Conditions for the solvability of the Cauchy problem for linear first-order functional differential equations

Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family and are necessary for the solvability of the Cauchy problem for all equations from the family. In contrast to many known articles, we consider equations with functional operators acting into the space of essentially bounded functions

On solvability of periodic boundary value problems for second order linear functional differential equations

The periodic boundary value problem for second order linear functional differential equations with pointwise restrictions (instead of integral ones) is considered. Sharp sufficient conditions for the solvability are obtained

On solvability of focal boundary value problems for higher order functional differential equations with integral restrictions

Sharp conditions are obtained for the unique solvability of focal boundary value problems for higher-order functional differential equations under integral restrictions on functional operators. In terms of the norm of the functional operator, unimprovable conditions for the unique solvability of the boundary value problem are established in the explicit form. If these conditions are not fulfilled, then there exists a positive bounded operator with a given norm such that the focal boundary value problem with this operator is not uniquely solvable. In the symmetric case, some estimates of the best constants in the solvability conditions are given. Comparison with existing results is also performed

On the solvability of the periodic problem for systems of linear functional differential equations with regular operators

Systems of two linear functional differential equations of the first order with regular operators are considered. General necessary and sufficient conditions for the unique solvability of the periodic problem are obtained. For one system with monotone operators we get effective necessary and sufficient conditions for the unique solvability of the periodic problem

On periods of non-constant solutions to functional differential equations

We show that periods of solutions to Lipschitz functional differential equations cannot be too small. The problem on such periods is closely related to the unique solvability of the periodic value problem for linear functional differential equations. Sharp bounds for periods of non-constant solutions to functional differential equations with Lipschitz nonlinearities are obtained

Obstacles to Variational Quantum Optimization from Symmetry Protection

The quantum approximate optimization algorithm (QAOA) employs variational states generated by a parameterized quantum circuit to maximize the expected value of a Hamiltonian encoding a classical cost function. Whether or not the QAOA can outperform classical algorithms in some tasks is an actively debated question. Our work exposes fundamental limitations of the QAOA resulting from the symmetry and the locality of variational states. A surprising consequence of our results is that the classical Goemans-Williamson algorithm outperforms the QAOA for certain instances of MaxCut, at any constant level. To overcome these limitations, we propose a nonlocal version of the QAOA and give numerical evidence that it significantly outperforms the standard QAOA for frustrated Ising models