You don’t have to be a mathematician to understand quantum computing, but when things get tricky, math is your ally.
In classical computing, we are used to thinking in logical spaces. The term is true or false. The bit is off (0) or on (1). That’s it. It is the basis of all our algorithms.
With Boolean logic and only a very few operators, such as No, andand or, we can develop quite complex programs. If the term is true, do this, otherwise do it. Since this term is true, repeat doing a certain thing.
As a programmer, you probably have habit sheets in mind all day.
amount cwhen computed, our bit is qubit (short for quantum bit). It is not 0 or 1, but in a complex (such as complex numbers) linear combination of 0 and 1.
Suppose we have a coin. When you put a coin on the table, you have a classic bit. It’s either heads-up or tail-up. You can configure your logic logically. You specify what happens if the coin is heads-up or what you do with it up. And of course you can work with a coin. You can put it on the table in any direction you want. Your instructions look like this: “If the coin is heads-up, take it and put it on the table.”
In quantum calculation, we throw a coin. It spins in the air. It is a combination of head and tail. If only if you catch up and look at it, it will decide the value. After landing, it is a normal coin with heads-up or tail-up.
Coins have two spaces when lying on a table, at the ends, or at the tails. In the air, our quantum coin is in a state of overlap between the two states. Let’s take a closer look at this coin. It constantly turns between the heads and the tail. As the coin lands, this rotation determines whether we see the heads or tails. Take a closer look. This coin also rotates along the edge.
The problem is that we can no longer describe this state with a simple logical value. We need to describe its spin and how it affects the probability of landing heads-up or tail-up.
In quantum computation, we therefore represent the state of the reaction. When the kbite is superimposed (the coin is in the air), we still don’t know if it will be measured as 0 or 1 (whether it will fall on the head or on the tail). But we know that certain spins affect the probability of it being measured at both values. And we know that the sum of all probabilities is one.
Therefore, we represent the rotation of the cubit with a normalized vector. We call it 𝜓⟩ (“psi”). This is a Dirac notation and simply represents a column vector.
This formula means that the qubit state of superbitation is a normalized vector. All possible states start at the same point and are the same length. Therefore, they form a circle, as shown in the following figure.
The distance of the vector end from the columns means the probability of measuring this specific praise, such as 0 (distance to the South Pole or proximity to the North Pole) or 1 (distance to the North Pole or proximity to the South Pole). .
The medical status vector can point directly to the North Pole. This is the status 0⟩. Then there is a probability of 1 (= 100%) to measure it 0. If the qubit state vector points to the south pole, there is a probability of 1 to measure it 1. And the qubit state vector can point to any point around the circle with the probability of a circle | 𝛼 | ² measure it as 0 and | 𝛽 | ² to measure it as 1. We separate the state vectors by 𝜃, the angle between the North Pole | 0⟩ and state vector.
The angle 𝜃 represents the rotation over the sides of the coin. It constantly turns from head to toe.
Like a coin that can additionally rotate along its edge, qubits can also rotate in different directions. We use the Greek letter “Phi” (𝜙) in the center of the second circle, which is perpendicular to the first circle.
These two circles form a sphere around the center. This ball is known as the Bloch Sphere. Bloch Sphere provides a visual reference for a single cubic. And Bloch Sphere is a great tool that makes sense out of qubit operators.
The Bloch ball has three axes. The Z axis is vertical and includes hubs. Remember that space 0⟩ represents the north pole. We measure the volume in this state to 0 – always. And measure the cubit always as a state 1⟩, South Pole. We draw the Y-axis horizontally. Finally, the X-axis is diagonal to the pattern.
One way to represent a sphere is with a three-dimensional coordinate system. However, in quantum computing, we use only two-dimensional vectors. But the number inside the vector is a complex number.
A complex number is a number that can be expressed in the form 𝑎 + 𝑏⋅𝑖, where a is the real part and 𝑏⋅𝑖 is the imaginary part. 𝑖 represents a satisfactory imaginary unit of the equation 𝑖² = −1. Since no real number satisfies this equation, the parts 𝑎 and 𝑏⋅𝑖 are independent of each other. Therefore, the complex number is two-dimensional.
For 𝛼 and 𝛽 are complex numbers, they extend to the plane. And a two-dimensional vector of two-dimensional numbers can form a three-dimensional sphere.
And of course we can also work with qubit. But we don’t directly control whether the receipt is 0 or 1. Instead, we control the spin of the cubit in different directions. We are in the air with it. And things get a little more complicated here. We can increase or decrease the wheels. We can translate them completely. And we can do this from all different perspectives. There are countless ways to change spins. Therefore, there are many more operators for chips than traditional bit operators.
Suppose we have an arbitrary qubit | 𝜓⟩. How can we change that?
First, we can mirror it on one axis. These operators usually have the name of the axis that we use as a mirror. For example, the X operator mirrors the qubit state vector on the X axis. The following figure graphically illustrates the effect of this operator.
An important effect of the X operator is that it reverses the probabilities of measuring praise 0 and 1. Therefore, we also know the X operator as the NOT operator.
Another way to change the qubit mode is to rotate. We can rotate the qubit state vector by any arbitrary degree about any arbitrary axis. Of course, we usually use one of the three major axes (X, Y, or Z). But we are not just them.
As an example, let’s rotate around the standard axis, the Z axis, of our receipts. We can rotate it counterclockwise in a quarter of a circle, like this:
We call this operator the S-operator. And we can also turn it backwards (clockwise). this is S ^ T operator. Superscript T is intended for “transferred”. Transposition is the operator we use to transform the matrix. It flips the matrix across the diagonal. Simply put, it swaps the row and column indices of the matrix. The reason why we use matrix notation to name thank operators is simple. They are matrices.
As mentioned above, the qubit state is a vector. And there are many ways to work with vectors. But there is one particular method we use in quantum computing. This is a matrix repetition. If the matrix is multiplied by an input vector, the output is another vector. Matrix multiplication allows us to change the superposition state of a cubit without collapsing it.
Basically, this is the reason why there is so much math in quantum computing. Quantum states are vectors, operators are matrices, and we use complex numbers.
All states and their changes require linear algebra and matrix calculations. This differs from classical computation, where the state is binary (0 or 1), and from operators (No, and, oretc.) are very intuitive and you find it easy to justify.
The good news is that we have computers. No one needs to tell the matrices manually. Even our classic computers are pretty good at it. As the two examples show, we can consider quantum operators without solving a formula or telling a matrix.
So, you don’t have to be a mathematician to understand quantum computing. But you have to cope with linear algebra because it is the basis of quantum computing. With a little practice, you can explain the effects of many changes without consulting math. However, when things get tricky, math is your ally.