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I am trying to implement the Mixer of the Max-Independent Set from The Quantum Alternating Operator Ansatz. From this paper: https://arxiv.org/pdf/1709.03489.pdf in Chapter 4.2 page 15 to 17.

For every verice $v$ in the Graph this operator $H_{CX,v}$ is definded. ($nbhd(v)$ are all the neighbors of v) \begin{align*} H_{CX,v} &= X_{v} H_{NOR(x_{nbhd(v)})} \\ &= 2^{-D_{v}}X_{v}\prod_{w \in nbhd(v)}(I + Z_{w})\\ \end{align*}

Then the operators for all vertices are summed up. Are there any rules how to sum up operators? $$H_{CX} = \sum_{i = 0}^{n-1}H_{CX,v_{i}}$$

After that the exponential of the Matrix is calculated. I have no clue how to do that. $$U_{sim-CX}(\beta) = e^{-i\beta H_{CX}}$$

Should I maybe do a Hamiltonian-simulation or are there other ways for doing this?

Thanks for reading :)

Hannah
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  • Yes, but you can simulate it with Trotter approximation. Look at this answer: https://quantumcomputing.stackexchange.com/questions/5567/circuit-construction-for-hamiltonian-simulation Afterward, you will see that it is not so bad to construct a circuit to do such simulation. – KAJ226 Oct 25 '20 at 19:44
  • @KAJ226 Thankyou :) So you think that a Hamiltonian simulation is what Hadfield indendet in the paper? (if you have read the paper) – Hannah Oct 26 '20 at 00:05
  • In a sense, yes. Hamiltonian simulation is essentially simulating the Hamiltonian $H$ as $e^{-iHt}$. Hence you can see the resemblance here. – KAJ226 Oct 26 '20 at 03:05
  • Thank you so much @KAJ226 – Hannah Oct 26 '20 at 12:54
  • @KAJ226 Are there any better methods for this specific case to simulate this Hamiltonian than the Trotter approximation? – Hannah Nov 16 '20 at 07:51

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