6

As requested in the comments, here is a worked example. The main body deals with minimizing $f(x)$ for a specific problem. At the bottom follows a brief discussion of constraints then a brief discussion about the general case. Let's solve the Weighted Maximum Cut problem since this Is a relatively straight-forward example Is hard classically Is a ...


5

For the specific linear function you are interested in, the solution turns out to be trivial: you can take the channel to be $N_{X\rightarrow Y}(\rho) = \operatorname{Tr}(\rho) |\psi\rangle\langle \psi|$ for $|\psi\rangle$ being an eigenvector of $\sigma_Y$ having the largest possible eigenvalue. More generally, however, you can optimize any real-valued ...


4

Short Answer: It is potentially hard (as bRost03 indicates in the comments). To be precise, coNP-hard. Longer Answer: In adiabatic quantum computation, the ground-space of the final Hamiltonian is typically determined by the optimum solution to some constraint satisfaction problem (CSP). If the CSP is perfectly solvable, the ground-space is spanned by (...


4

The proof of the variational theorem (the theorem that the ground state energy is the lowest possible energy you can get from $\frac{\langle \psi|H|\psi\rangle}{ \langle \psi | \psi \rangle}$) is very simple: https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics) If you get a lower energy, it means you don't actually have $\frac{\langle \psi|...


3

If you are looking for a more complete implementation of a quantum variational algorithm in the context of Cirq, I would recommend looking at the second example in the OpenFermion-Cirq notebook found here. It uses a custom ansatz for hydrogen in a minimal basis, but makes a bit more explicit all the required pieces. Another good example, perhaps without ...


2

In the article you mentioned it is said that classical algorithms can beat some cases of (quantum ) QAOA's as is proved in this article. So finding cases where quantum QAOA can still beat classical algorithms and can run on NISQ devices with low depth circuits is still exciting and promissing. The article uses plausible conjectures from complexity theory to ...


2

There is currently no way to check the status of a job in Qiskit Aqua: https://github.com/Qiskit/qiskit-aqua/issues/545 However, it looks like it is a feature that is coming.


2

So in your example, you try to find the quantum circuit representing the Toffoli operation. I would then change my objective/fitness function and compare the unitary matrix representing the operation. You can use an minimization objective like : $$ \mathcal{F} = 1-\frac{1}{2^n} |\operatorname{Tr}(U_aU_t^{\dagger})| $$ with $ U_a $ is the unitary of the ...


1

Let's answer my own question: it is not possible. After some research I ended up computing the "truth table" for the two possible cases: $b = 0$: $\vert 00 \rangle\rightarrow\vert 00 \rangle$ $\vert 01 \rangle\rightarrow\vert 10 \rangle$ $\vert 10 \rangle\rightarrow\vert 10 \rangle$ $\vert 11 \rangle\rightarrow\vert 01 \rangle$ $b = 1$: $\vert 00 \...


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