Polylogarithmic-depth controlled-NOT gates without ancilla qubits.

Controlled operations are fundamental building blocks of quantum algorithms. Decomposing n-control-NOT gates (C(X)) into arbitrary single-qubit and CNOT gates, is a crucial but non-trivial task. This study introduces C(X) circuits outperforming previous methods in the asymptotic and non-asymptotic regimes.

Quantum Spatial Search with Electric Potential: Long-Time Dynamics and Robustness to Noise.

We present various results on the scheme introduced in a previous work, which is a quantum spatial-search algorithm on a two-dimensional (2D) square spatial grid, realized with a 2D Dirac discrete-time quantum walk (DQW) coupled to a Coulomb electric field centered on the the node to be found. In such a walk, the electric term acts as the oracle of the algorithm, and the free walk (i.e.

Dirac Spatial Search with Electric Fields.

Electric Dirac quantum walks, which are a discretisation of the Dirac equation for a spinor coupled to an electric field, are revisited in order to perform spatial searches. The Coulomb electric field of a point charge is used as a non local oracle to perform a spatial search on a 2D grid of points. As other quantum walks proposed for spatial search, these walks localise partially on the charge after a finite period of time.

Quantum walk hydrodynamics.

A simple Discrete-Time Quantum Walk (DTQW) on the line is revisited and given an hydrodynamic interpretation through a novel relativistic generalization of the Madelung transform. Numerical results show that suitable initial conditions indeed produce hydrodynamical shocks and that the coherence achieved in current experiments is robust enough to simulate quantum hydrodynamical phenomena through DTQWs. An analytical computation of the asymptotic quantum shock structure is presented.

Nonlinear optical Galton board: Thermalization and continuous limit.

The nonlinear optical Galton board (NLOGB), a quantum walk like (but nonlinear) discrete time quantum automaton, is shown to admit a complex evolution leading to long time thermalized states. The continuous limit of the Galton board is derived and shown to be a nonlinear Dirac equation (NLDE). The (Galerkin-truncated) NLDE evolution is shown to thermalize toward states qualitatively similar to those of the NLOGB.

Effective dissipation and turbulence in spectrally truncated euler flows.

A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3D Euler equation with a high-wave-number spectral truncation is characterized. Large-scale dissipative effects, caused by the thermalized modes that spontaneously appear between a transition wave number and the maximum wave number, are calculated using fluctuation dissipation relations. The large-scale dynamics is found to be similar to that of high-Reynolds number Navier-Stokes equations and thus obeys (at least approximately) Kolmogorov scaling.