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My master thesis was on using machine learning techniques to synthesise quantum circuits. Since any operation on a QC can be represented as a unitary matrix my

My master thesis was on using machine learning techniques to synthesise quantum circuits. Since any operation on a QC can be represented as a unitary matrix my research topic was that, using ML, given a set of gates, how many and in what arrangement of them you could generate or at least approximate this matrix. Another aspect was, given a unitary matrix, could a neural network predict a number of gates needed to simulate that matrix as a QC and thereby give us a measure of complexity. It was a lot of fun to test different algorithms from genetic algorithms to neural network architectures. Back then NNs were a lot smaller and I trained them mostly on one GPU and since the matrices get exponentially bigger with the amount of qbits in the circuit it was only possible for me to investigate small circuits with less than a dozen qubits but it was still nice to see that in principle this worked quite well.

sigmoid10 2y ago
At which university? I literally know a guy who did exactly the same thing for his master thesis. I'm wondering if the world is so small or this specific topic is so common.
- Escapado 2y ago
  University of Hamburg in Germany.
IIAOPSW 2y ago
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.
- westurner 2y ago
  Solvay-Kiteav theorem: https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem /? Solvay-Kiteav theorem Cirq QISkit: https://www.google.com/search?q=Solvay-Kiteav+theorem+cirq+qiskit https://www.google.com/search?q=Solvay-Kiteav+theorem+cirq+q... qiskit/transpiler/passes/synthesis/solovay_kitaev_synthesis.py: https://github.com/Qiskit/qiskit/blob/main/qiskit/transpiler/passes/synthesis/solovay_kitaev_synthesis.py https://github.com/Qiskit/qiskit/blob/main/qiskit/transpiler... qiskit/synthesis/discrete_basis/solovay_kitaev.py: https://github.com/Qiskit/qiskit/blob/stable/1.2/qiskit/synthesis/discrete_basis/solovay_kitaev.py#L24-L185 https://github.com/Qiskit/qiskit/blob/stable/1.2/qiskit/synt... SolovayKitaevDecomposition: https://docs.quantum.ibm.com/api/qiskit/qiskit.synthesis.SolovayKitaevDecomposition https://docs.quantum.ibm.com/api/qiskit/qiskit.synthesis.Sol... What are more current alternatives to the Solvay-Kiteav theorem for gate-based quantum computing?
  - IIAOPSW 2y ago
    The more current alternatives (iirc) are usually specific to the particular hardware and the gates its capable of rather than a general solution. Though to be fair this topic was tangential to my focus and I'm a bit rusty on it so I'd have to look around to answer that.
- Escapado 2y ago
  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.
OscarCunningham 2y ago
I would be interesting if finding quantum circuits was one of the things that quantum computers were good at.
Xcelerate 2y ago
> given a unitary matrix, could a neural network predict a number of gates needed to simulate that matrix as a QC and thereby give us a measure of complexity That's really interesting. I'm curious—did you explore whether the predictivity of the neural network was influenced by any hidden subgroup structure in the unitary matrix? Seems like the matrix symmetries could play a significant role in determining the gate complexity.
- Escapado 2y ago
  I didn’t include this in my thesis but from what I remember looking at hundreds if not thousands of matrices and their QC solutions some symmetries would immediately make it so that way less gates would be needed but it could also be deceptive and completely depended on which gate set you have available. If you take the matrix for the quantum furier transform for example with a gate set of phase gates and a hadamard gate then for the 100% solution you n hadamard gates and n! phase gates for an n qubit circuit even though the matrix is highly symmetrical. If your gate set was Clifford + toffli you would be able to do it with n*log(n). And then depending on how close you wanna approximate it you could get away with even less. But I have not gone into further analysis on which symmetries would have which effect on which gate set and whether it would influence predictivity. But it would be fun to investigate for sure!

sigmoid10 2y ago

At which university? I literally know a guy who did exactly the same thing for his master thesis. I'm wondering if the world is so small or this specific topic is so common.

Escapado 2y ago
University of Hamburg in Germany.

IIAOPSW 2y ago

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.

westurner 2y ago
Solvay-Kiteav theorem: https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem https://en.wikipedia.org/wiki/Solovay%E2%80%93Kitaev_theorem /? Solvay-Kiteav theorem Cirq QISkit: https://www.google.com/search?q=Solvay-Kiteav+theorem+cirq+qiskit https://www.google.com/search?q=Solvay-Kiteav+theorem+cirq+q... qiskit/transpiler/passes/synthesis/solovay_kitaev_synthesis.py: https://github.com/Qiskit/qiskit/blob/main/qiskit/transpiler/passes/synthesis/solovay_kitaev_synthesis.py https://github.com/Qiskit/qiskit/blob/main/qiskit/transpiler... qiskit/synthesis/discrete_basis/solovay_kitaev.py: https://github.com/Qiskit/qiskit/blob/stable/1.2/qiskit/synthesis/discrete_basis/solovay_kitaev.py#L24-L185 https://github.com/Qiskit/qiskit/blob/stable/1.2/qiskit/synt... SolovayKitaevDecomposition: https://docs.quantum.ibm.com/api/qiskit/qiskit.synthesis.SolovayKitaevDecomposition https://docs.quantum.ibm.com/api/qiskit/qiskit.synthesis.Sol... What are more current alternatives to the Solvay-Kiteav theorem for gate-based quantum computing?
- IIAOPSW 2y ago
  The more current alternatives (iirc) are usually specific to the particular hardware and the gates its capable of rather than a general solution. Though to be fair this topic was tangential to my focus and I'm a bit rusty on it so I'd have to look around to answer that.
Escapado 2y ago
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.

OscarCunningham 2y ago

I would be interesting if finding quantum circuits was one of the things that quantum computers were good at.

Xcelerate 2y ago

> given a unitary matrix, could a neural network predict a number of gates needed to simulate that matrix as a QC and thereby give us a measure of complexity That's really interesting. I'm curious—did you explore whether the predictivity of the neural network was influenced by any hidden subgroup structure in the unitary matrix? Seems like the matrix symmetries could play a significant role in determining the gate complexity.

Escapado 2y ago
I didn’t include this in my thesis but from what I remember looking at hundreds if not thousands of matrices and their QC solutions some symmetries would immediately make it so that way less gates would be needed but it could also be deceptive and completely depended on which gate set you have available. If you take the matrix for the quantum furier transform for example with a gate set of phase gates and a hadamard gate then for the 100% solution you n hadamard gates and n! phase gates for an n qubit circuit even though the matrix is highly symmetrical. If your gate set was Clifford + toffli you would be able to do it with n*log(n). And then depending on how close you wanna approximate it you could get away with even less. But I have not gone into further analysis on which symmetries would have which effect on which gate set and whether it would influence predictivity. But it would be fun to investigate for sure!