he computers we use today are engineering and
technical marvels. They employ powerful mathe-
matical abstractions about information and computa-
tion to create virtual machines that can do everything
from simulating a thunderstorm to engaging you in
real-time battle with a Fre-breathing dragon.
But for all their power, today’s computers can only do
one thing at a time. Sure, we can put two computers side
by side and carry out out
, but to have
streams of parallel computation, we need to have
computers. And parallel computing is hard to do efF-
ciently: if we double the number of processors, the speed
of computation goes up, but very rarely by a factor of 2.
These are the realities of
There’s a whole new approach on the horizon, called
. The idea is to harness some of the
strange properties from the world of quantum physics
to build a new breed of computers. The scientiFc and
engineering challenges to building a real quantum com-
puter are formidable. But tiny, 3-bit quantum comput-
ers have been built and they prove that the theory works.
In the last issue, I introduced some of the ideas of
quantum computing. Here I’ll begin with a quick recap
of those ideas, and then dig into the notation and ter-
minology. In the next issue we’ll see some tools and algo-
rithms central to quantum computing.
You’ll probably Fnd the notation here a little unusu-
al, but basically all we’ll be doing is manipulating vectors
and matrices. I’ll use physics language and symbology
here because it’s the language of quantum computing.
A quick review
As a quick refresher, let’s look at the characteristics of
the quantum world I discussed last time. Then I’ll quick-
ly review some linear algebra, because we’ll be talking
about familiar ideas with unfamiliar notation.
In the following, electrons and photons are typical
examples of quantum particles:
A quantum particle can exist simultaneously in many
incompatible configurations, or
. We call this
We can operate on a quantum particle while it’s in a
superimposed state and affect all the states at once.
a quantum particle, the very act of
observation causes the particle to take on one and
only one state. If we repeat the same observation
before otherwise affecting the particle, we’ll see the
The particle and the measuring apparatus determine
the possible states that result from a measurement.
As we saw last time, quantum mechanics often con-
tradicts our intuition. After all, how can a particle exist
simultaneously in several incompatible states? What
could that possibly mean? Nothing in our everyday
experience works like that—a bird is ﬂying or it isn’t, a
tree is alive or dead, and a bit is one or zero. Certainly
a tree can be thriving or dying, but it can’t be
dead simulaneously. But in the quantum realm it’s a
different story, and that opens the door to quantum