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Quantum Mechanics 103 Quantum Implications for Computing

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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

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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 During that time, there is a 50% chance that one of the nuclei will decay and trigger a Geiger Counter

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Schrödinger’s Famous Cat Until an observer opens the box to make a “measurement” of the system, The nucleus remains both decayed The Geiger counter remains both triggered and undecayed 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

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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

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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”

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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

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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

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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

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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

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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

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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

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

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

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Quantum Storage Consider a quantum dot capacitor, with sides 1 nm in length and 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

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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

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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

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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

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

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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

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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

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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)

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