This article explores search strategies for the design of parameterized quantum circuits. We propose several optimization approaches including random search plus survival of the fittest, reinforcement learning both with classical and hybrid quantum classical controllers, and Bayesian optimization as decision makers to design a quantum circuit in an automated way for a specific task such as multi-labeled classification over a dataset. We introduce nontrivial circuit architectures that are arduous to be hand-designed and efficient in terms of trainability. In addition, we introduce reuploading of initial data into quantum circuits as an option to find more general designs. We numerically show that some of the suggested architectures for the Iris dataset accomplish better results compared to the established parameterized quantum circuit designs in the literature. In addition, we investigate the trainability of these structures on the unseen dataset Glass. We report meaningful advantages over the benchmarks for the classification of the Glass dataset which supports the fact that the suggested designs are inherently more trainable.
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This work is partially supported by the Defense Advanced Research Projects Agency as part of the project W911NF2010022: The Quantum Computing Revolution and Optimization: Challenges and Opportunities.
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Appendix A: Hyperparameters and optimization schemes for R-QCDS
There are two separate parts in the R-QCDS environment, a controller and a PQC. Regardless of having a fully classical controller (DNN) or a Q-controller, the optimization part is on classical optimization methods. When we have a Q-controller, then a fixed small structure shown in Fig. 3 will be used, one for any decision to be made. That is why we limited the Q-controller to the ones where there is only one layer and that layer repeats itself. The structure is fixed while the rotation 𝜃 s are parameters of the controller and are going through optimization scheme to be optimized. (Similar to the parameters of the DNN if the controller was a classical one)
In this study for R-QCDS, we use Adam optimization algorithm (Kingma and Ba 2014) with β1 = 0.9, β2 = 0.999 and 𝜖 = 0.001 to optimize the parameters of both controller network and PQC; However, we use separate learning schemes with different learning rates. This allows independent learning paces for the two structures of the framework. Our studies show that the learning rate for controller network can be larger than the PQC learning rate. For instance, in Fig. 7 controller learning rates are 0.1, 0.2, 0.02, and 0.02 for the different R-QCDS approaches in the same order written in the plot legend while PQC learning rate is 0.01 for all the cases. Table 1 presents the hyperparameters for the R-QCDS framework. After a design is being suggested by the algorithm, then the focus is on the PQC and the controller hyperparameters are out of use.
Appendix B: New datasets and the structure of the discovered circuits
Generally, in the literature of PQC, there are two approaches towards dealing with the features of a dataset: (1) using preprocessing and reduction techniques to bring down the size of the features in the dataset to make it manageable for NISQ devices and (2) considering one qubit per one data feature. We point out that in our study we consider one qubit per one data feature. For the first approach though, for instance, Farhi et al. (2014) simply crop 4 × 4 = 16 feature sections out of 28 × 28 = 784 MNIST images. However, such modifications completely change the dataset. The remaining dataset is no longer a representation of MNIST. In general, the performance of the whole framework would be greatly affected by how we reduce the feature dimension.
In our study, to show the robustness of our claims, we use datasets that are already manageable in size by quantum simulators and there is no need for feature reduction. That is why we choose Iris and Glass datasets. Their original feature spaces equal to four and nine and we further consider four and nine qubit circuits for them respectively to consider a qubit per feature.
In addition, We aim to show the structures that we found are inherently strong in learning and capturing the information of other unseen datasets over which they have not been trained in comparison with standard benchmark structures. So, we do not want to change the structures, however, the implementation of these designs for the new task is not straightforward because the designs we discovered so far are for four qubit circuits and now we want nine qubit circuits and as we said earlier we prohibit ourselves from changing the Glass dataset to fit to our circuit. Hence, we set this simple rule that if needed (like this case) we repeat the structure of a discovered circuit to the point that we fill out the new circuit widthwise.
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Pirhooshyaran, M., Terlaky, T. Quantum circuit design search. Quantum Mach. Intell. 3, 25 (2021). https://doi.org/10.1007/s42484-021-00051-z
- Parameterized quantum circuits
- Quantum circuit design
- Circuit design search
- Random quantum circuits
- Multi-labeled classification
- Reinforcement learning