# What is the difference between Bloch's sphere and IBM's Q-sphere?

I'm new to Quantum Computing and I've been trying to understand single-qubit operations, quantum phases etc through Bloch's Sphere visualization. However, in IBM's Circuit Simulator, they seem to be using something called Q-sphere. Why are we using Q-sphere instead of Bloch's sphere? Wouldn't Bloch's sphere be a more accurate representation?

To follow up on hizqial's answer, these two spheres represent two different things.

• The Bloch sphere is a way to visualize the state of a single qubit in every way possible. On the image below you see that the North Pole of the sphere represents the $$|0\rangle$$ state, whereas the South Pole represents the $$|1\rangle$$ state. Every point between the two represents a superposition with relative phase $$\phi$$ and amplitude angle $$\theta$$. So every state can be represented as $$|\psi\rangle = cos(\frac{\theta}{2})|0\rangle + e^{I\phi}sin(\frac{\theta}{2})|1\rangle$$, as seen on the image below : • The Q-sphere is different. It is there to represent transformations between different multi-qubits states, until 5 qubits. The North Pole of the Q-sphere represents the $$|0\cdots 0\rangle$$ state and the South Pole represents the $$|1\cdots 1\rangle$$ state. Then the other states are put between them with the ones with more 1s are put closer to the South Pole. The latitudes is defined as the Hamming Distance. The vector on the sphere represents the state vector of the system, the same way as the Bloch sphere. The phase is not represented physically as with the Bloch sphere, but with a color code. You can see that on the image below, which is an example for a 4-qubit system : Here, I hope this can help, these are both visualization tools, to gain intuition about quantum systems, but the Q-sphere is about visualizing transformation to up to 5 qubits, and the Bloch sphere is about visualizing a single qubit. The Q-sphere can be seen as a generalization of the Bloch sphere, but there is a less good visualization, the Bloch sphere on the other hand, only represents one qubit, but with better intuition.

A single qubit can be written down as $$\cos(\theta/2)|0\rangle + \mathrm{e}^{i\varphi}\sin(\theta/2)|1\rangle$$. So, the qubit is described by two angles ($$\theta$$ and $$\varphi$$). Hence, it can be mapped to a unit sphere (or with any other radius but then unit is used as a convention). This is a Bloch sphere, a visualization of a sigle qubit.

A Q-spehere is a visualization of multi-qubit states introduced by IBM. Assuming our state is composed of $$n$$ qubits, state $$|00\dots00\rangle$$ is shown on north pole, whereas $$|11\dots11\rangle$$ is south pole. A parallel of latitude is used for a state with same number of zeros. Going from north to south, the number of 1 is increasing. As mentioned in other answers, a phase is expressed with colors. So, a Q sphere is a way how to show $$n$$-qubit states.

Overall, these visualizations are more or less complements, not substitutes.

A Bloch Sphere is the state space of all possible points to which a state vector can point to. A Bloch sphere can only demonstrate the state of one qubit while a Q-sphere can demonstrate multiple qubits. We use Q-sphere more often than not because quantum circuits are generally more than one qubit.

• Hi and welcome to Quantum Computing SE. I would not say that we use Q sphere more often, it depends on situation. Both visualization shows different pictures, so this part of your answer is speculation. The rest is OK and thanks for providing it. – Martin Vesely Apr 23 at 22:19
• Thank you for clarifying! – hizqial Apr 24 at 4:15