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I'm trying to evolve quantum circuits using genetic algorithms as they did in this paper Decomposition of unitary matrices for finding quantum circuits: Application to molecular Hamiltonians (Daskin & Kais).

So given a unitary target matrix ($U_t$), the task is to evolve a circuit that approximates $U_t$ (let's call it $U_a$). This is the function I'm maximizing:

$$F = \frac{1}{2^n}|\operatorname{Tr}(U_aU_t^{\dagger})|$$ (where n is the number of qubits)

My circuit encoding is a little bit different, I'm using a json string representation, here's an example of a random circuit:

[{0: ('Y', -1, None), 1: ('Td', -1, None), 2: ('Y', -1, None)}, 
{0: ('Z', 2, None), 1: ('Y', -1, None)}, 
{0: ('Vd', 1, None), 2: ('V', -1, None)}, 
{1: ('X', -1, None), 2: ('Y', 0, None)}, 
{0: ('Td', -1, None), 2: ('V', 1, None)}, 
{1: ('S', -1, None), 2: ('X', 0, None)}]

Every dictionary item in the array is a column in the format {target: (gate, control, param)}. The value -1 means no control, and None means there are no parameters. Here's a visual representation of this circuit:

+--------+-------+-------+-------+-------+-------+-------+
| qubit  | col_0 | col_1 | col_2 | col_3 | col_4 | col_5 |
+--------+-------+-------+-------+-------+-------+-------+
| q0-->  |   Y   |  Z•2  |  Vd•1 |   I   |   Td  |   I   |
| q1-->  |   Td  |   Y   |   I   |   X   |   I   |   S   |
| q2-->  |   Y   |   I   |   V   |  Y•0  |  V•1  |  X•0  |
+--------+-------+-------+-------+-------+-------+-------+

So far, I've been successful at finding very small circuits (with 2 "columns"). The problem comes when trying to evolve solutions for more complex circuits. For example, the paper shows this solution for the Toffoli unitary:

enter image description here

My algorithm is not able to find this solution or any other exact solution to this problem. I tried changing the selection method, crossover and mutation rates, but the fitness value never exceeds 0.85. After some generations, it seems that all individuals are converging to the same non-optimal circuit (this happens even with uniform crossover and random selection).

For crossover, I've tried one-point crossover and uniform crossover. The mutation operation can do these things:

  1. If it's a controlled gate, it can change the type of the target gate, can change the position of the control, can swap target/control, can remove the control, or can shuffle a column.
  2. If it's not a controlled gate, it can change the type of gate or introduce a control.

I don't see what else can I do to improve these operators. In the paper, they also have a continuous angle phase parameter, but this particular circuit solution does not use it anyway.

Another curious fact: for this particular problem, the empty circuit has a fitness of 0.75. If a take the exact solution (from the image) and change a single gate, I can get a fitness of 0.5. So, an empty circuit has better fitness than a circuit which is a small step from the exact solution. This is very confusing, I don't see where the GA will find "optimal sub-circuits" to exchange during crossover.

It should be noted that the paper uses a more sophisticated version of a GA, but for the reasons stated above, I don't think it will make a difference in my case. So what I'm missing or how can I improve this? Thank you!

Edit

It seems the GLOA did it's work. The algorithm found this for the Toffoli unitary:

+--------+-------+-------+-------+-------+-------+
| qubit  | col_0 | col_1 | col_2 | col_3 | col_4 |
+--------+-------+-------+-------+-------+-------+
| q0-->  |   I   |   I   |   I   |   I   |   I   |
| q1-->  |  Z•2  |   I   |   I   |   I   |  S•0  |
| q2-->  |   I   |  Vd•0 |  Z•1  |  V•0  |   I   |
+--------+-------+-------+-------+-------+-------+

But I limited the kind of gates it could use, now I'll try with all the gates.

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  • $\begingroup$ @Blue I'm implementing GLOA, let's see what happens. $\endgroup$
    – Fernando
    Commented Apr 2, 2019 at 19:16
  • $\begingroup$ Well, it worked. The GLOA found this for the Toffoli unitary: [{1:('Z', 2, None)}, {2:('Vd', 0, None)}, {2:('Z', 1, None)}, {2:('V', 0, None)}, {1:('S', 0, None)}] $\endgroup$
    – Fernando
    Commented Apr 2, 2019 at 21:07
  • $\begingroup$ Great! Oh, so you weren't using the GLOA before? Did you use some other genetic algorithm? $\endgroup$ Commented Apr 2, 2019 at 21:10
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    $\begingroup$ I think that the GLOA works like a ensemble of GAs, just like a ML ensemble of models. When there are few knobs to turn, all individuals in a classic GA tend to become equal, the GLOA try to avoid this I think. $\endgroup$
    – Fernando
    Commented Apr 2, 2019 at 21:19

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