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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 ...


6

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

What motivated this construction is mentioned in the original paper (section VI): adiabatic quantum computing. This construction is basically a Trotterized version of the evolution by the time dependent hamiltonian: $$ H(t) = (1-t/T)B + (t/T) C $$ where T is the total runtime. The Trotterized evolution consists of alternately applying $U_{C}$ and $U_{B}$. ...


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|...


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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 (...


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 ...


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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.


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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

I will try to give you a hint how to rewrite your problem as an binary optimization problem which can be solved on quantum annealer, i.e. a single purpose quantum computer for solving optimization task (see more about annealers here). Your problem as an binary assignment problem: Lets denote $x_{ik} \in \{0;1\}$ a binary variable meaning whether a person $...


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The objective of the portfolio optimization problem is to trade off expected return ($\mu^T x$) with the risk taken ($x^T \Sigma $x). This could be achieved by introducing a constraint on the risk, e.g. $x^T \Sigma x \leq R$, for an acceptable risk level $R$ and then maximize the return under this constraint. However, this is not a QUBO, i.e., it cannot be ...


1

From what I understand, $x_{i,t}$ are the binary variables. So your QUBO matrix should not be indexed as Q[i][t]. If you do this way, this means you have a binary variable $x_i$ and a binary variable $x_t$ and they have a real coefficient, so representing a term $Q[i][j] *x_i x_j$. In this case, if you really want a QUBO matrix with a correct indexing, you ...


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Based on comment by DaftWullie and my experience with the algortihm, it seems that a title of the article is misleading. The authors claim that algorithm they proposed is efficient. However, this is true only partialy. The authors devised only part of an algorithm for solving TSP. In particular, they are able to calculate length of a Hamiltonian cycle ...


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