# How to design the circuit of VQLS?

I am very confused about how to design the VQLS circuit. For example, I know the A is: X is pauli matrix. And how to add these three X pauli gates into circuit? I know the overall circuit should look like this: I don't know how to design it, so...: I think this circuit is error, because I can't get the right result... Thanks for any help!

Note that, $$A_l$$ is followed by $$A_{l^{\color{red} \prime}}^{\dagger}$$ not $$A_l^{\dagger}$$. Otherwise, they will cancel each other and your circuit will be just the ansatz.

• I feel very shame...I misunderstanded these matrix...Al is orginal matrix, A†l is conjugate transpose matrix of Al. So what is the A†l′ ? According to the formual of VQLS, I've only seen these two matrices. Is there something wrong with my understanding? Thanks for your reply! Sep 1, 2022 at 4:18

If you have a general linear system $$Mx=b$$, then a quantum computer can at best solve the corresponding $$A\left|x\right> = \left|b\right>,$$ where $$\left|b\right> = \frac{b}{||b||}$$, $$A=\frac{M}{||M||}$$ is the original matrix normalized so that $$||A|| \leq 1$$ and $$\left|x\right> = \lambda x$$.

For VQLS specifically you must further decompose the matrix as a linear sum of unitary matrices $$A=\sum_{l} c_l A_l$$. In your case, $$A = c_0 A_0 = 0.5 (X \otimes X \otimes X).$$ As $$A_0A_0^{\dagger}=I$$ the operations cancel out indeed and the measurement result will be $$P(0)=1$$.

The circuit you present in your question is only to compute $$\left<\psi|\psi\right>$$ though, which in your case is 1. As you need to evaluate a normalized cost $$C = \frac{\left< \psi | H | \psi \right>}{\left<\psi|\psi\right>}$$ you have to evaluate the Hamiltonian, global or local, $$\left<\psi| H |\psi \right>$$, in which case the $$A_l$$'s never cancel out.

$$A_l$$ = $$A_{l^\prime}^\dagger$$ in your question and you can try it.

• Hi and welcome to Quantum Computing SE. Once you have enough reputation you can post comments. Please use the comment for the short answers like this one. Nov 18, 2022 at 6:40