# Bloch sphere and quantum operations [closed]

Evidently the Bloch Sphere is used as a graphical representation for any single qubit system, although what does it mean at an intuitive level? Moreover, the manipulation of qubits still seems unclear, regarding certain aspects. Hence I've attempted to concretely list my questions and preferably non-mathematical answers are desired. Thanks in advance and I look forward to hearing from you soon.

Bloch Sphere related

• Since quantum states always contain two possible states: $$|1\rangle$$ and $$|0\rangle$$, then why does the Bloch Sphere depict three separate axes, namely Z, X, and, Y. What is the meaning of each axis?
• Why would it be relevant to represent a quantum state via the three-axis if the quantum state possesses merely twee possible states?

Quantum operations related

• Apparently an infinite number of quantum operations are possible on a qubit, even though it seems only a few quantum logic gates exist to manipulate a qubit in a fixed and predetermined way. What is meant by an infinite amount of quantum operations and what is a quantum operation (perhaps to merely alter the complex coefficients or change the phase of a qubit)?

• If it's true that quantum algorithms can use destructive and constructive interference for destroying states en amplifying certain states, then how are the specific quantum states selected to for example destructively interfere, and others to constructively interfere. In passing, I do comprehend the concept of interference, however which parts of qubits interfere with one another to amplify and eliminate particular quantum states?

• Is the phase of a qubit only of importance for inducing destructive and constructive interference, if so, which gates are used for modifying the phase of one or more qubits?

Entanglement

• A quantum register could implement, for instance, 12 qubits, would it be possible to create different groups of entangled qubits, e.g. 6 - 4 - 2 (three separate entangled qubit pairs). Furthermore, does measuring one qubit of an entangled group of qubits result in a total collapse of all the qubits' superpositions or could still a few quantum states remain in superpositions while certain other states have collapsed?
• Are entanglement and interference always used together during quantum computations?

Unfortunately, I am not fully acquainted with the required mathematics of quantum computing, apart from the Dirac notation.

• Hint: asking one question per a post usually results in better answers. Dec 3 '19 at 13:30
• I would recommend reading for example this source ibm.com/quantum-computing. Hope this helps. Dec 3 '19 at 13:57
• hi, sorry but I'm voting to close this as too broad. Each post should contain a single, focused question, to improve searchability and reusability of the posts. If you have many question you can ask them in separate threads. Unless you can edit this question to focus on a single aspect of it, it will probably be closed.
– glS
Dec 3 '19 at 15:12
• Welcome to Quantum Computing SE! Note that you need not write your questions in the form of letters; we prefer to cut out the noise as much as possible. Dec 3 '19 at 16:29
• Dec 3 '19 at 16:34

I hinted in the comments of your previous question that the answers to the above questions have a shared basis for an answer, and since you're already asking about that too I guess you have found that: the Bloch sphere. The Bloch sphere is a method of visualizing the state of a qubit. However, to really understand this visualization, you really need to use some mathematics.

## The state of a qubit

The state $$|\psi\rangle$$ of a qubit, in its most general form, can be written in ket notation as:

$$\begin{equation} |\psi\rangle = \alpha |0\rangle + \beta |1\rangle \end{equation}$$ with $$\alpha$$ and $$\beta$$ complex numbers.

If you are not familiar with complex number, I strongly advise you to get acquainted with them (for instance by watching this amazing video on youtube), but for now you can think of them as a kind of 'two-dimensional number'. The best way to conceptualize that is to think of them as some point in the $$xy$$-plane. They have two separate defining parameters:

• The distance to the origin, commonly referred to as their magnitude or modulus or absolute value. We write the magnitude of a complex number $$z$$ as $$|z|$$.
• The angle they make with the $$x$$-axis, commonly referred to as their phase. We write the phase of a complex number $$z$$ as $$arg(z)$$, and we often denote it with the letter $$\phi$$.

The important thing to realize is that these two parameters are independent of each other.

For reasons that I wont go into here, there is another constraint on $$\alpha$$ and $$\beta$$: their moduli squared together should add up to unity: $$|\alpha|^{2} + |\beta|^{2} = 1$$. Moreover, we only care about the phase difference between $$\alpha$$ and $$\beta$$, which we call the relative phase.

This culminates in us writing the state of a qubit as:

$$\begin{equation} |\psi\rangle = \cos(\theta)|0\rangle + e^{i\phi}\sin(\theta)|1\rangle. \end{equation}$$

where $$\theta \in [0,\pi)$$ and $$\phi \in [0, 2\pi)$$. This can be mapped to a sphere with radius $$1$$, where $$\theta$$ is the angle between the point on the sphere and the north pole (e.g. the latitude of a point on earth), and $$\phi$$ is the angle of the point in the $$xy$$-plane (e.g. the longitude of a point on earth).

We map all possible states of the qubit on a sphere, which is a surface (mind you, the state of the qubit is on the sphere, not in!). This surface is a $$2D$$ structure so can be parametrized by $$2$$ parameters (in our case, $$\theta$$ and $$\phi$$).

## Operation on qubits

Apparently an infinite number of quantum operations are possible on a qubit, even though it seems only a few quantum logic gates exist to manipulate a qubit in a fixed and predetermined way. What is meant by an infinite amount of quantum operations and what is a quantum operation (perhaps to merely alter the complex coefficients or change the phase of a qubit)?

An operation or gate on a qubit is the tranformation of the state $$|\psi_{1}\rangle$$ of a qubit to some other state $$|\psi_{2}\rangle$$. Since these states are defined by continuous parameters, there are an infinite number of different mappings or operations. Thus, there are an infinite number of gates possible. Luckily, we don't need all of them for sensible quantum computing.

If it's true that quantum algorithms can use destructive and constructive interference for destroying states en amplifying certain states, then how are the specific quantum states selected to for example destructively interfere, and others to constructively interfere. In passing, I do comprehend the concept of interference, however which parts of qubits interfere with one another to amplify and eliminate particular quantum states?

This is a though question, and can only be really answered by very precise mathematical description. In principle, it is the task of the quantum algorithm to invoke this interference in such a manner that certain states get more likely to be measured.

Is the phase of a qubit only of importance for inducing destructive and constructive interference, if so, which gates are used for modifying the phase of one or more qubits?

Again, this is a though question. you need a very thorough understanding of the role of the phase in a qubit's state description. In essence, the fact that the qubit has a phase makes it a quantum bit instead of a classical (statistical) bit. In one sense all the phase does is help in the interference, but in the same sense one could say that quantum computing is all about said interference.

A quantum register could implement, for instance, 12 qubits, would it be possible to create different groups of entangled qubits, e.g. 6 - 4 - 2 (three separate entangled qubit pairs). Furthermore, does measuring one qubit of an entangled group of qubits result in a total collapse of all the qubits' superpositions or could still a few quantum states remain in superpositions while certain other states have collapsed?

Yes, there can exist such subgroups of underlying entangled pairs. Moreover, the measurement of a qubit that is entangled with other qubits will always collapse some entanglement, but not necessarily all 'entanglement'. So the remaining qubits might still be entangled with each other. The measured qubit, however, will be completely disentangled from all other qubits; we say the qubit to be in a separable state.

Are entanglement and interference always used together during quantum computations?

Without elaborating very much, I would say that they are not on the 'same level' within the concepts of quantum computation, and I would therefore say that they are not strictly 'used together'. However, entanglement and interference both play a necessary role in quantum computations.

## Final sidenote

This many subquestions in one post is generally unfavourable. Please try to ask them separate posts next time.

• In reference to my previous questions on quantum gates, which you answered thoroughly, I would like to also greatly thank you for providing answers to this thread. Furthermore I'll post less clustered questions, sorry for the inconvenience. Dec 4 '19 at 20:26