanics is one of the most-tested and best-verified theories of all time • But it seems counter-intuitive since we live in a macroscopic world where uncertainty on the order of  is not noticeable

6. Quantum paradox #2 • Einstein-Podolsky-Rosen (EPR) paradox • Consider two electrons emitted from a system at rest; measurements must yield opposite spins if spin of the system does not change • We say that the electrons exist in an “entangled state”

7. More EPR • If measurement is not done, can have interference effect since each electron is superposition of both spin possibilities • But, measuring spin of one electron destroys interference effects for both it and the other electron; • It also determines the spin of the other electron • How does second electron “know” what its spin is and even that the spin has been determined

8. Interpreting EPR • Measuring one electron affects the other electron! • For the other electron to “know” about the measurement, a signal must be sent faster than the speed of light! • Such an effect has been experimentally verified, but it is still a topic of much debate

9. Interference effects • Remember this Mach-Zender Interferometer? • Can adjust paths so that light is split evenly between top U detector and lower D detector, all reaches U, or all reaches D – due to interference effects • Placing a detector (either bomb or non-destructive) on one of the paths means 50% goes to each detector ALL THE TIME

10. Interpretation • Wave theory does not explain why bomb detonates half the time • Particle probability theory does not explain why changing position of mirrors affects detection • Neither explains why presence of bomb destroys interference • Quantum theory explains both! • Amplitudes, not probabilities add - interference • Measurement yields probability, not amplitude - bomb detonates half the time • Once path determined, wavefunction reflects only that possibility - presence of bomb destroys interference

11. Quantum Theory meets Bomb • Four possible paths: RR and TT hit upper detector, TR and RT hit lower detector (R=reflected, T=transmitted) • Classically, 4 equally-likely paths, so prob of each is 1/4, so prob at each detector is 1/4 + 1/4 = ½, independent of path length difference • Quantum mechanically, square of amplitudes must each be 1/4 (prob for particular path), but amplitudes can be imaginary or complex! • This allows interference effects

12. What wave function would give 50% at each detector? • Must have |a|2 = |b|2 = |c|2 = |d|2 = 1/4 • Need |a + b|2 = |c+d|2 = 1/2

13. If Path Lengths Differ, Might Have • Lower detector: • Upper detector: Voila, Interference!

14. When Measure Which Path, • Lower detector: • Upper detector: Voila, No Interference!

15. Quantum Storage • Consider a quantum dot capacitor, with sides 1 nm in length and 0.010 microns between “plates” • How much energy required to place a single electron on those plates? • Can make confinement of dot dependent upon voltage • Lower the voltage, let an electron on –> 1 • Lower voltage on other side, let the electron off -> 0

16. What must a computer do? Deterministic Turing Machine still good model • Two pieces: • Read/write head in some internal state • “Infinite” tape with series of 1s, 0s, or blanks • Follows algorithms by performing 3 steps: • Read value of tape at head’s location • Write some value based on internal state and value read • Move to next value on tape

17. Can we improve this model? • Probabilistic Turing Machine sometimes better • Multiple choices for internal state change • Not 100% accurate, but accuracy increases with number of steps • Can solve some types of problems to sufficient accuracy much more quickly than deterministic TM can • Similar concept to Monte Carlo integration

18. Limits on Turing Machines • Some problems are solvable in theory but take too long in practice • e.g., factoring large numbers • Can label problems by how the number of steps to compute grows as the size of the numbers used grows • addition grows linearly • multiplication grows as the square of digits • Fourier transform grows faster than square • factoring grows almost exponentially

19. Examples of factoring time • MIP-year = 1 year of 1 million processes per second • Factoring 20-digit decimal number done in 1964, requiring only 0.000009 MIP-years • 45-digit decimal number (1974) needs 0.001 MIP-years • 71-digit decimal number (1984) needs 0.1 MIP-years • 129-digit decimal number (1994) needs 5000 MIP-years

20. Quantum Cryptography • Current best encryption uses public key for encoding • Need private key (factors of large integer in public key) to decode • Really safe unless • Someone can access your private key • Quantum computers become prevalent

21. Quantum Cryptography II • Quantum Computers can factor large numbers near-instantly, making public key encryption passe • But, can send quantum information and know whether it has been intercepted

22. What problems face QC? • Decoherence: if measurement made, superposition collapses • Even if measurement not intentional! • i.e., if box moves, cat becomes alive or dead, not both • Quantum error correction • No trail of path taken (or else no superposition) • Proven to be possible; that doesn’t mean it’s easy! • HUGE Technical challenges • electronic states in ion traps (slow, leakage) • photons in cavity (spontaneous emission) • nuclear spins in molecule (small signal in large noise)