Quantum Phase Estimation - Calculating an Eigenvalue

We continue our series of fundamental and ongoing quantum algorithms. In today's post, we'll look at the eigenvalue calculation that is familiar to anyone in both the basic sciences and engineering. We will consider a slightly different situation, of course, our work is quantum.

Let's consider the quantum state of a system!
When we make an observation, we obtain real observation values by calculating the eigenvalues and eigenvectors of the state in that system, the mathematical calculation that helps us to have information about the position, momentum, or energy of that system.

So, how can we find the eigenvalues of the unitary operators, which act on a system without changing its size, that is, preserve its physical size? To solve this problem, we will examine the Quantum Phase Estimation solution.

In order to find the eigenvalues of the unit operator U, we need to find the value of the phase ϕ, which is inside the exponential value. However, it is not that easy. A quantum circuit is available to solve this problem. Now let's look at this quantum circuit in detail!

The main thing to note here is that we get different probabilities for different ϕ phases. Let's take a look at the possibilities at a few different phase values.

For Φ = 0, P₀= 1 and P₁=0

For Φ = π, P₀= 0 and P₁=1

For Φ = 2, P₀= 0.99969 and P₁=0.00031

For Φ = 10, P₀= 0.9924 and P₁=0.0076

As can be seen, this process, which takes different probability values in different phases, is known as quantum phase estimation. If we want to obtain a precise result with high accuracy, this process must be repeated many times and use multiple qubits.

In my next article, I will revisit quantum phase estimation using multiple qubits.

Stay Curious. :)

Reference

https://quantum-computing.ibm.com/composer/docs/iqx/guide/quantum-phase-estimation

https://www.youtube.com/watch?v=dmgfy0CxVLI&list=PLqNc_xpYGu775P7iJA7Kvfxmv_Fm9wwHj&index=13

Comments

Bloch Sphere – Geometric Representation of Quantum State

It is very difficult to visualize quantum states before our eyes. The Bloch sphere represents quantum state functions quite well. The Bloch sphere is named after physicist Felix Bloch. As you can see in the Bloch sphere figure below, it geometrically shows the pure states of two-level quantum mechanical systems. The poles of the Bloch sphere consist of bits |0⟩ and |1⟩. Classically, the point on the sphere indicates either 0 or 1. However, from a quantum mechanics point of view, quantum bits contain possibilities to be found on the entire surface of the sphere. Traditionally, the z-axis represents the |0⟩ qubit, and the z-axis the |1⟩ qubit. When the wave function in superposition is measured, the state function collapses to one of the two poles no matter where it is on the sphere. The probability of collapsing into either pole depends on which pole the vector representing the qubit is closest to. The angle θ that the vector makes with the z-axis determines this probabilit...

Statement of Bell's Inequality

One of the most important applications of quantum mechanics is Bell's inequality. In 1965, John Stewart Bell actually thought that Einstein might be right and tried to prove it, but proved that quantum logic violated classical logic. It was a revolutionary discovery. Let's look at its mathematically simple explanation. To create the analogy, let's first write a classical inequality. This inequality above is a classic expression. The quantum analogy of this expression represents the Bell inequality. With just one difference. Bell's inequality shows that the above equation is violated. Expressing Bell's inequality by skipping some complicated mathematical intermediates; While there is no such situation in the classical world that will violate the inequality we mentioned at the beginning, it is difficult to find a situation that does not violate this inequality in the quantum world. Reference https://www.youtube.com/watch?v=lMyWl6Pq904 https://slideplay...

Bernstein–Vazirani Algorithm

Quantum computers currently available are not sufficient to solve every existing classical problem due to their own characteristics. This does not mean that they are inferior to classical computers, or that, on the contrary, quantum computers should give all kinds of advantages over classical computers. Popular quantum algorithms solved in quantum computers are algorithms created by taking advantage of the superposition and entanglement properties of particles. So, they are algorithms created to show the prominent features of quantum properties. Today I will talk about an enjoyable algorithm that demonstrates the efficiency and speed of these quantum features. Known as the Bernstein - Vazirani algorithm is a quantum algorithm invented by Ethan Bernstein and Umesh Vazirani in 1992. The purpose of this algorithm, which is a game, is to find a desired number. To put it more clearly, let's keep in mind a string of binary numbers, for example, 1011001. Next, let's write an algori...

Era Quantic | Quantum Computing and Information Articles

Search This Blog