# Applying Group Leaders Optimization to Quantum Belief Systems

Context: I am particularly interested in quantum cognition & would like to use a tool like pyZX to perform the following types of optimizations.

In Preparing a (quantum) belief system they "establish a theoretical result showing that in the absence of constraints on measurements, any state can be obtained as the result of a suitable sequence of measurements."

It is also stated in the paper that "achieving the desired belief state may require a sequence of measurements that is not practically feasible."

How could the Group Leaders Optimization Algorithm be applied in the above context to generate practically feasible sequences of measurements?

The GLOA is just an optimization algorithm (another genetic algorithm actually). So as long as your problem translates into an objective function you seek to minimize/maximize, this would be possible (even by another genetic algorithm).

I suggest to first think how you encode your problem for the optimization. For example, a sequence of discrete and/or real variables. The objective function is also important to specify. I am guessing here you would look at the fidelity with the desired belief state.

The only thing however you would have to consider is the definition of being practically feasible as I don't know what the authors refer to

a sequence of measurements that is not practically feasible.

However, let us take the example of the GLOA applied to find the decomposition of a unitary operator U into a sequence of gates (or circuit) within a predefined set, which you can find the explanation in this article. This was discussed in this question before. Maybe it will help you relate to your problem. The idea is to represent a circuit as a sequence of variables taking integer or real values. For instance, let's try to find the decomposition of the Toffoli gate using the set of gates $$\{V,Z,S,V^{\dagger},R_x(\theta) \}$$ and we allow to use maximum 5 gates. We start by randomly creating candidate circuits. Here is one example of candidate circuit (showing you an encoding of the problem) :

1 3 2 0.0; 2 3 1 0.0; 3 2 1 0.0; 4 3 2 0.0; 5 1 3 0.75

You see you have 5 sequences of 4 numbers (3 integers and one real). Each sequence represent an application of a gate. 1 3 2 0.0, in this case, means apply the $$V$$ gate (index 1 in the set) on qubit $$3$$ controlled by qubit $$2$$ with an angle $$0.0$$ (note that with the $$V$$ gate there is no angle but say you were playing with like rotation gates, this becomes relevant). So it is like a problem of 20 variables to tweak for the optimization scheme.

The optimization will consist of finding the solution circuit represented by a unitary operator $$U_a$$ close to the unitary operator of interest $$U_t$$; using GLOA which maximize the trace fidelity (here serving as an objective function telling us how close the candidate is to the solution): $$\mathcal{F} = \frac{1}{N}|\operatorname{Tr}(U_aU_t^{\dagger})|$$

Say you were looking for quantum states, the objective function would be for instance to minimize a distance between two quantum states with one being the desired solution. In the book of Nielsen and Chuang, you have chapter 9.2 which is about quantifying how two quantum states are close.

• Could you potentially elaborate on encoding methods, objective functions, fidelity & feasibility? – user820789 Oct 29 '18 at 18:46
• I can take the example of the GLOA used to find a decomposition of a unitary operator if you want and you can relate to quantum belief state yourself. For the feasibility, it is just that I don't know what authors refer to as a sequence of measurements that is not practically feasible. And I am not ready to go into the details of the subject now as my efforts are directed on another topics. Would that be enough? – cnada Oct 29 '18 at 20:38
• Any additional information you would be willing to provide would be greatly appreciated! – user820789 Oct 29 '18 at 20:45