5 - Where's that noise coming from?

In the last articles, we learned some basics that we need before studying error correction. In order to learn about error correction, we first need to learn what are these errors? where are they coming from? And how do we mathematically describe them?

Interaction with the Environment

Errors are the factor that is corrupting the information and changing the state of our qubits in a way that is corrupting the final result of any algorithm. As long as the qubit is not interacting with anything, the state will remain the same and will not be changed.

Problems are arising when an interaction is happening. Any interaction can be described as an operator that is acting on the whole system. Any operation E is reversible, meaning that it is theoretically possible to operate with the reversed operation E†, and to return to the original state:

Where I is the identity matrix.

It sounds very easy thing to do, but there are a few problems with this idea.

1. We don't know which E happened.

2. This E came from a system that is bigger and more complicated than the single qubit, so there might be a piece of information that leaked from the qubit to the outer environment.

3. This E might also be probabilistic, from the chaotic behavior of the environment, so again, we can't know which E happened.

4. Even if we did know which E happened, we can't engineer a system that can't operate with exactly the same reversed operation E† that we want, it will always be a little bit different.

Let's have a look now at 2 types of errors :

Coherent Errors

Those errors refer to errors that can be described as unitary operations, without a probabilistic nature. The source of such errors is mostly an inaccurate control on the qubit.

Mathematically can be described as a unitary operation on the state:

Or on the density matrix:

For example, in Linear Optic Quantum Computing (LOQC) , gates are implemented with couplers between two waveguides [1]. The parameters of the waveguides and the coupling are determining a gate U that we want to make. Since fabrication always comes with artifacts, the U planned will be slightly different:

So the error operator is:

A graphical way to describe the coherent error is as a rotation on the Bloch sphere. Since any unitary operation can be described as a rotation matrix. Since the coherent errors are usually small, meaning nearly Identity operator, we can also say that their rotation angle is usually small. Finally, we can say that coherent error is a relatively small rotation on the Bloch sphere.

In-Coherent Errors

In-Coherent errors are processes that cause the quantum system to lose information to the environment, and therefore can't be corrected using an inverse unitary operation which is unknown to us [2].

Usually, we describe such errors as a set of unitary operations, that might happen in some according to a set of probabilities. Mathematically we describe it as an operation on the density matrix that is summing unitary operations with different probabilities that sum to 1.

For example, a bit-flip error is causing X gate to happen in probability p (and I otherwise):

The same can be done with Phase Flip error, where Z happens in some probability.

Kraus Representation

Generally speaking, we can describe in-coherent error with Kraus representation, which is a set of K matrices [A{0},..., A{K-1}] such that the evolution of a density matrix is given by [3]:

Notice that coherent errors can be described as a Kraus set of one unitary operator. Also notice that it is not possible to write in-coherent as a representation of coherent error:

To Summerize

Awesome! We now know what noise is, I invite you to Google it ("Amplitude Damping", "Phase Damping", "Depolarizing"...) and to read more about noise. Ref[2] is very recommended for that.

Please vote:

💡 If you learned anything new

🤔If it was all a piece of cake for you, but you still want to learn more about QEC

👏If you are professional ducks, that liked the explanation

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In the next article, we will finally start to learn how to deal with that noise.

Stay tuned...

References

[1] Greener, Hadar, Elica Kyoseva, and Haim Suchowski. "Detuning modulated universal composite pulses." arXiv preprint arXiv:2012.04401 (2020).

[2] Boone, Kristine. "Concepts and methods for benchmarking quantum computers." (2021).

[3] Wood, Christopher J., Jacob D. Biamonte, and David G. Cory. "Tensor networks and graphical calculus for open quantum systems." arXiv preprint arXiv:1111.6950 (2011).