Translate the frequency spectrum in the frequency domain


I had a question on whether the frequency spectrum encoded in the PQC can be translated in the frequency domain.

For example, if I have a PQC encoding a frequency spectrum of {0, 1, 2}, could I do some operations such that the spectrum becomes {4, 5, 6}? I think such operations would be very helpful.

Thank you

or is there a way to add a phase of pi/2 to all terms in a fourier series expressed by a PQC? for eg-> e^(-i2x) + e^(-ix) + constant + e^(ix) + e^(i2x) becomes e^(-i(2x + pi/2)) + e^(-i(x + pi/2)) + constant + e^(i(x + pi/2)) + e^(i(2x + pi/2))

Hi @Daniel_Wang,
I’m not sure I’m understanding your question.

You can represent a quantum circuit using a Fourier series. We actually have a super cool interactive demo on this.

We also have a Fourier module in PennyLane which contains functions to analyze the Fourier representation of quantum circuits.

You can also do some arithmetic with the quantum Fourier transform (QFT). This is probably not what you’re looking for though.

You could try using different circuits and seeing how changes affect their Fourier series representation.

Does this help? Or could you maybe rephrase your question if not?

Hi Cataline,

Thanks for replying to me and sorry for not getting the question in a clear way.

I understand that the PQC in the data-reuploading fashion can be seen as a truncated Fourier series with the frequency encoded by the data-encoding block. However, one problem of such model is that the encoded frequency has to start from 0 to a certain value in a staircase way just as shown by the paper of Schuld “effect of data encoding on the expressiveness of the quantum circuit”.

So what if I want the minimum frequency included in the frequency spectrum encoded by the PQC not to start with 0 but from some higher value? So that would mean that if there is a way to shift the some frequency spectrum starting with 0 towards the direction of higher frequency (basically lifting a frequency spectrum by a certain number). So this is what I am wondering.

Besides this, I also had a question of phase shifting. So for eg, if I have a truncated Fourier type sum expressed by PQC being:

e^(-i2x) + e^(-ix) + c_0 + e^(ix) + e^(i2x) = cos(2x + phase_1) + cos(x + phase_2) + c_0

I want to change all the cosine terms into sine terms which would mean that there is an additional phase shift of pi/2. I wonder if there is a way to make this happen. It would mean that if it is possible to do convert the previous expression into the following:

e^(-i(2x + pi/2)) + e^(-i(x + pi/2)) + c_0 + e^(i(x + pi/2)) + e^(i(2x + pi/2)) = sin(2x + phase_1) + sin(x + phase_2) + c_0

I thought about this problem and I thought the phase gate Z(\theta) = [e^(-i\theta /2) 0;0, e^(i\theta /2)] might help. But I am not too sure how to do that…

Thank you so much for your and possibly your colleage’s help!

Hi! interesting question :smile:
Off the top of my head I would say that simply replacing x with x + Pi/2 in the circuit should work. However without testing it I’m not sure. Could you provide a minimal script to be able to see the results you get?

Hi! I guess this would not work because if you take e^(-i(2x)), and you replace x with x + pi/2, it would become e(-i(2x + pi)), not e(-i(2x + pi/2)).

This is because simple modifying x would be also influenced by the factor in the front which is 2 for this case.

And I also had another question translating the frequency spectrum in the original post. Do you have an idea for that?

Thank you

You are right, it doesn’t look like an easy task, there is no variable change of the x that will solve your problem. Still one could ask the question if it is possible to do it. Could it be the case that there is no unitary such that this property is fulfilled for an arbitrary x?

I am not too sure about this as right now which is why I ask for help. From the expressions, the hope to solve the problem either comes from a unitary to change the phase or replace the variable x which does not seem to work now…

But if PQC can only be expressed as a truncated Fourier type sum with frequencies jumping in a staircase from 0, it sounds a bit limited…

Without being an expert on the subject, I would say that these techniques are more useful from a theoretical point of view than a practical one. However, I invite you to continue working on the topic, it is a very interesting field!

hi! Thanks for keeping catching up. Well, this question does have quite some significant applications such as low-to-high compact frequency signal reconstruction and many others. There are many areas that only compact high-frequency range signals need to be captured. If we do just standard PQC, we might need many qubits to reach the high frequency region in a staircase way. Now I formulated an idea that if we can undertstand to how convert cosine terms into sine terms (have a phase shift of (4n + 1)pi/2, n is any integer), then we can translate the frequency spectrum. However, this is probably indeed a bit difficult due to the parallelization and superposition, which makes the analysis tricky.

If anyone knows how to do this, always feel free to leave a message :>

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Thanks for sharing your thoughts here @Daniel_Wang ! Let us know if you figure out how to do this. Or as Daniel said, if someone else knows the answer please let us know here in the thread!