[4.2.4] Travelling Salesman Problem with QPE: Error in the Hamiltonian Cycles

[nquetschlich]

Describe the bug

https://qiskit.org/textbook/ch-paper-implementations/tsp.html#general_strategy

At the end of the "General Strategy" Section, all three Hamiltonian cycles are listed as:

Sequence path	Eigenstates
1−2−3−4	|11000110 ⟩
1−4−2−3	|10001101 ⟩
1−3−2−4	|11001001 ⟩

I think that the last two sequences describe the same Hamiltonian cycle, because a Hamiltonian Cycles ends at the starting node. As a consequence, cycle three is equal to cycle two in reversed order.

I suggest to substitute cycle three with the sequence 1-2-4-3 and its eigenvalue |10000111 ⟩.

To Reproduce

The error becomes clear when drawing the cycles as shown in the screenshot provided under "Expected behavior".

Expected behavior

As one can see, the Hamiltonian cycles two and three are equivalent. If cycle 3 is substituted as suggested, the resulting three cycles would be:

Version info

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Additional context

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