Based on our previous works, and in order to relate them with the theory of quantum graphs and the quantum computing principles, we once again try to introduce some newly developed technical structures just by relying on our toy example, the coordinate ring of
n×n quantum matrix algebra
Mq(n), and the associated directed locally finite graphs
G(Πn), and the Cuntz-Krieger
C∗-graph algebras. Meaningly, we introduce a
(4i−6)-qubit quantum system by using the Cuntz-Krieger
G(Πi)-families associated to the
4i−6 distinct Hamiltonian paths of
G(Πi), for
i∈{2,⋯,n}. We also will present a proof of a claim raised in our previous paper concerning the graph
C∗-algebra structure and the associated Cuntz-Krieger
G(Πn)-families.
Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit’s generators. The barren plateau problem represents one of the major bottlenecks for parametrized quantum circuits algorithms. Here, the authors study the known sources of BP using the lens of Lie algebraic theory, finding an expression of the variance of the loss function depending on the dynamical Lie algebra of the circuit.
We show the relationship between the Fourier coefficients and the barren plateau problem emerging in parameterized quantum circuits. In particular, the sum of squares of the Fourier coefficients is exponentially restricted concerning the qubits under the barren plateau condition. Throughout theory and numerical experiments, we introduce that this property leads to the vanishing of a probability and an expectation formed by parameterized quantum circuits. The traditional barren plateau problem requires the variance of gradient, whereas our idea does not explicitly need a statistic. Therefore, it is not required to specify the kind of initial probability distribution.