Symmetry exploitation in quantum circuits

We observe the high permutative symmetry in quantum machine learning and use it as a tool to restrict the Hilbert space necessary for finding the solution.

Quantum Machine Learning (QML) as an intermix of machine learning and quantum computing uses tunable quantum circuits to train models. These models exhibit a number of unique problems, in particular a heightened presence of barren plateaus[1], high noise and expensive training, both in time and financial cost.

Our approach relies on the high permutation symmetry in machine learning. Permuting the input vectors and matrices inside the weights of each layer returns the same model prediction. In the context of QML this can be exploited to order the states within the model by the value of their corresponding weights. Simple counting arguments can be used to restrict the solution space to ½^n of the total Hilbert space, n the number of available states. This follows from the realization that the weights have either negative or positive values. With a permutation of the values we can ensure that they are always located in their respective halves of the Hilbert space.

Our work is a necessary stepping stone to increasing the applicability of QML and countering some of the problems that are experienced in practice. Our solution is a simple yet powerful restriction in the search space without losing any of the generality that makes machine learning so broadly applicable.

[1] McClean, J. R. et al. Barren plateaus in quantum neural network training landscapes. Nat. Comm. 9, 1