  In a conventional circuit for quantum machine learning, the quantum gates
used to encode the input parameters and the variational parameters are
constructed with a fixed order. The resulting output function, which can be
expressed in the form of a restricted Fourier series, has limited flexibility
in the distributions of its Fourier coefficients. This indicates that a fixed
order of quantum gates can limit the performance of quantum machine learning.
Building on this key insight (also elaborated with examples), we introduce
indefinite causal order to quantum machine learning. Because the indefinite
causal order of quantum gates allows for the superposition of different orders,
the performance of quantum machine learning can be significantly enhanced.
Considering that the current accessible quantum platforms only allow to
simulate a learning structure with a fixed order of quantum gates, we reform
the existing simulation protocol to implement indefinite causal order and
further demonstrate the positive impact of indefinite causal order on specific
learning tasks. Our results offer useful insights into possible quantum effects
in quantum machine learning.
