Understanding the Bloch Sphere: A Visualisation of Qubits in Quantum Computing

In my previous blog post I briefly described the fundamental advantages of quantum computing, such as superposition and entanglement. The key player behind these powerful features is the qubit, which serves as the fundamental unit of quantum information. Unlike classical bits that represent only 0s and 1s, qubits can exist in multiple states at once due to the principles of superposition and entanglement. This allows quantum computers to solve certain problems much faster than classical computers.

But how do we actually represent a qubit? One of the most useful ways to visualise qubits is through the Bloch sphere, which is what I will explain in this post. The Bloch sphere gives us a geometrical representation of a qubit’s state, helping us better understand its behaviour and how it can be manipulated during quantum computations.

Qubit States on the Bloch Sphere

The Bloch sphere is a geometrical representation of a qubit’s state space. Named after physicist Felix Bloch, who contributed to nuclear magnetic resonance (NMR) theory, and was awarded a Nobel prize for this work, it is essentially a unit sphere in a three-dimensional space. Each point on this sphere represents a possible state of a qubit, with the surface of the sphere encoding all possible superpositions of the quantum states and .

In the traditional representation of a qubit, its state can be written as:

where and are the basic states (think of them as the quantum version of 0 and 1), and a and b are complex numbers that describe the probabilities of the qubit being in state or .

The coefficients a and b can be written using complex numbers, which introduce both magnitude and phase. But rather than get bogged down in complex numbers, we can think of the Bloch sphere as a way of mapping these values into three dimensions that correspond to specific aspects of the qubit’s state.

The Three Dimensions of the Bloch Sphere

The Bloch sphere has three key axes that correspond to different aspects of a qubit’s state:

Z-axis (Computational Basis)

This dimension is the simplest to understand and relates to the “classical” states of a qubit: and . 

• If the qubit is at the north pole of the Bloch sphere, it represents .
• If the qubit is at the south pole, it represents .

This is why the z-axis is often referred to as the computational basis, since these states are the fundamental building blocks of classical computing.

X-axis (Superposition on the Equator)

This dimension captures superposition, where the qubit exists in a state between and . A qubit in superposition might lie on the equator of the Bloch sphere, representing a mix of both classical states.

For example, the states and lie on the equator. These states are key to many quantum algorithms because they describe qubits that are equally likely to be measured as or .

The x-axis helps us understand how qubits are manipulated during quantum computations, particularly when using gates like the Hadamard gate, which create superposition.

Y-axis (Phase)

The y-axis deals with the phase of a qubit. Phase is a bit tricky—it’s not something we can directly measure, but it plays a critical role in quantum computations. Phase helps differentiate states that might have the same probabilities of being or , but which behave differently when subjected to certain quantum gates.

For example, the qubit can have a positive or negative phase ( or ), which describes how the qubit oscillates over time. This oscillation (or rotation on the sphere) affects how qubits interfere with each other during quantum operations.

Putting It All Together

At any given moment, a qubit’s state can be represented by a point on the surface of the Bloch sphere. This point is determined by two angles:

• θ (theta): This angle tells us how far the state is from the north or south pole (i.e., the computational basis). A qubit at the poles is in a classical state, while a qubit on the equator is in a superposition.
• φ (phi): This angle defines where on the equator the qubit lies, describing the phase of the superposition.

We can mathematically represent any qubit state () using these two angles:

In this equation, θ and φ map the qubit’s state to a specific point on the Bloch sphere, where we use trigonometry to describe its position. And since we have come to the point where we are talking about trigonometry, we can finally plot the Bloch sphere.

This model gives us an intuitive way to visualise qubits and the operations we perform on them. For example:

• Quantum gates can be thought of as rotations of the qubit on the Bloch sphere. When we apply a quantum gate, we essentially rotate the point that represents the qubit’s state.
• Superposition is clearly seen when a qubit lies on the equator, halfway between and .
• Phase changes are rotations around the z-axis, altering the angle φ but keeping the qubit’s probabilities the same.

Most importantly, the Bloch sphere shows us how quantum operations work in three dimensions, even though we might only measure the final state in one dimension (the z-axis). It helps explain why quantum algorithms can achieve speedups by harnessing the full power of quantum mechanics.

Conclusion

The Bloch sphere is an incredibly useful tool for visualising the state of a qubit and understanding how quantum operations work. By representing qubit states as points on a sphere, we can grasp key quantum properties like superposition and phase, which are crucial for quantum computations. As we continue exploring quantum computing, the Bloch sphere will serve as a fundamental concept for understanding how qubits behave and how quantum gates manipulate them.

In the next post, I’ll dive deeper into the computational basis and how it’s used in quantum algorithms. If you’re curious about the notation used to represent qubit states, check out this article on bra-ket notation until then!