You’ve heard a bunch about this Bloch sphere throughout this article and it is just a sphere that is used to visualize the state of a qubit. So how does any of this relate to qubit logic gates? Well, logic gates correspond to rotations on the Bloch sphere, which modify the equation previously mentioned. Each gate also corresponds to a transformation matrix, but I won’t be talking about that in this article. As you can see both theta and phi correspond to angles.

Finally, for the Bloch sphere

Qubits are the quantum equivalent of bits, but they work very differently from classical bits (the bits used in our computers). Qubits, like classical bits, can be in states of 0 and 1, also known as spin up and spin down, but what makes qubits useful is the concept of quantum superposition.

Qubits

This is the Bloch sphere in the 0 state. Both theta and phi are at 0 right now.cos(0/2) = 1sin(0/2) = 0This means that there is a 100% chance that this qubit is in the state of 0. This is generally the starting state of qubits when it comes to quantum computing.The gray circle at the bottom corresponds to phi.A gate will cause the vector (the arrow in this case) to rotate in a certain way

Depiction

First we will look at the Pauli-X gate,also known as the NOT gate,and works very similarly to the NOT gate in classical computer science.All it does is switch the probabilities of |0>and |1>. So if we start the qubit with a 100%possibility of being0,then this gate will change it to a 100% possibility of being in the 1 state.This is how the Blochsphere representation of the qubit will look if the NOTgate was applied on the previously shown gate of 0.Notice the 180degree rotation around thex-axis.The Pauli-Ygate corresponds to a 180degree rotation around they-axis,similar to thePauli-Xgate

The Pauli-X And Pauli-Y Gates

I hope that this article was a nice introduction to quantum gates and some of the math behind it. There are many things I could not mention or explain in this article for the sake of time and staying on-topic but this is the groundwork to start learning more concepts.

Conclusion

In Brief,Superposition is essentially when a qubit is in a state somewhere in between 0 and 1 that then becomes either 0 or 1 when observed. A very common example is one of a spinning coin. When you flip a coin, there is a mixture of heads and tails (like a qubit being in the states of 0 and 1) but when it stops, it is either heads or tails (this is what happens when we observe the qubit to get the result of our calculation).Calculations can be performed while the qubit is in a superposition, which is what makes quantum computers so powerful.

Superposition

The Hadamard gate is very important because it puts the qubit into a state of superposition. It corresponds to a 90-degree rotation around the x and the z axes. After applying the Hadamard gate you are in between the 0 and 1 states, meaning there is a 50% chance that the qubit will result in a 0 and a 50% chance that the qubit results in a 1. There are in-between states. For example, 25% chance for 0 and 75% chance for 1.

The Hadamard Gate

Qubits can be represented using an equation, which is what we will do.This equation is very useful when working with the Bloch sphere, which is a great tool for visualizing the state of a qubit.Now let me explain what is going on. The |0> and |1> are just the states of 0 and 1, using a notation commonly used in quantum equations called the bra-ket notation. e is <a href="https://www.youtube.com/watch?v=AuA2EAgAegE">Euler’s number </a>and <a href="https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:complex/x2ec2f6f830c9fb89:imaginary/v/introduction-to-i-and-imaginary-numbers">i is the imaginary unit </a>.

Now some math

If you know a bit about computers, you may know that computers use bits, which can either be on or off, usually referred to as 1 and 0 respectively. The states of these bits can be modified using logic gates like <a href="https://www.tutorialspoint.com/computer_logical_organization/logic_gates.htm">AND, OR, XOR </a>, etc. Quantum computers also have bits and logic gates, though they get a bit more complicated.

BITS And LOGIC GATES

One important thing to remember when it comes to the Bloch sphere and quantum gates is the phase of the qubit. The phase will generally “point” in the same direction as the arrow. If we look at the equation, phi is the angle of the arrow on the 2D (x and y axes) plane. The phase is very important in quantum computing but it is a separate subject altogether, so I will link more resources at the bottom. So now let’s look at some common gates.

The phase

The Z gate was applied after a Hadamard gate. The Z gate changed the phase to π The T and S gates both have inverse gates that correspond to a rotation in the opposite direction. For example, if the inverse of the S gate was applied after a Hadamard gate, the phase would be 3π/2. The Pauli-Z gate does not have an inverse because applying it twice brings it back to the original position. (180-degrees + 180-degrees = 360-degrees which is a full rotation)

In Continuation...

For this article, you won’t need to worry too much about these. Just know that they are important when representing the states of the qubit because a lot of quantum mechanics uses imaginary numbers.The theta and phi symbols are used to describe a vector on the Bloch sphere. In quantum mechanics, the theta influences the magnitude and phi determines the phase of the qubit. Again, you don’t need to understand this terminology for the purposes of this article as they can be thought of as angles on the Bloch sphere but I will link more information at the end of the article.

Introduction 

These three gates correspond to rotations around the z-axis by different amounts, with T being a 45-degree rotation, S being a 90-degree rotation and Pauli-Z being a 180-degree rotation. The T gate was applied after a Hadamard gate. The T gate changed the phase to π/4

The T, S and Pauli-Z Gates

If you know a bit about computers, you may know that computers use bits, which can either be on or off, usually referred to as 1 and 0 respectively. The states of these bits can be modified using…

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Quantum Logic Gates

A <a href="http://www.astronomytrek.com/the-bootstrap-paradox-explained/">Bootstrap Paradox </a>is a type of paradox in which an object, person, or piece of information sent back in time results in an infinite loop where the object has no discernible origin, and exists without ever being created. A bootstrap paradox involving a person: a 20 year old male time traveler who goes back 21 years, meets a woman, has an affair, and returns home three months later without knowing the woman was pregnant. Her child grows up to be the 20 year old time traveler, who travels back 21 years through time, meets a woman, and so on.

2: Bootstrap/ Ontological Paradox

They finish what they are doing and return to the present, but everything is gone. They reappear in a wild world with no humans, and no signs that they ever existed. They fall to the floor of their platform, the only man-made thing in the whole world, and lament “Why? We didn’t change anything!” And there on the heel of the scientist’s shoe is a crushed butterfly.

The Story Of The Paleontologist Cont.

The Butterfly Effect is a reference to Chaos Theory where seemingly trivial changes can have huge cascade reactions over long periods of time. Consequently, the Timeline corruption hypothesis states that time paradoxes are an unavoidable consequence of time travel, and even insignificant changes may be enough to alter history completely.

Are Time Paradoxes Inevitable?

In one story, a paleontologist, with the help of a time travel device, travels back to the Jurassic Period to get photographs of Stegosaurus, Brachiosaurus, Ceratosaurus, and Allosaurus amongst other dinosaurs. He knows he can’t take samples so he just takes magnificent pictures from the fixed platform that is positioned precisely to not change anything about the environment. His assistant is about to pick a long blade of grass, but he stops him and explains how nothing must change because of their presence.

The Story of The Paleontologist 

1) Closed Causal Loops, such as the Predestination Paradox and the Bootstrap Paradox, which involve a self-existing time loop in which cause and effect run in a repeating circle, but is also internally consistent with the timeline’s history.2) Consistency Paradoxes, such as the Grandfather Paradox and other similar variants such as The Hitler paradox, and Polchinski’s Paradox, which generate a number of timeline inconsistencies related to the possibility of altering the past.

2 Categories Of Time Travel Paradoxes 

American theoretical physicist Joseph Polchinski proposed a time paradox scenario in which a billiard ball enters a wormhole, and emerges out the other end in the past just in time to collide with its younger version and stop it going into the wormhole in the first place.Solutions have been formulated to avoid the inconsistencies Polchinski suggested, essentially involving the billiard ball delivering a blow which changes its younger version’s course, but not enough to stop it entering the wormhole. This solution is related to the ‘timeline-protection hypothesis’

5: Polchinski’s Paradox

There is nothing in Einstein’s theories of <a href="http://www.astronomytrek.com/black-holes-relativity-and-time-travel/">relativity </a>to rule out <a href="http://www.astronomytrek.com/interesting-facts-about-time-the-fourth-dimension-and-time-travel/">time travel </a>, although the very notion of <a href="http://www.astronomytrek.com/is-traveling-back-through-time-possible/">traveling to the past </a>violates one of the most fundamental premises of physics, that of causality. With the laws of cause and effect out the window, there naturally arises a number of inconsistencies associated with time travel, and listed here are some of those paradoxes which have given both scientist and <a href="http://www.astronomytrek.com/10-notable-time-travel-movies-of-the-21st-century/">time travel movie </a>buffs alike more than a few sleepless nights over the years. The <a href="http://www.astronomytrek.com/7-profound-questions-about-time/">time </a>travel paradoxes which follow fall into two broad categories:

Background 

Similar to the Grandfather Paradox which paradoxically prevents your birth, the Killing Hitler paradox erases your reason for going back in time to kill him. Furthermore, while killing Grandpa might have a limited “butterfly effect”, killing Hitler would have wider consequences for everyone in the world, even just studying him in school. The paradox itself arises from the idea that if you were successful, then there would be no reason to time travel in the first place. If you killed Hitler then none of his actions would trickle down through history and cause you to want to make the attempt.

4: Let’s Kill Hitler Paradox

A self-fulfilling prophecy is only a causality loop when the prophecy is truly known to happen and events in the future cause effects in the past, otherwise the phenomenon is not so much a paradox as a case of cause and effect. Say, for instance, an authority figure states that something is inevitable, proper, and true, convincing everyone through persuasive style. People, completely convinced through rhetoric, begin to behave as if the prediction were already true, and consequently bring it about through their actions.

Are Self-fulfilling Prophecies Paradoxes?

Scientists have suggested ingenious ways to avoid time travel paradoxes, including–The Solution: time travel is impossible due to the very paradox it creates–Self-healing hypothesis: successfully altering events in the past will set off another set of events which will cause the present to remain the same–The Multiverse /“many-worlds” hypothesis: an alternate parallel universe or timeline is created each time an event is altered in the past–Erased timeline hypothesis : a person traveling to the past would exist in the new timeline, but have their own timeline erased

Solutions

For example, if you traveled to the past and killed your grandfather, you would never have been born and would not have been able to travel to the past – a paradox. Let’s say you did decide to kill your grandfather because he created a dynasty that ruined the world. You figure if you knock him off before he meets your grandmother then the whole family line (including you) will vanish and the world will be a better place.

3: Grandfather Paradox

A <a href="http://www.astronomytrek.com/time-travel-the-predestination-paradox-explained/">Predestination Paradox </a>occurs when the actions of a person traveling back in time becomes part of past events, and may ultimately causes the event he is trying to prevent to take place.Imagine that your lover dies in a hit-and-run car accident, and you travel back in time to save her from her fate, only to find that on your way to the accident you are the one who accidentally runs her over. Your attempt to change the past has therefore resulted in a predestination paradox. Any attempts to stop something from happening in the past will simply lead to the cause itself.

1: Predestination Paradox

I have always been interested in the concept of time and time travel, as it is always as mind boggling as it is simple, and this article I found useful to introduce paradoxes which arise as part of time travel.

Time paradoxes, whether to the past or future, are an inherent feature of time travel, exemplified by the grandfather paradox and the bootstrap paradox.

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5 Bizarre Paradoxes Of Time Travel Explained

Sulfur

Mercury

The symbol for the philosopher's stone

Since the beginning of alchemy, alchemists have used symbols to represent different elements. Alchemy symbols sometimes contain hints of the qualities the element was thought to have, as well as the history of the element. Using symbols helped alchemists keep their work secret from non-alchemists who wouldn’t recognize the symbols. Because early alchemy also drew much of its information from astrology, many alchemy element symbols are connected to planets or other celestial bodies. Alchemy symbols continued to be used until the 18th century, becoming more standardized as time went on.

How Have Alchemy Symbols Been Used?

Salt is now known to be a chemical compound comprised of sodium and chloride, but alchemists believed it was a single element. Salt represents the body, as well as physical matter in general, crystallization, and condensation. Salt is often impure when first collected, but through chemical processes it can be dissolved and purified, which some alchemists compared to purification processes the human body can undergo. Its symbol is a circle bisected by a horizontal line.

Salt

Mercury (which is also one of the seven planetary metals) can mean both the element and the planet. In either case, this alchemy symbol represents the mind, as well as a state that could transcend death. In ancient times, mercury was known as quicksilver, and it was believed to be able to shift between liquid and solid states. Therefore, in alchemy, mercury was believed to shift between life and death.

Mercury 

Sulfur represents properties such as dryness, heat, and masculinity. In alchemy, it could also represent evaporation, expansion, and dissolution. In terms of human body, it represented the soul. In terms of the tria prima, sulfur was seen as the middling element connecting salt (high) and mercury (low).Sulfur’s symbol is typically a triangle atop a Greek cross (seen above), but it can also be represented by a Cross of Loraine atop an ouroboros/infinity symbol. This symbol has come to be known as Satan’s Cross and is sometimes used as a satanic symbol.

Sulfur

Salt

Alchemy is an area of study, sometimes described as a science, sometimes described as a branch of philosophy, that was practiced in Europe, Asia, and Africa. Alchemy originated in the early centuries AD, primarily in Egypt, Greece, and Rome, but eventually spread as far as India, China, and England by the 12th century.

What is Alchemy? 

The three primes, also known as the tria prima, were named by Paracelsus, a Swiss philosopher, in the 16th century. He believed the tria prima contained all the poisons that caused disease, and that by studying them, alchemists could learn how to cure disease. He also believed the tria prima defined humans, and he assigned each of the elements to a different part of the human identity.

The Three Primes

What are alchemy symbols? Check out our complete guide to the most important alchemy symbols and meanings.

philosophy

Philosophy

Alchemy Symbols and Their Meanings

When a theory’s predictions are confirmed, it becomes an essential tool in the further practice of science, but even good theories may someday be superseded by theories more comprehensive or more accurate.

Good theories may be superseded by better theories.

As Einstein demonstrated, conceptions of time and space based on everyday life don’t apply accurately to all speeds and all realms of space.

The Reality ft. Einstein

There is more to the world than we have dreamed of in our philosophies. There is more to be discovered, and principles today unknown will someday yield their secrets to continued inquiry, if the culture of the future remains educated enough to permit science to survive.

Last but the BEST!

The microworld of the atom, and realms even smaller, obey “quantum” laws completely at odds with common sense, and notions of cause and effect and the very nature of reality are inherently blurred on that scale.

The subatomic realm is weird.

The way a thing works is often influenced by its connections to other things and the ways that they work, a principle that applies to everything from the networks of cells in the brain and the body’s other organs, to ecological and economic systems, to human interactions and social institutions.

Networks Are Us.

When scientific disputes arise, the ultimate arbiter is not expert authority or common sense but experimental evidence, guided by theory.

The ultimate arbiter is experimental evidence, guided by theory

Scientific theories are not “guesses” but are logi­cally rigorous attempts to explain the observed facts of nature and to predict the results of new observations.

Scientific theories are not “guesses”

Science successfully explains natural phenomena through rational investigation and logical reasoning rather than by recourse to superstition and mysticism.

Science explains nature rationally and logically

The universe is vast and old, with our sun one of bil­lions of stars in a local galaxy, joined by billions of similar galaxies occupying the depths of space beyond.

There are billions and billions of stars.

Life has changed over the eons, with complex creatures evolving from simpler precursors, and human beings therefore occupy one branch of an immense fam­ily tree of living organisms — all sharing a common molecular machinery driving basic life processes.

All life is related.

Little is certain in science but much is highly probable, and the proper quantification of probabilities is essential for inferring facts, drawing conclusions and formulating sound judgments.

Learn some damn statistics.

Much of scientific knowledge can be condensed into a few basic principles that every educated person should know.

Top 10 things everybody should know about science

describe the angular momentum of an electron. An electron spins around an axis and has both angular momentum and orbital angular momentum. Because angular momentum is a vector, the Spin Quantum Number (s) has both a magnitude (½) and direction (+ or -)(+½) ===> in the direction of clock (-½) ===> against the direction of clockThe electrons take place in orbs in a specific way.Imagine the orbs like beds and electrons like persons. Every person tend to sleep alone in bed (+½) unless there is no bed left so the other person have to sleep upside down (-½).photo owner: Byju's

Spin Quantum Numbers (ms)

The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital.Every azimuthal number has his own capacity of electrons<ul><li>S(v=0) ==> 1 orb ==> 2 electrons</li><li>P(v=1) ==> 3 orbs ==> 6 electrons</li><li>D(v=2) ==> 5 orbs ==> 10 electrons</li><li>F(v=3) ==> 10 orbs ==>12 electrons</li></ul>Ps:open the photo to see the whole image. ((v = value))photo owner: Byju's

Azimuthal Quantum Numbers (ℓ)

We have to define the exact place of the last electron for the element Neon¹⁰<ol><li>We have to configure the electrons 1s²,2s²,2p⁶</li><li>Now we focus on the last number 2p⁶</li></ol>• n (the number before the letter) = 2• ℓ ( = P ) = 13. We define P's orbitals2p⁶ ==> 3 orbs ==> 6 electronsEvery orb contains 2 electrons4. We have to define spin|(+½)¹ (-½)⁴|.|(+½)² (-½)⁵ |.|(+½)³ (-½)⁶|.-----[-1]------. -----[0]------. -----[1]------The order of spin is marked on top of the electron¹• mℓ (last electron place) = 1• ms = -½Hope you enjoy the explanation :)See you in the next post 👋.

Practice 

A quantum number is a value that is used when describing the energy levels available to atoms and molecules.It defines the exact position of electron in atom.photo owner: Universiteit Leiden

What Is A Quantum Number

There is four quantum numbers that are used to describe completely the movement and trajectories of each electron within an atom.<ol><li class="ql-align-justify">Principal</li><li class="ql-align-justify">Azimuthal</li><li class="ql-align-justify">Magnetic</li><li class="ql-align-justify">Spin</li></ol>photo owner: ThoughCo

Quantum Numbers

describe the energy of an electron and the most probable distance of the electron from the nucleus.To know the maximum amount of electron that an energy level can handle, Use this formula 2n² (does not include O , P , Q).photo owner: All About Circuits

Principal Quantum Numbers (n)

Specifie the orientation in space of an orbital of a given energy (n) and shape (l). This number divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Every orbital capacity is 2 electronsMagnetic values are {-3,-2,-1,0,1,2,3}<ul><li>S ==> 1 orb ==> 0</li><li>P ==> 3 orbs ==> -1,0,1</li><li>D ==> 5 orbs ==> -2,-1,0,1,2</li><li>F ==> 7 orbs ==> -3,-2,-1,0,1,2,3</li></ul>photo owner: unknown

Magnetic quantum Numbers (mℓ)

A total of four quantum numbers are used to describe completely the movement and trajectories of each electron within an atom. The combination of all quantum numbers of all electrons in an atom is …

Quantum Logic Gates

5 Bizarre Paradoxes Of Time Travel Explained

Alchemy Symbols and Their Meanings

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Top 10 things everybody should know about science

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