

The Square Root of Not - ColinWright
http://www.americanscientist.org/issues/id.3362,y.0,no.,content.true,page.1,css.print/issue.aspx

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michael_nielsen
If you enjoyed this article, you may enjoy my series of short videos on
"Quantum Computing for the Determined", which go deeper into the subject than
this article, while remaining accessible to anyone with some basic background
in linear algebra. The videos provide an intro to quantum mechanics, and cover
the basic quantum computing model, as well as quantum teleportation and
superdense coding:

[http://michaelnielsen.org/blog/quantum-computing-for-the-
det...](http://michaelnielsen.org/blog/quantum-computing-for-the-determined/)

(Apologies for the self-plug, but this particular article seems like a good
prelude to my course.)

~~~
cft
A quick note: in the first or second lecture when you multiply a state vector
by a constant and use number 2 as an example, it's a bit confusing: it does
not preserve the unitarity. A better example would be a complex number
exp(i\phi).

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lisper
Here's a much simpler and more intuitive (IMHO) way of understanding this:
imagine you're encoding bits as polarization. Vertically polarized is 1 and
horizontally polarized is 0. Rotating the polarization by 90 degrees is then a
logical NOT operator. Rotating by 45 degree is then the "square root of not"
because if you do it twice you get a NOT operator. (This exactly analogous to
how you can define the square root of -1 by thinking of multiplying by -1 as a
rotation through 180 degrees on a number line. The square root of -1 then
becomes a rotation by 90 degrees.)

BTW, you can actually do this experiment yourself at home using very
inexpensive materials: a laser pointer, polarized sunglasses, and a few
dollars worth of "quarter wave plate" material that is easily found on the
internet.

~~~
CatMtKing
Is it correct to say that the x-axis in your example is the imaginary number
line and the y-axis is the real number line? What about other degrees of
rotation?

~~~
lisper
Your question is a little ambiguous, but I think you'll find the answer here:

[http://betterexplained.com/articles/a-visual-intuitive-
guide...](http://betterexplained.com/articles/a-visual-intuitive-guide-to-
imaginary-numbers/)

Note that the two examples (polarization versus imaginary numbers) differ in
one important respect: in the polarization example, negation is rotation by 90
degrees, e.g. from vertical to horizontal. In the imaginary numbers example,
negation is rotation by 180 degrees (e.g. from positive X to negative X -- see
the link above).

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fennecfoxen
This is a fascinating article but the formatting is atrocious, mostly with the
closing >-like character in the |0> notation, which seems to be rendered as an
image and then repositioned in an inappropriate way so that it leaves its
context.

People with Chrome developer tools or Firebug will find it marginally improved
by disabling the stylesheet rules for the .imageRight class (and maybe setting
the p.font-size to 16 or so to boot, so the > isn't grossly oversized relative
to the text)

~~~
tantalor
Much better: [http://bit-player.org/wp-content/extras/bph-
publications/AmS...](http://bit-player.org/wp-content/extras/bph-
publications/AmSci-1995-07-Hayes-quantum.pdf)

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ColinWright
Non-print friendly version, possibly with better formatting, but in five
pages:

[http://www.americanscientist.org/issues/pub/the-square-
root-...](http://www.americanscientist.org/issues/pub/the-square-root-of-not)

------
waynecochran
"In all known classical factoring algorithms, the amount of time needed to
find the prime factors of a number grows as an exponential function of the
size of the number, making the algorithms impractical for very large numbers."

General number field sieve is sub-exponential. Factoring is interesting
because it is not NP-complete, but not known to be in P either. It is my
understanding that NP-complete problems remain so in the quantum world.

~~~
drostie
Saying "NP-complete problems remain so in the quantum world" is a bit
imprecise because NP-complete is simply defined as a set of problems which can
be solved by classical nondeterministic computers. Of course it "remains so"
in the quantum world; it's a mathematical definition, it remains so in any
world.

What's very interesting is that Scott Aaronson proved that normal quantum
computers could have solved NP problems (in fact, something stronger: BQP =
PP) if quantum mechanics allowed nonunitary linear operations or was based on
the norm

    
    
        norm = lambda vec: sum(abs(component) ** p for component in vec)
    

for p > 2 (quantum mechanics is based on p = 2, classical probability is based
on p = 1). So if it were any larger, the quantum computer could do brute force
with ease.

Right now it's not known whether BQP contains NP-complete or not; it might
very well contain both problems in NP and problems not in NP while deftly
evading NP-complete.

It is known that PostBQP = PP and therefore contains NP-complete. PostBQP is
the set of quantum algorithms which are allowed to destroy the universe when
they fail, under the anthropic principle reasoning that "humans will only live
on in universes where the computation succeeds."

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nsns
>>Since quantum mechanics appears to be a true theory of nature, it governs
all physical systems...

That's quite naïve. I'd say quantum mechanics seems to be the best
_description_ we currently have for the laws of physics on a micro level.

~~~
eoinmurray92
All macro systems are consistent with blown up versions of quantum systems.
Chemistry, then biology. Whats left?

~~~
dvanduzer
When you're studying the philosophy of science, it's more accurate to say that
our theories and equations describe reality rather than to say that they
_govern_ reality.

That's not necessarily arguing that a more fundamental theory might exists
that could overturn socially unpopular theories higher up the chain. Sure,
that happens in certain kind of debates rather often, but I'm not sure that's
what's happening here.

~~~
eoinmurray92
Yes you correct of course. I thought the point nsns was making that quantum
was holding on the micro scale, and not-above that. I didn't get that he was
commenting on the difference between _govern_ and _describe_.

------
dmlorenzetti
_Also note that the probabilities in each column and each row sum to 1,
indicating that every possible combination of input and output has been
accounted for._

Perhaps a _quantum_ probabilistic NOT gate requires the columns to add up to
1, but it's certainly not a general requirement of any physical device that
imperfectly implements a NOT gate.

Consider the following probability matrix:

    
    
          0    1
     0  0.1  0.9
     1  0.8  0.2
    

For any input (say, 0, in the top row), the output will be interpreted as
either a 0 or 1, so the row sum must be 1.

However, there's nothing forcing the probability of 0 as an output in one row
to depend on its probability in another row. One could implement the gate
above in software, with no problems, and a hardware equivalent could certainly
be produced.

[Edit-- fixed row probabilities in second row.]

~~~
d4vlx
I don't think you logic works. Your table says that if a 0 is in-putted then
there is a .1 chance of getting a 0 and a .8 chance of getting a 1. That
leaves a probability of .1 or something else happening, but it you add a third
state it is no longer a not gate.

In other words, a not gate, regardless of the implementation must have it's
columns sum to 1. This is a fundamental principle of probability and the
definition of a not gate.

~~~
dmlorenzetti
Your comment supposes that the column headers are the inputs. However, I used
the definition from the article-- "Here the numbers along the left edge,
labeling the rows of the matrix, are the inputs to the gate..."

So, taking the columns as the outputs, my point is that adding down the
columns corresponds to no particular causal relationship in the physics of the
gate.

Your comment did, however, make me realize that I hadn't modified the row
probabilities for the second row to sum to one (11-month old under foot). So
thanks for that.

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nullc
Ugh. Promotes (or, at least, inadequately rebukes) the bad and common
misunderstanding that quantum computing provides a _general_ exponential
speedup. It does not. QC is only even theorized to provide a exponential
speedup for a few things, for other things QC cannot provide a better than
sqrt() speedup (and this is tightly bounded for general non-linear search).

I strongly recommend Scott Aaronson's thesis
<http://www.scottaaronson.com/thesis.pdf> for anyone who doesn't want to be
made QC-stupid by articles like this. The beginning is interesting and
generally accessible as it goes into the details you need to be comfortable
with linear algebra to follow along.

~~~
ars
Did you read the penultimate paragraph? I suspect you didn't.

