Beyond Binary: Quantum and Biologically-Inspired Computing
CS 301 Lecture, Dr. Lawlor, 2005/11/23
This pre-Thanksgiving break lecture is purely for your own edification--this material won't show up on the test.
Quantum computers seem like contrived nonsense until you understand the
background physics, so we'll start with a physics lesson.
The fact that light acts like a wave is well-known: interference phenomena are common, from the colors shining off a CD to oil drops on concrete to the double-slit experiment a peacock's tail to Newton's Rings. You can play with wave and interference phenomena online using Paul Falstad's Ripple Tank Applet. and Olympus Microscope's double-slit experiment.
The weirdest part about wave phenomena is destructive interference: two
out-of-phase waves can cancel each other out to give nothing (like in noise-cancelling headphones).
Destructive interference is what cancels out most colors in an oil
sheen, and what creates the dark spots in the double-slit experiment.
In 1924 Louis de Broglie (in a 9-page PhD thesis in physics) proposed that even matter acts as waves, and gives an equation for the wavelength. This bizarre idea is
now universally accepted--interference of electrons was demonstrated
experimentally three years later by Davisson, Germer, and Thomson. 1926 Erwin Schrödinger invented the "Schrödinger Equation"
to describe how wavefunctions for light and matter change as time progresses.
This is still the defining equation in quantum mechanics.
This was a very curious time, because Millikan's oil-drop experiments
(in 1909-1913) had shown very clearly that electrons were particles,
not waves--he'd even measured the mass and charge of an electron.
Einstein had shown in 1905 that the photoelectric effect was best
explained by considering light as a series of particles (called
photons), not waves. So we've got the strange situation where
experiments are telling us two totally contradictary things.
In a nutshell, the wave/particle contradiction boils down to this:
Particle-like properties makes macroscopic sense. There are lots
of things, like ball bearings, that come in discrete chunks. It's
a little weird to imagine a ball bearing that you can't split into
pieces (or melt down), but it's not totally unnatural.
- Wavelike: Electrons have to be waves because they can interfere
with one another. Particles shouldn't ever cancel each other out
(destructive interference), but electrons and photons do cancel each
other out (e.g., in the double-slit experiment).
- Particlelike: Electrons have to be particles because they're
"quantized": the mass and charge of a clump of electrons is always
measured to be an integer multiple of the mass and charge of a single
electron. Waves should be able to dissipate smoothly, so you
should be able to capture half an electron wave, but in real
experiments, this just never happens!
Wave-like properties seem insane when applied to matter. You
can't somehow roll together two ball bearings (at the right angles
perhaps?) so that when they collide you're left with zero ball bearings
The current explanation of these two separate facts is, essentially, two separate theories:
The "many worlds" interpretation of particlelike behavior is that when
you observe it, the particle in effect just extends its own indecision
to you the observer. So when you ask where the particle "is", you
actually split into a superposition of an infinite number of copies
that each receive different answers. Since you're just one of
those copies, you think the particle just gave you one definitive
answer. But from that point onward, you should exhibit wavelike
behavior--you'd have to avoid certain activities because they'd
destructively interfere with those of your other copies. This is
of course insane, and nobody has ever been able to experimentally
demonstrate wavelike behavior on macroscopic objects, but that still might
be exactly what happens! There are other interpretations of
particlelike behavior, including Roger Penrose's theory that the
particlelike wavefunction collapse is a relativistic effect that occurs
when superposition effects would distort Einstein's spacetime into
topologically incompatible shapes--the wavefunction collapse is in
effect what keeps spacetime intact.
- Wavelike: Electrons and photons really are described by a wavefunction that evolves through time according to Schrödinger's
equation. A single electron's wavefunction can cancel itself out
in places, as can the (coupled) wavefunctions of two or more
"entangled" electrons. Thus electrons and photons really act like
waves. The official name for this process is "unitary evolution".
- Particlelike: Electrons and photons are always measured as discrete particles, with probability according to the square of the wavefunction.
That is, when acting in a wavelike way, the particle extends across
space and sits in a superposition of different locations, but when you
look at it, it makes up its mind and is actually observed in a single
spot. This means the very act of observation ("measuring")
perturbs the system, and makes it act a different way (the Heisenberg
Uncertainty Principle). This amazing fact can itself be
experimentally verified--adding an electron detector to the double-slit
experiment destroys the interference pattern! Nobody's clear on
exactly what constitutes a "measurement", but it's clear anything that
we can see or directly measure causes stuff to act like particles; it
only acts like waves when "nobody's watching". The official name
for this process is "state-vector reduction" or "wavefunction collapse".
Luckily, for quantum computation, all we need is some wavelike and
particlelike behavior, and (somehow!) we do indeed get it. The
idea is to build a "quantum register" made of "quantum bits" that store
not just 0 or 1; but the superposition of the two! For example,
one qubit can be built out of a block of negatively charged silicon
with two small positively-doped "wells". Electrons like to stay
in the wells, and avoid the (repulsive) negative region. If an
electron is in the left well, that marks a 0. If it's in the
right well, that marks a 1. If the electron's wavefunction
extends to both wells, that marks the superposition of 0 and 1.
Here's a picture of two adjacent wells, where both have trapped
electrons (click the picture to go to the IBM research site describing
If we connect wires between two adjacent wells, the electrons that
(maybe) are in the wells interact as waves. If we connect
circuitry between the wells, the electrons will seep out into the
circuitry--so we can hence perform calculations on these quantum
bits! If we're very careful (and this is the hard part), the
calculations will stay wavelike.
Why do you care if your quantum computation stays wavelike? Well,
imagine you've got a 1000-qubit register where all the bits are in the
superposition of 0 and 1. If you feed that register into a
quantum circuit, the circuit's output is the superposition of all the
outputs of all 1000-bit inputs. If you now carefully collapse the
circuit's output to a particular value, the inputs will also collapse
due to entanglement. But this means you can now trivially find
the input for any output, or invert any function!
For example, consider multiplication (Peter Shor described a theoretical quantum multiplication circuit
in 1994). On a normal machine, it's easy (polynomial time in the
number of digits) to multiply two long numbers, but it's really hard
(exponential time in the number of digits) to find the factors of the
product of two numbers. But on a quantum machine, it's polynomial
time for either one!
Practical Difficulties with Quantum Computers
If many inputs correspond to the same output, the wavefunction will
collapse to a randomly chosen valid input. This means even in a
perfectly designed and manufactured quantum computer, you'd have to run
your programs several times through to make sure you're getting the
unique right answer, and not one of several possible answers. For
example, a quantum multiplication circuit might give you the inputs X
and 1 for the output X, even if X is composite! A quantum
computer is also limited by the size of its inputs, although techniques
exist (similar to multiple precision arithmetic) for splitting a big
problem into smaller machine-sized pieces.
The real practical problem with quantum computers is that of
"decoherence"--premature wavefunction collapse. If the electrons
involved interact with too much of the outside world, they cease to
behave in a wavelike fashion too early, and then you're back to normal
circuitry. The largest quantum computers actually built so far
have only two or three qbits, which means they're not at all
useful--even grade schoolers can already factor 3-bit numbers.
With larger quantum computers manufacturing defects, thermal noise, and
leakage current destroy wavefunctions before they can do useful
work. But a huge variety of electron (and other particle)
containment devices are being tried and built,
so useful quantum computers may someday be built. Or perhaps
there is some physical limitation on the scale of wavelike effects, and
hence quantum computers will always be limited to a too-small number of
bits or too-simple circuitry.
As components get smaller and faster, ordinary computers are getting
more and more sensitive to small disturbances. Already, cosmic rays can impact server uptimes. Anecdotally, there's a curious fact about space travel. Due to the higher radiation:
Human brain cells can be killed in large absolute numbers by chemicals
such as alchohol and a variety of viruses and bacteria (e.g.,
meningitis). Yet the brain remains operational (at degraded
efficiency) despite such abuse, and rarely suffers significant
permanent damage from these injuries. That is, the human brain is
a suprisingly fault-tolerant computer. Unfortunately, we have no
idea how the human brain operates (see Gerstner's online neuron reverse-engineering attempts). But we can infer it's at least as smart as the reactions of a single cell.
- Typical computers lock up about once a day (including flight computers, laptops, digital cameras, etc.) and must be rebooted.
- Human beings have never been observed to lock up, and have never
needed to be rebooted. This is a good thing, because a spaceship
with an insane or catatonic crew might have trouble returning to Earth.
A single cell uses a number of interesting design principles.
Many of these same principles are shared by healthy ecosystems, economic markets,
functioning democracies, and piles of gravel. I've come to call
these principles "ecosystem design":
As an example, I claim an automobile, CPU, dictatorship, and stack of
bricks use "machine design", not ecosystem design: these systems have
crucial decisionmaker parts (e.g., the braking system, control unit,
dictator, or middle brick) whose failure can dramatically change the
entire system. Machine design requires somebody to go in
and fix these crucial parts now and then, which is painful.
- Many independent decision-makers. Examples: A cell is a
complicated collection of separate but interacting proteins. A
market is a collection of interacting buyers and sellers. A
gravel pile is a collection of interacting pebbles. Results: no
central decision-maker means no single point of failure, so the loss of
any few small pieces doesn't affect the overall result.
- Dynamic equilibrium--lots of small-scale stuff is changing all
the time, but the overall averages remain quite constant.
Examples: the metabolic rate of a cell is the result of all its pieces,
but it's quite predictable overall. A market's overall prices
tend to remain fairly stable. A gravel pile's average slope can
be predicted to within a few degrees.
Ecosystem Design Example
Consider this design problem: you're responsible for maintainance of a
city park's pond. Sadly, the pond is right next to a middle
school and convenience store, so the pond continually fills up with
plastic bottles the junior high schoolers toss into the water.
How do you keep the pond free of floating plastic bottles?
The standard machine design responses are top-down social control
measures like signs, fines, fences, periodic cleanup drives, or guards;
these are extremely expensive per bottle collected. One could
also imagine complicated mechanical devices like a bottle-collecting
pump, dredge or skimmer, which has the problem that it'll immediately
clog up with non-bottle stuff (algae, plants, sticks, rocks).
Ecosystem design offers many alternatives. One is a to design a
small bottle-eating critter, like an insect; but living things have
their own drawbacks, like escape, extinction, disease, etc.
Consider the option of periodically dumping a box of "sticky BBs" into
the pond. These would be tiny (1mm) spheres made of a special
plastic that sticks only to polyethelene (which the bottles are made
of). How would this help eliminate bottles?
Does this solve the problem? Yes! The pond now has far fewer
floating bottles. Instead, it's now got a big clump of
stuck-together bottles, which isn't the same thing. A big clump
can then be harpooned and hauled out once a month; or it might
accumulate dirt and sink; or turned into a planter, etc.
Crucially, the inevitable loss of a few dozen, or a few hundred out of
the thousands of BBs will have only a minor effect, and the supply is
- Wind blows the stuff in the pond around.
- Whenever a bottle runs into a BB, the BB sticks to the bottle.
- Hence after a while most of the bottles in the pond accumulate a few BBs.
- When two (BB-covered) bottles run into each other, the BBs stick to
each other. Since the BBs are stuck to the bottles, that means the
bottles stick together.
- After a while, all the bottles in the pond will be glued together into one big glob. (See the bizarre video game Katamari Damacy.)
This is exactly the strategy your body uses to eliminate known invaders
like viruses and bacteria (bottles). Your immune system engineers
a sequence that literally physically sticks to these invaders, and
attaches two copies of the sequence to a Y-shaped protein called an antibody
(BB). Each side of the antibody sticks to one invader, which (a) coats
invaders in antibodies, which may keep them from operating properly and
(b) sticks invaders together into bigger clumps of invaders and
antibodies, which can't cause as much trouble as separate invaders and
are easier to eliminate completely.
Computing With Biology
Leonard Adleman demonstrated how to use DNA to quickly perform an exhaustive search in parallel.
The idea is to dump all the input molecules into a well-mixed test
tube. The inputs stick together in various ways, some of which
represent the outputs you're looking for. You then have to use
chemistry to winnow out the (huge number of) incorrect outputs until
you're left with the correct outputs, which you can read off. The
advantage is density--since each input letter can be represented by a
single molecule, and the computation proceeds in parallel and in 3D,
you can theoretically search enormous search spaces quickly.
People are also trying to combine quantum and biological
computing--using electrons on biological molecules to represent quantum
states. There is even speculation (from Rodger Penrose, see "The
Emperor's New Mind" and "Shadows of the Mind") that mammal brains use
some sort of quantum computation to do their amazing work.