 # VQLS: accuracy of x

Good morning,

I am trying the VQLS demo and have a question on the output x.

In the demo, the results of Quantum probabilities and Classical probabilities match well. But when I am trying to access x, I found that it is not accurate enough:

Classical result: x = [0.25253814 0.25253814 0.35355339 0.35355339 0.25253814 0.25253814 0.35355339 0.35355339]

Quantum result: x = [0.29177217 0.2907112 0.4064591 0.40583864 0.29186469 0.29102234 0.40754877 0.40554531]

Since the cost function is already very small, I am curious if there is any way to improve the accuracy?

Thank you so much for any suggestions!

Sincerely

Hey @Dobby and welcome to the forum!

I believe the differences in the vectors you provide are due to the normalization. For example, take the classical (unnormalized) result x and find x / np.linalg.norm(x). The result will be much closer to the quantum value you provide.

More generally, the quantum system can only prepare a normalized state, and so the vectors x that we derive from the state’s amplitudes are also normalized. That is why we then normalize the target x vector for a fairer comparison.

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Hi Tom,

Thank you for the clarification! Still, I am curious can the accuracy of VQLS be controlled someway? If the difference is required to be smaller than some threshold (for example, 10^-6), is keeping minimizing the cost function the only way?

Thank you for any suggestions!

Hey @Dobby!

I have to admit that I’m not deeply familiar with the VQLS algorithm so this may be a bit of a learning process for us both!

Still, I am curious can the accuracy of VQLS be controlled someway?

Yes, indeed this should be the objective of the cost function - to act as a quantitative measure of the accuracy in this problem. From this part, you can see that we are picking as a cost function C_G = 1 - \vert\langle b | \Psi \rangle\vert^{2}. Minimizing this means that we are maximizing the overlap between |b\rangle and |\Psi\rangle.

If the difference is required to be smaller than some threshold (for example, 10^-6), is keeping minimizing the cost function the only way?

Yes, if we have set up the cost function in a way that captures the accuracy, then minimizing it further should help increase the accuracy. There a couple of options to help things along as the cost function becomes very small, for example we could use a more advanced optimizer that has a variable step size.

Two questions I have for you are:

• What measure of accuracy are you interested in? The cost function is an overlap/fidelity-based measure of accuracy.
• For me, the optimization is already doing quite well with a cost of order 10^{-6} after 30 iterations. What target accuracy are you thinking of aiming for?

Cheers,
Tom

Hi Tom,

For my problem, I may wish that the relative error of x could be smaller than 1e-8. Anyhow, I achieved this target now.

Thank you again for all your suggestions!

Dobby

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