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Quantum Mechanics 103
Quantum Implications for Computing
Schrödinger and Uncertainty
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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 c
for a period of time
• During that time, there is a 50%
chance that one of the nuclei will
decay and trigger a Geiger
Counter
• If the Geiger Counter
triggers, a gun is
discharged and the cat is
killed
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Schrödinger’s Famous Cat
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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
Paradox?
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Paradoxical as it may seem, the concept of “superposition
of states” is borne out well in experiment
пЃ¶ Like superposition of waves producing
interference effects
Quantum Mechanics is one of the most-tested and bestverified theories of all time
But it seems counter-intuitive since we live in a
macroscopic world where uncertainty on the order of пЃЁ is
not noticeable
Quantum paradox #2
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Einstein-Podolsky-Rosen (EPR) paradox
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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
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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!
пЃ¶ Such an effect has been experimentally
verified, but it is still a topic of much debate
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Interference effects
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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
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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!
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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
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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 = ВЅ,
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!
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This allows interference effects
What wave function would
give 50% at each detector?
пЃ™ пЂЅ a TR пЂ« b RT пЂ« c RR пЂ« d TT
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Must have |a|2 = |b|2 = |c|2 = |d|2 = 1/4
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Need |a + b|2 = |c+d|2 = 1/2
пЃ™пЂЅ
1пЂ« i
2 2
TR пЂ«
1пЂ­ i
RT пЂ«
2 2
aпЂ«b
2
пЂЅ
1пЂ« i
2 2
2
2
пЂЅ
2
пЂЅ
2
2 2
4
пЂЅ
8
2 2
cпЂ«d
RR пЂ«
2
пЂЅ
4
8
1
2
пЂЅ
1
2
1пЂ­ i
2 2
TT
If Path Lengths Differ, Might
Have
пЃ™пЂЅ
1
TR пЂ«
пЂ­1
2
RT пЂ«
2
пЃ¶
Lower detector:
RR пЂ«
2 2
2
пЃ¶
1пЂ« i
пЃ™ пЂЅ
1
пЂ«
Upper detector: пЃ™ пЂЅ
2
1пЂ« i
2 2
TT
2 2
пЂ­1
2
1пЂ« i
2
пЂЅ0
2
пЂ«
1пЂ« i
2 2
Voila, Interference!
2
пЂЅ
2 пЂ« 2i
2 2
2
пЂЅ1
When Measure Which Path,
пЃ™пЂЅ
пЃ™пЂЅ
1
2
1
1
TR пЂ«
2
1
RR пЂ«
2
пЃ¶
TT
RT
2
Lower detector:
пЃ™
2
пЂЅ
1
2
пЂЅ
2
2
пЃ¶
Upper detector:
пЃ™
2
пЂЅ
1
2
1
2
пЂЅ
1
2
Voila, No Interference!
Quantum Storage
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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
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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
пЃ¶ Two pieces:
Read/write head in some internal state
 “Infinite” tape with series of 1s, 0s, or blanks
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Follows algorithms by performing 3 steps:
Read value of tape at head’s location
пЃ¶ Write some value based on internal state
and value read
пЃ¶ Move to next value on tape
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Can we improve this model?
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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
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Some problems are solvable in theory but
take too long in practice
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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
пЃ¶ multiplication grows as the square of digits
пЃ¶ Fourier transform grows faster than square
пЃ¶ factoring grows almost exponentially
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Examples of factoring time
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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 MIPyears
71-digit decimal number (1984) needs 0.1 MIPyears
129-digit decimal number (1994) needs 5000
MIP-years
Quantum Cryptography
Current best encryption uses public key for
encoding
пЃ¶ Need private key (factors of large integer in
public key) to decode
пЃ¶ Really safe unless
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Someone can access your private key
пЃ¶ Quantum computers become prevalent
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Quantum Cryptography II
Quantum Computers can factor large
numbers near-instantly, making public key
encryption passe
пЃ¶ But, can send quantum information and
know whether it has been intercepted
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What problems face QC?
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Decoherence: if measurement made, superposition
collapses
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Quantum error correction
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Even if measurement not intentional!
i.e., if box moves, cat becomes alive or dead, not both
No trail of path taken (or else no superposition)
Proven to be possible; that doesn’t mean it’s easy!
HUGE Technical challenges
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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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