How did it do compared to the baseline of the solvoy kitaev algorithm (and the more advanced algorithms that experts have come up with since)?
How (if at all) can the ML approach come up with a circuit when the unitary is too big to explicitly represent it as all the terms in the matrix but the general form of it is known? Eg it is known what the quantum fourier transform on N qubits is defined to be, and consequently any particular element within its matrix is easy to calculate, but you don't need (and shouldn't try) to write it out as a 2^n x 2^n matrix to figure out its implementation.
I have not investigated this as much as I should have but if I remember correctly there were cases where the NN approach was yielding smaller solutions. Would be a great follow up to the thesis.
The way this was turned into an optimization problem was to assume a NN to input an identity matrix and then have a custom layer in there to generate S(2) unitaries (exponential form) for which the phase parameters are then the learned parameters in the NN. Similarly for global XX gates. Then from this the final unitary can be computed and compared against the desired unitary and a loss function can be derived. I remember it was a little fiddly to implement these custom layers in tensorflow since many of the functions didn’t work for imaginary numbers. But yeah in short a circuit structure was assumed (alternating between a global XX and single qubit operations) for which the phases of their generating matrices were the learned parameters of the network. Then multiple topologies (how many steps in the QC) could be validated. I think the coolest result was that NNs consistently outperformed other optimisation strategies to learn those parameters.
Interesting. I agree a follow up result would be fascinating.
w.r.t. imaginary numbers and tensor flow, as you may have come across at some point, you can always map a quantum circuit over to one that only uses real valued amplitudes at the cost of just a single ancila qubit. Just replace i with |1> on the ancila and all multiplications by a phase with a controlled rotation on the ancila. In other words, the phase is a qubit the universe gives you for free.
How (if at all) can the ML approach come up with a circuit when the unitary is too big to explicitly represent it as all the terms in the matrix but the general form of it is known? Eg it is known what the quantum fourier transform on N qubits is defined to be, and consequently any particular element within its matrix is easy to calculate, but you don't need (and shouldn't try) to write it out as a 2^n x 2^n matrix to figure out its implementation.