17 research outputs found
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
Better bounds for low-energy product formulas
Product formulas are one of the main approaches for quantum simulation of the
Hamiltonian dynamics of a quantum system. Their implementation cost is computed
based on error bounds which are often pessimistic, resulting in overestimating
the total runtime. In this work, we rigorously consider the error induced by
product formulas when the state undergoing time evolution lies in the
low-energy sector with respect to the Hamiltonian of the system. We show that
in such a setting, the usual error bounds based on the operator norm of nested
commutators can be replaced by those restricted to suitably chosen low-energy
subspaces, yielding tighter error bounds. Furthermore, under some locality and
positivity assumptions, we show that the simulation of generic product formulas
acting on low-energy states can be done asymptotically more efficiently when
compared with previous results