Quantum Mechanics 103

Quantum Implications for Computing

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

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

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

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

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”

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

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

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

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

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

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

TTRRRTTR dcba

TT22

1RR

22

1RT

22

1TR

22

1 iiii

2

1

8

4

22

22

2 ba

2

1

8

4

22

22

2 dc

If Path Lengths Differ, Might Have

Lower detector:

Upper detector:

TT22

1RR

22

1RT

2

1TR

2

1 ii

02

1

2

12

2

122

22

22

1

22

122

2

iii

Voila, Interference!

When Measure Which Path,

Lower detector:

Upper detector:

TT2

1TR

2

1

2

1

2

12

2

Voila, No Interference!

RT2

1RR

2

1

2

1

2

12

2

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

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

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

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

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

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

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

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)