No-faster-than-light-signaling implies linear evolutions. A re-derivation

There is a growing interest, both from the theoretical as well as experimental side, to test the validity of the quantum superposition principle, and of theories which explicitly violate it by adding nonlinear terms to the Schr\"odinger equation. We review the original argument elaborated by Gisin (1989 Helv. Phys. Acta 62 363), which shows that the non-superluminal-signaling condition implies that the dynamics of the density matrix must be linear. This places very strong constraints on the permissible modifications of the Schr\"odinger equation, since they have to give rise, at the statistical level, to a linear evolution for the density matrix. The derivation is done in a heuristic way here and is appropriate for the students familiar with the textbook quantum mechanics and the language of density matrices.Comment: 17 pages, 7 figure

Simulating quantum circuit expectation values by Clifford perturbation theory

The classical simulation of quantum circuits is of central importance for benchmarking near-term quantum devices. The fact that gates belonging to the Clifford group can be simulated efficiently on classical computers has motivated a range of methods that scale exponentially only in the number of non-Clifford gates. Here, we consider the expectation value problem for circuits composed of Clifford gates and non-Clifford Pauli rotations, and introduce a heuristic perturbative approach based on the truncation of the exponentially growing sum of Pauli terms in the Heisenberg picture. Numerical results are shown on a Quantum Approximate Optimization Algorithm (QAOA) benchmark for the E3LIN2 problem and we also demonstrate how this method can be used to quantify coherent and incoherent errors of local observables in Clifford circuits. Our results indicate that this systematically improvable perturbative method offers a viable alternative to exact methods for approximating expectation values of large near-Clifford circuits