Grammatical Fixes in qaoa.ipynb

[davisclarke]

1. Added proper punctuation for the opening bullet points (bullet point punctuation with longer bullet points.
2. QAOA (Quantum Approximate Optimization Algorithm) introduced by Farhi et al .[1] -> QAOA (Quantum Approximate Optimization Algorithm) was introduced by Farhi et al [1].
3. For that we would first define
   the underlying graph of the problem shown above. -> For that, we would first define the underlying graph of the problem as shown above.
4. Such a state, when number of qubits is 4 ($n=4$), can be prepared by applying Hadamard gates starting from an all zero state as shown in
   the circuit below. -> Such a state, when the number of qubits is 4 ($n=4$), can be prepared by applying Hadamard gates starting from an all-zero state as shown in the circuit below.
5. So far we have seen that the preparation of a quantum state during QAOA is composed of three elements -> So far, we have seen that the preparation of a quantum state during QAOA is composed of three elements:
6. Such an expectation can be obtained by doing measurement in the Z-basis. -> Such an expectation can be obtained by doing measurements in the Z-basis.
7. Hence the general form a combinatorial optimization problem is given by -> Hence, the general form a combinatorial optimization problem is given by
8. This cost function can be mapped to a Hamiltonian that is diagonal in the computational basis. Given the cost-function $C$ this Hamiltonian is then written as -> This cost function can be mapped to a Hamiltonian that is diagonal in the computational basis. Given the cost function, $C$, this Hamiltonian is then written as
9. The cost function to be optimized is in this case the sum of weights of edges connecting points in the two different subsets, crossing the cut. -> The cost function to be optimized is, in this case, the sum of weights of edges connecting points in the two different subsets, crossing the cut.
10. This can of course be turned again into a decision problem if we ask where there exists a bit string that satisfies more than $\tilde{m}$ of the $m$ clauses, which is again $NP$-complete. -> Of course, this can be turned again in to a decision problem if we ask where there exists a bit string that satisfies more than $\tilde{m}$ of the $m$ clauses, which is again $NP$-complete. (Redundancy)
11. Both the previously considered problems $MAXCUT$ and $\text{MAX 3-SAT}$ are actually known to be a NP-hard problems 3. -> Both the previously considered problems $MAXCUT$ and $\text{MAX 3-SAT}$ are known to be a NP-hard problems 3. (Redundancy)
12. In fact it turns out that many combinatorial optimization problems are computationally hard to solve in general -> In fact, it turns out that many combinatorial optimization problems are computationally hard to solve in general
13. In light of this fact, we can't expect to find a provably efficient algorithm, i.e. an algorithm with polynomial runtime in the problem size, that solves these problems. This also applies to quantum algorithms. There are two main approaches to dealing with such problems. First approach is approximation algorithms that are guaranteed to find solution of specified quality in polynomial time. The second approach are heuristic algorithms that don't have a polynomial runtime guarantee but appear to perform well on some instances of such problems. -> In light of this, we can't expect to find a provably efficient algorithm, i.e., an algorithm with polynomial runtime in the problem size, that solves these problems; this also applies to quantum algorithms. There are two main approaches to dealing with such problems. First approach is approximation algorithms that are guaranteed to find solution of specified quality in polynomial time. The second approach, heuristic algorithms, don't have a polynomial runtime guarantee, but appear to perform well on some instances of such problems. (Clarity, redundancy, comma rules)
14. For the $MAXCUT$ problem there is a famous approximate algorithm due to Goemans and Williamson 2 .-> For the $MAXCUT$ problem, there is a famous approximate algorithm due to Goemans and Williamson 2.
15. This algorithm is based on an SDP relaxation of the original problem combined with a probabilistic rounding technique that yields an with high probability approximate solution $\textbf{x}^$ that has an approximation ratio of $\alpha \approx 0.878$. This approximation ratio is actually believed to optimal so we do not expect to see an improvement by using a quantum algorithm.-> This algorithm, based on an SDP relaxation of the original problem, is combined with a probabilistic rounding technique that yields a high probability approximate solution $\textbf{x}^$ that has an approximation ratio of $\alpha \approx 0.878$. This approximation ratio is believed to be optimal. So, we do not expect to see an improvement by using a quantum algorithm.
16. The Quantum approximate optimization algorithm (QAOA) -> The Quantum Approximate Optimization Algorithm (QAOA)
17. QAOA takes the approach of classical approximate algorithms and looks for a quantum analogue that will likewise produce a classical bit string $x^$ that with high probability is expected to have a good approximation ratio $\alpha$. -> QAOA takes the approach of classical approximate algorithms and looks for a quantum analogue that will likewise produce a classical bit string $x^$. The outputted bit string is expected, with a high probability, to have a good approximation ratio $\alpha$. (Run-on sentence)

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[davisclarke]

Sorry. For some reason, an already merged commit showed up here. Second, should I list the changes I made in the comments? It tends to format oddly outside of Jupyter and would be a bit of a timesaver.

[frankharkins]

Thanks! Most changes are great, I've just left a couple of comments.

[frankharkins]

Looks good, thanks!

[frankharkins]

Tests are failing but I'm merging as they're not related to this PR, and fixing them is going to be very involved.