## A tale of two qubits: how quantum computers work

Quantum information is the physics of knowledge. To be more
specific, the field of quantum information studies the implications that quantum
mechanics has on the fundamental nature of information.
By studying this relationship between quantum theory and information,
it is possible to design a new type of
computer—*a quantum computer*.
A largescale,
working quantum computer—the kind of quantum computer some
scientists think we might see in 50 years—would be capable
of performing some tasks impossibly quickly.

To date, the two most promising uses for such a device are quantum search and quantum factoring. To understand the power of a quantum search, consider classically searching a phonebook for the name which matches a particular phone number. If the phonebook has 10,000 entries, on average you'll need to look through about half of them—5,000 entries—before you get lucky. A quantum search algorithm only needs to guess 100 times. With 5,000 guesses a quantum computer could search through a phonebook with 25 million names.

Although quantum search is impressive, quantum factoring algorithms pose a legitimate, considerable threat to security. This is because the most common form of Internet security, public key cryptography, relies on certain math problems (like factoring numbers that are hundreds of digits long) being effectively impossible to solve. Quantum algorithms can perform this task exponentially faster than the best known classical strategies, rendering some forms of modern cryptography powerless to stop a quantum codebreaker.

Quantum computers are fundamentally different from classical
computers because the physics of quantum information is also the
physics of *possibility*. Classical computer memories are
constrained to exist at any given time as a simple list of zeros and
ones. In contrast, in a single quantum memory many such
combinations—even *all possible* lists of zeros and ones—can all exist *simultaneously*.
During a quantum algorithm, this symphony of possibilities split and
merge, eventually coalescing around a single solution.
The complexity of these large quantum states made of multiple
possibilities make a complete
description of quantum search or factoring a daunting task.

Rather than focusing on these large systems, therefore, the goal of this
article is to describe the most fundamental, the most intriguing,
and the most *disturbing* consequences of quantum
information through
an in-depth description of the smallest quantum systems. By
learning how to think about the smallest quantum computers, it
becomes possible to get a feeling for how and why larger
quantum computers are so powerful. To that end, this article is
divided into three parts:

**Single qubits.**
The quantum bit, or qubit, is the simplest unit of quantum
information. We look at how single qubits are described, how they
are measured, how they change, and the classical assumptions about
reality that they force us to abandon.

**Pairs of qubits.** The second section deals with two-qubit systems,
and more importantly, describes what two-qubit systems make possible:
*entanglement*. The crown jewel of quantum mechanics, the phenomenon
of entanglement is inextricably bound to the power of quantum
computers.

**Quantum physics 101.** The first two sections will focus on the
question of *how qubits work*, avoiding the related question
of *why they work they way they do*. Here we take a crash course
in qualitative quantum
theory, doing our best to get a look at the man behind the curtain.
The only prerequisites for this course are a little courage and a
healthy willingness to ignore common sense.

## Single qubits

Bits, either classical or quantum, are the simplest possible units of
information. They are oracle-like objects that, when asked a
question (i.e., when *measured*), can respond in one of only two ways. Measuring a bit,
either classical or quantum, will result in one of two possible
outcomes.
At first glance, this makes it sound like there is no difference
between bits and qubits. In fact, the difference is not in the
possible *answers*, but in the possible
*questions*. For normal bits, only a single measurement is
permitted, meaning that only a single question can be asked: *Is this bit a zero or a one?*
In contrast, a qubit is a system which can be asked
many, many different questions, but to each question, only one of
two answers can be given.

This bizarre behavior is the very essence of quantum mechanics, and the goal of this section is to explain both the bounds that quantum theory places on such an object and the consequences that such bounds have for our classical assumptions. Given how counterintuitive this behavior seems, I will first explain in some detail how polarized light provides the perfect example of a qubit. Using a little light, some polarized sunglasses, and a 3D screening of "Avatar," I'll use that specific example to describe how all single-qubit states can be thought of as points on or inside a sphere, and finally how the fundamental operations of quantum measurement, rotation, and decoherence can be visualized and understood using that sphere.

Before continuing, I should define a word that I'll be using frequently:
*state*. A system's state is a complete description
of that system; every system (including a single qubit) is in a
particular state, and any systems
that would behave completely identically are said to have the same state. Classical
bits, therefore, are always in one of exactly two states, "zero" or
"one."

With that out of the way, our first step is to find an object which always gives one of exactly two answers, but which can be measured in many different ways. Here's where you're going to need those polarized sunglasses. Polarized sunglasses are different from normal sunglasses because they are designed to block the glare from horizontal surfaces, like a long stretch of desert highway or the surface of a lake on a sunny day.

How do they work? Light is in fact made of photons—the smallest indivisible unit of light—and every photon creates a tiny, oscillating electric field as it travels. Light from the sun (and most other sources of light) is composed of photons oscillating in all sorts of directions. However, light which is reflected off a horizontal surface (like glare off a lake) will become horizontally polarized. When the light reaches the sunglasses, the photons are either transmitted or absorbed. If a photon's electric field oscillates horizontally, polarized sunglasses absorb it. If it oscillates vertically, it will pass right through the same sunglasses.

These polarized lenses provide our first example of a *quantum
measurement*, as they show a way to distinguish between
horizontally polarized and vertically polarized photons (based on
which gets transmitted and which gets absorbed). They can, of course, be used
to ask a different question (make a different measurement) if they
are tilted. By tilting your head 90 degrees, you make a measurement which is the opposite
of the first, as the sunglasses transmit
all of the glare you were trying to avoid. By tilting your head 45
degrees to one side (diagonally) or the other side (antidiagonally),
they will transmit only half the glare.

Does this mean that the types of questions you can ask are limited
to the angles at which you can tilt your head? That may seem reasonable,
but if you went to see the 3D showing of *Avatar*, you
might have guessed that this isn't true. In order to create the
illusion of three-dimensional objects on a two-dimensional screen,
movie theaters need to control exactly which photons go to each of
your eyes. For decades, this was done using color. (Remember the
3D glasses with one red lens and one blue lens?)

To get full-color 3D, we need another way to control which photons go in which eye. Once again there are only two answers—absorbed or transmitted—so we need new questions. You don't want the entire movie to change when you tilt your head, so using horizontally and vertically polarized lenses is out. Likewise, diagonally and antidiagonally polarized lenses won't work. (Test this out in a 3D movie—tilting your head won't ruin the effect.)

The solution is something completely
different, called *circular* polarization. The two lenses in modern
3D glasses each ask the question, *is an incoming photon
right-circularly polarized or left-circularly polarized?* Each lens
transmits only one of these two types of light (one of the two
answers to the question), allowing special projectors (which
transmit the same types of light) to control what image is seen by
each of your eyes, thereby creating the illusion of electric
blue warriors riding extra-terrestrial pterodactyls flying off the screen.

If the polarization of a photon is the perfect example of a quantum bit, what can the following three questions/measurments tell us about it?

- Is the polarization horizontal or vertical?
- Is the polarization diagonal or anti-diagonal? (In other words, will it pass through my polarized sunglasses when I tilt my head forty-five degrees to the left or to the right of vertical?)
- Is the polarization right-circularly or left-circularly polarized? (In other words, does it pass through the right or left lens of a pair of 3D glasses?)

If we performed the measurements that these three questions represent on the horizontally polarized photons generated by highway glare, we would learn that each photon always passes through a horizontal polarizer (question 1), but has only a 50% chance of passing through diagonal (question 2) or right-circular polarizers (question 3).

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