What does Superposition Really Mean?

Debunking the misconceptions surrounding superposition.

Source: https://www.machinedesign.com/

The basic building block of a classical computer is a bit, which can be either a 0 or a 1. Quantum computers, on the other hand, are made up of quantum bits (or qubits), which can be both 0 and 1 at the same time. The ability of quantum objects to be in two states at once is called superposition.

Most of that is BS. The idea that quantum particles can be in multiple states at once is one of the biggest misconceptions in quantum physics. While this idea isn’t entirely a lie, it’s a gross oversimplification that gives beginners the wrong idea.

If you’re planning to learn quantum computing, you may find yourself spending an infuriatingly long time making sense of misconceptions like these. So let’s break it down to the basics: What does “superposition” actually mean, in the context of quantum computing?

An Intuition for Superposition

Imagine you live in Classical City, a city consisting entirely of two perpendicular streets. One street runs directly north-south, while the other runs directly east-west. Thus, if you stand at the intersection of the two streets, the only directions you can go are the cardinal directions: north, east, south, and west.

Now, imagine you have a secret flying car that allows you to fly over buildings and trees, between the cardinal directions. Let’s say you fly four kilometers directly northeast, then decide you want to land. Well, you can’t just land on top of a building! You need to pick a street to land on: either the street running north-south or the one running east-west.

Which one do you land on? Let’s say you choose the east-west street. You go straight towards the east-west street and land. No one knows about your secret flying car, so anyone who sees you after you park thinks that you just drove two kilometers due east* from the intersection. They have no idea you flew northeast first!

*The reason it is two kilometers comes from the pythagorean theorem. We can visualize a right-angled triangle with your path northeast (light blue arrow) being the hypotenuse, and the net distance traveled to the east (dark blue arrow) being the base of the triangle.

The people of Classical City have never left the city, and as a result, they only know of the directions north, south, east, and west. They cannot visualize directions such as northeast and south-southwest. They simply cannot comprehend what these directions would mean, because their entire lives exist along two perpendicular streets!

What would happen if you told them about your flying car? How would you describe it? How would they visualize it? They might say that you have a magical car that can travel, say, both north and east at the same time! Then they might say that when you decide to land, you are forcing your car to pick one direction: either north or east. Or in other words, when you travel northeast, you are in a superposition of north and east, but choosing to land forces your car to collapse the superposition (pick a street to land on), with the result being the same as if you had just traveled due north or due east in the first place.

But would you really say that you had been travelling both north and east at the same time? No, you would think of it more like travelling somewhere between those two directions instead, then choosing one street to land on. That is, you were in a linear combination of a north vector and an east vector, and the landing process involved choosing one of the vectors (north or east) to project onto.

This is sort of what a superposition of 0 and 1 means: not really both 0 and 1 at the same time, but more like an in-between state. Only, saying that qubits can be in a state between 0 and 1 isn’t entirely accurate either. It’s not like they can be something like 0.5—superposition is more subtle than that.

Qubits in Superposition

Describing concepts in fields like quantum mechanics is where language begins to fail us. We can’t properly describe a superposition state in plain English without simplifying a certain aspect of the concept and making it at least a little inaccurate. Ultimately, the only way we can describe these concepts is with the universal language of math. In a nutshell, a superposition of a single qubit is a linear combination of the 0 state and the 1 state, with the coefficients being probability amplitudes.

Whoa, okay, what does that mean? First, let’s introduce some notation. The two states of a single bit in a classical computer are 0 and 1. Similarly, the computational basis states of a qubit are written as |0⟩ and |1⟩ (pronounced “ket-zero” and “ket-one”). Unlike a classical computer, these states can have coefficients, like this:

|0⟩ and |1⟩ represent the possible computational basis states. The coefficient "alpha" corresponds to the probability of collapsing into |0⟩ when the qubit is measured, while the coefficient "beta" corresponds to the probability of collapsing into |1⟩ when measured.

When a qubit is in superposition—in this “in between” state—it has a certain probability of collapsing into a |0⟩ state, and another probability of collapsing into a|1⟩ state. This “collapse” of the superposition is like when you pick a street to land your car on after flying over buildings in Classic City. However, in Classical City, you had to spend time flying towards the street you want to land on, whereas in quantum mechanics, the collapse of a superposition happens instantly, as soon as you measure the value of the qubit.

So how does the qubit decide which state to collapse into? Well, let’s go back to Classical City. Suppose, instead of flying directly northeast, you decided to fly north-northeast.

If you flew north-northeast, the north-south street would be closer than the east-west street.

Now, when you decide to land, you’re closer to the north-south street than the east-west street, so you’re more likely to choose to land on the north-south street. Of course, this doesn’t mean you have to land on north-street, you just have a higher probability of choosing to land there, as opposed to landing on the east-west street.

Because of this probabilistic nature of superpositions, qubits can be thought of in terms of their probability of collapsing into a |0⟩ or |1⟩ when measured.

Let’s go through some examples. How would we represent a qubit in a superposition with a 50% chance of collapsing to a|0⟩ state and a 50% chance of collapsing to a |1⟩ state? We would write its state mathematically like this:

The state of a single qubit in what is often called a |+⟩ superposition.

Notice the ½ fractions next to the |0⟩ and the |1⟩. These are the coefficients that correspond to the probability of the superposition collapsing into either basis state when you measure it. If you put a qubit in a superposition like this and measure it a bunch of times, half of the time you’ll get a result of |0⟩, and half of the time you’ll get a result of |1⟩.

But wait a minute, why are the fractions in square roots? Well, to understand this, we need to consider another way of representing qubits: visualizing them on a unit circle (a circle with a radius of one). The state of the qubit can be represented as a vector on this unit circle.

The image on the left represents a qubit in a basis state of |0⟩, and the image on the right represents a qubit in a basis state of |1⟩. The image in the middle represents a qubit in an equal superposition of |0⟩ and |1⟩.

We can see that our diagonal vector (the image in the middle) is a linear combination of the basis states. Since it’s on the unit circle, it still has a length of one, but the vertical and horizontal components each have a length of the square root of ½, because of the pythagorean theorem.

The reason the fractions are in square roots is because of the pythagorean theorem: the square of the horizontal component + the square of the vertical component = 1²

This is why the coefficients on our computational basis states involve square roots! The coefficients aren’t exactly the probability of collapsing into each state, but they’re related to these probabilities. These coefficients are called probability amplitudes. What probability amplitudes actually represent is a little complex, but for now, all you need to know is that the probability of collapsing into a certain basis state is the square of the absolute value of the amplitude: |α|² (where α represents the amplitude).

Note: Although we focused on qubit vectors in Quadrant I, the unit circle is a part of all four quadrants, implying that our amplitudes can be negative. In fact, they can even be complex, which is why qubits are usually represented on a 3D Bloch Sphere rather than a 2D unit circle. However, starting with a unit circle is a good starting point for gaining an intuition! Remember that the probability is always |α|² , so even if α is negative, its corresponding probability will be positive.

Why do we care?

Alright, superposition is cool and all, but what makes it so useful? Well, superposition allows quantum computers to solve certain problems exponentially faster than classical computers! You may have heard that this is because qubits can be “both 0 and 1 at the same time”, but now that we know better, let’s think about where this exponential speedup actually comes from.

First, let’s compare a classical 1-bit system to a quantum 1-qubit system. The state of a classical bit can be entirely described using only one number: either a 0 or a 1. On the other hand, to describe the state of a qubit, we would need two numbers: α and β (the coefficients, or amplitudes, of each computational basis state).

Now, let’s compare a 2-bit system to a 2-qubit system. There are four possible states the classical system can be in. However, the actual state of a system as a whole can be described with only two numbers: the state of the first bit and the state of the second bit. By contrast, in the case of the quantum system, we need to know the values of the coefficients α, β, γ, and δ. In other words, since the two qubits can exist in a superposition of four possible basis states, we need four numbers to describe the overall state of the system.

If you compare a 3-bit system to a 3-qubit system, you’ll find that you’ll only need three numbers to describe the state of the classical system, but eight numbers to describe the state of the quantum system. This is where the “exponential speedup” of quantum computers comes from! While a classical system with n bits can only hold n values of information, a quantum system can hold 2ⁿ values of information! The superiority of quantum computers as a result of properties like superposition is called quantum advantage.

Key Takeaways

• A qubit in superposition isn’t “both 0 and 1 at the same time.” Rather, it’s a linear combination of two basis states: |0⟩ and |1⟩.
• The coefficients on the basis states represent “probability amplitudes.”
• The probability of a superposition collapsing into a certain basis state when measured is the square of the absolute value of the probability amplitude.
• Quantum computers can solve certain problems exponentially faster than classical computers, because to describe the state of an n-qubit system, you need to keep track of 2ⁿ coefficients.

Quantum mechanics is one of the most difficult fields to understand, and the abundance of oversimplified pop-science articles and videos online make it even more difficult for a beginner to dive into. But now you have an intuition for what superposition actually means! This is one of the most fundamental concepts of quantum computing, so now that you have a solid grasp of this, you’re ready to tackle more advanced topics!