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

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