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).

In fact, every distinct type of polarization will produce photons that give a unique set of answers to these three questions. Not only that, but analogues of these three questions, which are really quantum measurements, exist for every type of qubit (all qubits, whether photons, electrons, or ions, are all mathematically equivalent). In fact, the answers to these questions are the key to showing that all single-qubit states can be represented as points on a sphere. Let’s start by plotting the possible answers to the first question on a line:

By placing a point somewhere on this red axis, we can indicate if a particular photon will be measured to be 100% horizontal, 100% vertical, or some combination of the two. Now let’s plot the answer to question 2 on a perpendicular green axis.

Now a single point can be used to read off the answer to both questions at the same time. Finally, let’s add a third blue axis to show the answer to the last question. Remember that the three questions are (1/Red) Horizontal or Vertical, (2/Green) Diagonal or Antidiagonal, and (3/Blue) Right or Left.

In this “space” of states that we’ve created, what types of single-qubit states are allowed by quantum mechanics? Every legal state falls somewhere on or inside a sphere. Let’s label six single-qubit states on this single-qubit sphere:

These six points each represent states that have a 100 percent chance of being transmitted through a particular type of polarizer—in order words, a 100 percent chance of being measured in a particular quantum state. (There is nothing special about these particular points, they’re just convenient.) In fact, every point on the surface of the sphere has a 100 percent chance of being measured if you pick the right measurement. This is because every possible measurement that can be performed on a qubit corresponds to a single line that passes through the center of the sphere.

The points where the line intersects the sphere’s surface are the states which give one of the two measurement results 100 percent of the time. Which measurements correspond to which lines? That isn’t always obvious, and depends on what type of qubit the sphere is describing. Tilting polarized sunglasses, for example, will always perform a measurement corresponding to an axis on the sphere’s equator.

How do you predict how a given quantum state (which always corresponds to a single point anywhere on or in the sphere) will react if subjected to a given quantum measurement (which always corresponds to a single axis)? To see how, draw a new line perpendicular to the measurement axis and passing through the point corresponding to the quantum state. The new line will divide the measurement axis into two segments whose lengths correspond to the probabilities of the two possible measurement results. An example:

This figure shows a measurement that corresponds to tilting your polarizer (for example, sunglasses) by 55 degrees, and gives a 20% chance of measuring state (2) and an 80% chance of measuring state (1).

This is now the third or fourth time we’ve encountered some type of measurement which gives random results. This is the first classical assumption that we have to let go:

Classical Theory: God does not play dice with the universe.
Quantum Theory: Quantum measurement can give random results.

Are these results truly random, or did we just not know the answer before performing the measurement? This is very similar to asking the question, “If you measure the same qubit many times, do you get the same answer?” Answer: If you perform the same measurement, you always get the same result. Only the first measurement result is (potentially) random.

Wait a minute. That doesn’t seem right.

Let’s say I measure a photon several times in a row using the red, green, and blue axes (horizontal, diagonal, and right-circular polarizers) and every time the photon is transmitted. 100 percent horizontal, 100 percent diagonal, 100 percent right-circular? Remember that we are plotting each point using the answers to those three questions. Plotting 100 percent/100 percent/100 percent will give a point far outside the sphere, a state forbidden by quantum mechanics. Can we really never obtain these results, or is quantum mechanics wrong about the sphere? Neither. This is a false choice, because it’s based on our second false assumption about classical information—an assumption that doesn’t apply to qubits:

Classical Theory: Reading the value of a bit doesn’t change the bit’s value.
Quantum Theory: Measuring a qubit changes its value to match the result of the measurement.

Using this newfound principle of quantum information, let’s walk through an example. Let’s start with a horizontally polarized photon, H. If we measure this photon using the H/V axis, it will stay horizontally polarized. If we then perform a measurement on the green axis, it is randomly transformed into either D or A (we’ll choose D for this example). Finally, we repeat our first measurement using the red axis. Instead of giving the same result as our first measurement, however, there’s now a 50 percent chance that our horizontal photon will be measured as vertical! The intervening measurement has changed the state of the qubit. (It’s worth noting here that measurement is a real process, the same process that polarizing sunglasses and 3D lenses perform. You can test all of these examples with just a few pairs of eyewear.)

Measurement appears, strangely, to be one way we can change the state of a qubit. For a quantum programmer wanting to adjust the qubits in a quantum computer, however, this may not be a good choice. After all, the results are random! Although some exotic quantum algorithms use measurement in the middle of a computation, most of the time measurement is reserved for the end, when the programmer learns the result of the computation.

How then, do we change qubits without introducing randomness? Physically, every species of qubit (photon, electron, ion, etc.) is changed differently. Photon polarization can be changed by directing a photon through quartz crystals or Scotch tape. All of these processes, regardless of the species of qubit or the type of change, have a simple interpretation on the single-qubit sphere: they act as rotations.

These rotations are defined by a single axis, just like measurements. But, instead of projecting all possible states into two possible outcomes, they rotate all states on an axis. Only the points on the axis of rotation will be unaffected. As an example, think of rotating the sphere by 90 degrees about the red axis. This kind of quantum operation leaves H and V unchanged, but transforms R > D, D > L, L > A, and A > R.

We can now summarize the important characteristics of single qubits:

All single-qubit states correspond to a point on or inside a sphere.
Every axis corresponds to a single quantum measurement, and
every measurement changes the state of the qubit to match the
result of the measurement.
Qubits can be changed by rotating them around an axis.

Although this succinctly describes the way in which a one-qubit quantum computer is supposed to work, what happens when things go wrong? For a classical bit, the only thing that can go wrong is for a bit to unexpectedly flip from zero to one or one to zero. The same type of thing could happen to qubits, in the form of unexpected or unwanted rotations. But there’s another type of process, one that researchers in quantum computing are constantly fighting to eliminate: decoherence.

Decoherence happens when something outside of the quantum computer performs a measurement on a qubit, the result of which we never learn. Let’s say we measure the state H in the D/A axis (the green) axis. There’s a 50% chance of measuring H in the state D and a 50% chance of measuring it in the state A. If we never learn which state the measurement resulted in, we’ll have no idea how to predict the result of another measurement.

This process is called decoherence, and, in fact, it’s how states inside the sphere are created. By measuring along an axis but never learning the result, all points on the sphere collapse to the measurement axis. By partially measuring something (with say, really thin polarized sunglasses), we can collapse only part of the way:

This sort of unwanted intrusion introduces randomness into a quantum computer. Because quantum bits can be single electrons, single ions, or single photons, all of which can be accidentally measured using a single stray atom, it can be exquisitely difficult to avoid decoherence. That’s the primary reason that a 100-qubit quantum computer has not yet been built.

Pairs of qubits

Pairs of qubits are much, much more than the sum of their parts.

Classical bits only become marginally more interesting when paired—it literally only makes the difference between counting to two and counting to four. Pairs of quantum bits, on the other hand, can be used to create entanglement, a phenomenon so… well, disturbing that one of the most controversial arguments in 20^th century physics revolved around whether it could exist at all.

Before talking about the strange things that can be done using pairs of qubits, let’s talk about the things that can’t. Like copying qubits. The most basic operation one can perform using classical bits is to copy the value of one bit into another bit. Simple, right?

Not really. When we want to copy a single classical bit, we really perform two operations in sequence:

Measure both bits.
If they don’t match, flip the second one.

Uh-oh. Not only can a single qubit take on a whole sphere full of values, it can only be measured along a single axis at a time. Not only that, but measuring it changes its state from whatever it was before the measurement to whatever state the measurement produced. That’s a problem. In fact, it can be proven that even in principle it’s not possible to copy an unknown qubit’s state. You can move it—that’s called quantum teleportation—but, just like in Star Trek, teleportation just moves the state from one place to another. It doesn’t make a copy.

Why, then, is classical copying allowed? Classical bits act exactly like quantum bits that never leave a single axis on the sphere. If an unknown qubit is constrained to a single axis (as it is after a measurement along that axis), the classical recipe of measuring and flipping will work fine. But it only works for that one axis. This leads us to another broken classical assumption:

Classical Theory: All information can be perfectly copied.
Quantum Theory: Only the results of a measurement can be copied.

OK, no copying. Strange—definitely strange—but that doesn’t sound worthy of a 75-year argument which had Nobel laureates on both teams. Let’s get to the good stuff. Let’s get to entanglement.

At the heart of entanglement is the concept of correlation, or how the results of measurements relate to each another. Specifically, it’s about whether the results of two measurements are the same (correlation) or different (anti-correlation).

This sounds too easy. For states that are on the surface of the sphere, the two measurements will correlate if the states fall on the same axis as the measurement. Right? Time to eliminate another fundamental assumption about information.

Classical Theory: The state of multiple bits is defined by the states of all of the individual bits.
Quantum Theory: The whole is greater than the sum of its parts.

Many, many two-qubit states cannot be completely described by the state of the first qubit and the state of the second qubit. We call the states which are just a combination of the individual qubit states separable; we call all other states entangled, because they exhibit extra correlations that simple, single-qubit descriptions miss.

Consider the “singlet state,” an example of an entangled two-qubit state. A singlet state has two defining characteristics:

Any single-qubit measurement performed on one half of the
singlet state will give a totally random result.
Any time the same single-qubit measurement is performed on
both qubits in a singlet state, the two measurements will give opposite
results.

The first characteristic sounds like a pair of single qubit states plotted at the origin, the point that divides every measurement axis in half. The second characteristic, that of perfect anti-correlation, is an entirely new phenomenon. This second “rule of singlet states” means that, if horizontal/vertical measurements are made on the two qubits, one qubit will always be measured as H and one will always be measured as V. Which is which will be completely random.

If you’ve read the last paragraph carefully, this should seem very, very strange. Even impossible.

Imagine if someone showed you a pair of coins, claiming that when both were flipped at the same time, one would always come up heads and one would always come up tails, but that which was which would be totally random. What if they claimed that this trick would work instantly, even if the coins were on opposite sides of the Universe.

You would probably say that’s impossible. Albert Einstein did.

In 1935, in one of the most famous scientific papers of all time, Einstein, Podolsky, and Rosen argued that because quantum mechanics allowed exactly this type of strange action at a distance, it must not be complete. Some part of the theory had to be missing.

In effect, they claimed that some extra information (called hidden variables) was programmed into the coins—although they seem random, they really only show correlation because of hidden instructions which tell the coins which way to flip. After all, dice seem random, but if you know precisely how a die is rolling, you can predict its outcome. This assumption—that, in principle, the outcome of any experiment is predictable—is called realism.

The EPR paper coupled this assumption with another basic assumption, locality, which states that events that are very far away can’t affect nearby outcomes (unless there’s enough time to for a signal to travel between the two events). They showed that as long as local realism is true, quantum mechanics can’t be the whole story.

For 30 years, the EPR paradox went unresolved. Finally, in 1965, John Bell proposed an experiment which could directly measure the paradox and, if performed, disprove local realism. He proposed creating a stream of identical singlet states and, for each state, separating the first qubit from the second. In separate locations, each qubit would be randomly subjected to one of two measurements:

If they exhibited too much of the right types of correlation and anti-correlation—as defined by John Bell’s equations—it would prove that a locally realistic universe could not exist. Over the past three decades, this experiment has been performed in many different settings using many different types of particles. The Bell experiment has most commonly been performed using polarized photons and the following procedure:

Create many copies of a singlet state (i.e., many pairs of entangled
photons).
Send the first photon in every pair through a polarizer. Randomly
choose, for every photon, whether to orient this polarizer at 90
degrees (V) or at 45 degrees (D). These two quantum measurements correspond to
the measurements (the red arrows) shown on the first sphere in Figure
8.
In the same way, send the second photon in every pair through a
polarizer. Randomly choose, for every photon, whether to orient this
polarizer at 22.5 degrees or at 67.5 degrees (corresponding to the red
arrows on the second sphere in Figure 8).
Count the number of times the measurement results matched
(exhibited correlation) and the number of times they didn’t
(anti-correlation).

When this experiment is performed, the results are incredibly surprising. To illustrate why the results are so suprising, I will describe an equivalent implementation of the Bell experiment which, to the best of my knowledge, has never been performed: I call it “The Nemesis Experiment”.

To perform this experiment, we’re going to need 1000 pairs of people to play the part of singlet states. Remember that the singlet state is the permanently anti-correlated entangled state, and so we can’t use just any pairs of people.

We need arch-enemies.

Luke Skywalker and Darth Vader. Harry Potter and Lord Voldemort. Mac and PC. Pairs who disagree with each other on such a deep, fundamental level that, when both are asked the same question, they will always try to disagree. When asked similar questions, they’ll disagree most of the time. Only when asked totally unrelated questions will they have no idea how to spite their nemesis.

Let’s imagine a trial run of this experiment, using the wrong measurement axes, to show why Bell chose the ones he did. First we separate all the heroes from all the villains, preferably by several light years so that it’s absolutely impossible for them to determine which question their partner is being asked. Every hero and every villain gets asked one and only one randomly selected question, which requires them to state which of two choices they prefer. Here the questions are going to play the part of quantum measurements. To show how, we’re going to redraw the spheres from Figure 8, this time showing only the equators for simplicity, with polarizer angles replaced with questions:

The questions have nothing to do with each other, so we should expect the arch-enemies to be able to disagree only when they are asked the same question. That means when they are both asked about love and war, one chooses “a” and one chooses “b,” every time. When they are both asked “Rome or Greece?” one choose “a” and one chooses “b,” every time. However, when the hero is asked “Rome or Greece?” and the villain is asked “Love or War?” it’s not clear which answer is the oppposite of which, so their answers match only half the time.

Does this experiment prove that a mystical, instantaneous connection allows the enemies to always know which question their partner is being asked, and therefore to always be able to oppose them?

Absolutely not!

This experiment has exactly the loophole that Einstein predicted: the heroes simply knew their partner’s preferences ahead of time. In the course of making all those commercials, PC simply remembered that Mac liked Greece and so he, villain that he is, chose Rome. These preferences could all have been set up ahead of time.

Einstein 1, quantum mechanics 0.

Because we are only allowed to ask one question of each person (remember, making measurements will destroy the singlet state so we only get one chance with each pair), you might wonder if it is ever possible, even in principle, to get around the preferences-written-down-ahead-of-time loophole.

If fact it is, and this was the great insight of Bell’s 1965 paper. Consider the following questions, now written on the Bell axes (again we’re showing just the equatorial planes of the two single-qubit spheres):

Now, the heroes and the villains are not asked exactly the same questions, and so they will only disagree most of the time, instead of all of the time. Let’s consider one villain’s dilemma as he tries his best to spite his nemesis, the hero. Let’s say that this particular villain knows that his nemesis is going to choose “Love” instead of “War” if asked the first question and “Rome” instead of “Greece” if asked the second question, but the villain doesn’t know which of those two questions the Hero will be asked.

If the villain is asked his first question, “Venus” or “Ares”, his task is simple. On the circle, the “Venus or Ares” question is pointing in roughly the same direction as both of the hero’s questions, so the villain needs to give the opposite letter answer as the hero. He needs an answer that is the opposite of both “(A) Love” and “(A) Rome.” That’s easy. “(B) Ares, Greek god of War” is roughly the opposite of both Love and Rome.

What if the villain is asked the second question, “(A) Gladiator” or “(B) My Big Fat Greek Wedding”? My Big Fat Greek Wedding is clearly a Greek-themed love story; it’s the opposite of “(A) Rome,” so if the hero is asked “Rome or Greece,” the villain should choose “(B) My Big Fat Greek Wedding.” Gladiator is clearly a Roman war movie; it’s the opposite of “(A) Love,” so if the hero is asked “Love or War,” the villain should choose “(A) Gladiator.” The villain is trapped. Without knowing which question the hero will be asked, the villain can’t decide what answer will oppose him. He has to guess, and he’ll be wrong half of the time.

Bell proved that, even with all of the information about the hero’s preferences, a villain can only thwart him, on average, 75 percent of the time. This is because half of the time the hero’s choices won’t present a conflict for the villain, and the other half of the time the villain has a 50/50 shot. 50 percent + (1/2)*50 percent = 75 percent.

The amazing thing is, when this experiment is done with entangled photons, or entangled ions, or any particles in a singlet state, the choices anti-correlate over 85 percent of the time. Even without working through the rigorous proof laid out by John Bell, this just seems impossible. Yet time and time again, experiment after experiment, the results show that something about local realism must be wrong. Either the events simply cannot be predicted, even in principle, or there is something fundamentally nonlocal about entanglement—an ever-present bond between entangled particles which persists across any distance.

So far, we’ve talked about a single type of entangled state, the singlet state. There are many, many more types of two-qubit states, however, which run the gamut from completely separable to completely entangled. How many more states are there?

To give you an idea, consider that single-qubit states can be represented by a point inside a sphere in 3-dimensional space. Two-qubit states, in comparison, need to be represented as a point in 15-dimensional space.

If this is how complicated things can get with only two qubits, how complicated will it get for 3 or 4, or 100? It turns out that the state of an N-qubit quantum computer can only be completely defined when plotted as a point in a space with (4^N-1) dimensions. That means we need 4^N good old fashion classical numbers to simulate it.

It’s no wonder, therefore, that quantum physicists talk about a 100-qubit quantum computer like it’s the holy grail. It’s simply much too complicated for us to simulate using even the largest conceivable classical computers. To be honest, we don’t have the faintest idea what it might be capable of. (OK, maybe a faint idea, but definitely not the whole story.)

One thing we do know is the primitive operations that are necessary in order to perform arbitrary computations on a working quantum computer:

You must be able to intialize all of your qubits to a known
state (H, for example).
You must be able to rotate individual qubits.
You must be able to measure individual qubits.
You must be able to perform an operation that entangles
pairs of qubits.
Your qubits must stay free of outside interference (aka decoherence) for as long as it takes to finish your
computation.

All of these capabilities exist, to varying degrees of precision, in the one or two dozen tiny quantum computers that exist in laboratories across the world. The largest obstacles to moving forward are making sure that all of these requirements can be fulfilled in larger systems (say, with more than 10 qubits).

Quantum physics 101

The first two parts of this article introduced quantum bits and showed how strangely they can act when paired together. I’ve shown you how quantum states work but, in order to to show you why quantum states work, I need to reveal the gears turning under the hood. Beware: the following is not for the mathematically faint of heart.

Let’s start by trying to name the states that, until now, we’ve only drawn pictures of. We’ve given six of the states on the single-qubit sphere names, but there is a much more general way to name an arbitrary state. For the sake of simplicity, let’s forget about the states inside the sphere and just think about states on the surface. Specifically, let’s think about a photon whose polarization is in the state |X>. (The vertical bar and right-angle bracket indicate the name of a quantum state.)

|X> can be defined by its latitude (theta degrees from |R>, the north pole) and its longitude (phi degrees from the H-R-V-L plane). Quantum physicists would write the state like this:

Let’s walk through this equation step by step. It casts all states as combinations, or more correctly superpositions, of the states |R> and |L>. The cos(theta) and sin(theta) terms indicate how much |R> and how much |L> are needed to make this state. In fact, cos²(theta) is the chance of measuring this state to be |R>, and sin²(theta) is the chance of measuring this state as |L>.

What does it mean that the state |X> is in a superposition of |R> and |L>? It means that while no one is watching, before any measurements are made, this photon exists as both R and L at the same time. The states |R> and |L> represent two possible existences for this particle, two possible realities that coexist until someone measures the state. It is the act of observing the state that finally forces only one possible reality to exist. This is the essence of the famous Schrodinger’s cat thought experiment.

But this is not the end of the story. Each of the possible realities, each of the possible existences for this photon, has a heartbeat.

You heard me.

Each state has a pulse, a never-ending drumbeat, the meaning of which is the very crux of the nature of energy and time. You can think of this pulse as the rotating hand of a clock or the tick of a metronome or the tone of a particular musical note. Technically it’s called an “oscillating phase” but this singularly quantum phenomenon is a heartbeat by any other name.

The higher the energy of a state, the faster its pulse beats. Double the energy of a state, double the beats per minute. For photons, which are essentially pure energy, their heartbeat has a direct, physical meaning: their color. Red light has a wavelength, which is just another word for color, of 600 or 700 nanometers. That means that at least once for every 700 nanometers that it travels, its electric field will oscillate up and down. Almost like a heartbeat.

For radio waves (which are just another, much lower energy form of light) this heartbeat might only occur every few meters. Regardless, the color of any type of light is simply the physical manifestation of this quantum heartbeat.

When two possible states of reality have the same energy, their quantum pulses tick away at the same rate, leaving only the question of whether or not they beat in time. In the equation above, the rather intimidating e^(i*phi) term has a very simple meaning: it shows whether the heartbeats for the |R> and |L> states are in or out of sync.

e ^(i*phi) is something called a “complex phase” but it is mathematically identical to the position of the second hand on a clock face, where the angle phi is just the angle of the second hand. When phi = 0 degrees, the pulses are perfectly in time. When phi = 180 degrees, the pulses are exactly out of sync. When phi = 4 degrees (which “My Cousin Vinny” has taught me is the correct ignition timing for a 1964 Bel Air Chevrolet), the beat for the state |L> happens every so slightly before the heartbeat for |R>.

Let’s look at a couple of different states. We could define |H> and |V> this way:

(As it turns out, e ^{(i*180 degrees)} = -1, which we can think of as the 6 o’clock position—in other words perfectly out of phase. e ^{(i*0 degrees)} = +1, is the 12 o’clock position—in other words perfectly in phase.) Exactly the same, except for the relative timing of the quantum heartbeat.

Can we fit rotations into this fantastic tale of phase, energy, and time? As it turns out, the only way to induce a rotation around the R/L axis is to create an energy difference between |R> and |L>, causing one state’s pulse to beat faster. This won’t change the |R> and |L> states themselves, because they are on the axis of rotation, but all of the other states (which are superpositions of |R> and |L>) will slowly start to rotate. After all, the latitude of the states on the sphere is determined by how much the incessant pulses of |R> and |L> are out of tune with each other. By quickening the beat of |R> for a tiny fraction of a second, you can force it out of sync and transform |H> into |D>.

Because any state can be thought of as a superposition of |R> and |L>, a qubit is often described as a superposition of the classical values of a bit, 0 and 1. By writing an arbitrary qubit as a combination of states called |0> and |1>, it makes a more direct connection to classical logic. The reason I haven’t mentioned that until now, however, is that, from the perspective of quantum mechanics, |0> and |1> form just one axis on the single-qubit sphere. As we’ve already seen, all axes are equivalent. Said another way, there’s nothing special about the R/L axis. It works exactly the same as every other axis.

Instead of describing the heartbeat of a quantum state, physicists would describe the rate at which it accumulates phase, that is, how fast its pulse beats and in turn wanders out of sync with other phases. A state accumulates phase at a rate equal to its energy. (Which, incidentally, is really the best definition of both energy and time that I can come up with.) What about when we look at entangled states? In an entangled state, two photons (or electrons, or whatever) exist in a joint state. We write the joint state of two photons, the first in the state |R> and the second in the state |L>, like this:

OK, simple. I haven’t written a phase in front of this state—not because it doesn’t have a pulse, but because I’m assuming it’s phase is 0 degrees (the second hand keeping time for this pulse is at 12 o’clock), and e^(i*0) = +1. (Incidentally, we generally ignore phases for states sitting by themselves because, if there’s only one instrument in the orchestra, it’s impossible to be out of sync with anyone.) It still has a pulse, however, which beats at a speed determined by the total energy of the state. Let’s explore this a little, and start by explicitly writing in the phase:

Now imagine that these two photons, one right-polarized and one left-polarized, are blue, which means they will pulse (i.e., accumulate 360 degrees of phase) exactly once every half a micron. If the state |X> travels a quarter of a micron, then both the |R> and |L> components should each accumulate a phase of e^(i*180) = -1. Written in the form of an equation it looks like this:

This. Is. Weird.

A quantum state can encompass many particles separated from each other by vast distances. Not only is this possible reality constantly… pulsing… but, whenever the energy of just one of those particles increases, the pulse of that entire potential reality beats faster.

How does this affect entangled states? Here’s how to write the singlet state:

This equation shows that there is an equal chance to measure either the state |R>|L> or the state |L>|R>. For two qubits, a measurement can yield four results, however. The states |R>|R> and |L>|L> aren’t listed here, because as we’ve already learned, the singlet state always gives anti-correlated results. |R>|R> and |L>|L> are correlated, and so have no place in the equation for the singlet. Those possible existences literally aren’t part of the equation.

Could this be the nonlocal connection between entangled particles and the source of spooky action at a distance? Here, there is no faster-than-light communication. There is instead a much simpler, much more disturbing underlying truth: two particles, even when separated, both contribute to the same potential reality and both determine the speed of that existence’s pulse. Let’s see what happens when we try to make a measurement using different axes. After all, we know that any point on the single-qubit sphere can be described using any axis. Why not for two qubits?

Substituting the following single-qubit equations:

into the two-qubit equation for the singlet state:

…we see that the singlet state is identical, even when written in terms of different states. Exactly the same anti-correlation, exactly the same state, written using different axes. All this, a simple mathematical consequence of the ability to describe objects in superposition. The ability to ask multiple questions of the same two-answer object.

Perhaps I should stop here.

The reason I should stop here, is that up ahead, just around the bend, there is a paradox. Not a paradox like the Einstein-Podolsky-Rosen paradox—for all intents and purposes, that has long since been resolved. An honest-to-goodness, one-of-the-most-important-questions-in-physics-today paradoxes. The reason I won’t stop here is that I just can’t in good conscience withhold one final, crucial piece of information.

You see, some quantum physicists aren’t convinced that quantum measurement exists.

I’m well aware it’s one of the two or three most fundamental operations in all of quantum theory, and pretty much the basis of the Copenhagen Interpretation. But I’m not sure it exists. In fact, I’d lay better than 3-to-1 odds that it doesn’t.

It’s called “The Measurement Problem” and, to explain, I need to talk about how a measurement actually works in the lab. You see, all the measurements we actually use are really just entangling operations, just like the kind you’d use in a quantum computer to entangle two qubits.

Let’s say you want to measure the polarization of a photon, and so you buy a polarizer (just like in the polarizing sunglasses). So I start with a state like this:

|R>

and I want to measure it using the H/V axis. So I put it through a polarizer. What that polarizer actually does is couple a polarization qubit to a spatial qubit, resulting in a superposition of two possible realities:

|H>|transmitted> + |V>|reflected>

That superposition is an entangled state. Using a different polarizer, it would be straightforward to unentangle it without ever making a measurement, effectively erasing the fact that the first “measurement” ever happened at all. Instead, a photodetector is placed in the path of the transmitted half of the entangled state. If there is a photon there, it will excite an electron:

That excited electron will cause an electron avalanche, which will cause a current to surge in a wire, which will be sent to a classical computer, which will change the data in that computer’s RAM, which will then finally be viewed by you: