We derive rigorous one-and two-dimensional mean-field equations for cigar-and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. We show how the dipolar interaction modifies the contact interaction of the strongly confined atoms. In addition, our equations introduce a nonlocal potential, which is anisotropic for pancake-shaped condensates. We propose to observe this anisotropy via measurement of the condensate aspect ratio. We also derive analytically approximate density profiles from our equations. Both the numerical solutions of our reduced mean-field equations and the analytical density profiles agree well with numerical solutions of the full Gross-Pitaevskii equation while being more efficient to compute. PACS numbers: 03.75.Hh, 75.80.+q, 67.85.-d arXiv:1006.4950v3 [cond-mat.quant-gas]

Current gate-based quantum computers have the potential to provide a computational advantage if algorithms use quantum hardware efficiently. To make combinatorial optimization more efficient, we introduce the Filtering Variational Quantum Eigensolver (F-VQE) which utilizes filtering operators to achieve faster and more reliable convergence to the optimal solution. Additionally we explore the use of causal cones to reduce the number of qubits required on a quantum computer. Using random weighted MaxCut problems, we numerically analyze our methods and show that they perform better than the original VQE algorithm and the Quantum Approximate Optimization Algorithm (QAOA). We also demonstrate the experimental feasibility of our algorithms on a Honeywell trapped-ion quantum processor.

We propose a scheme for parameter estimation with cluster states. We find that phase estimation with cluster states under a many-body Hamiltonian and separable measurements leads to a precision at the Heisenberg limit. As noise models we study the dephasing, depolarizing, and pure damping channels. Decoherence reduces the sensitivity but our scheme remains superior over several reference schemes with states such as maximally entangled states and product states. For small cluster states and fixed evolution times it remains at the Heisenberg limit for approximately 2 times as many qubits than alternative schemes.

We study numerically the long-time dynamics of a one-dimensional Bose-Einstein condensate expanding in a speckle or impurity disorder potential. Using the mean-field Gross-Pitaevskii equation, we demonstrate subdiffusive spreading of the condensate for long times. We find that interaction-assisted hopping between normal modes leads to this subdiffusion. A possible (partial) reason why the root-mean-square (rms) width saturates in the experiment [Nature (London) 453, 891 (2008)] is provided. We suggest that observing both the participation length and the rms width of a condensate, rather than only the rms width, could provide a more complete description of the long-time behavior of ultracold atoms in disorder potentials. Our study confirms subdiffusive spreading in spatially continuous disordered interacting models and highlights new features which spatially discrete models do not possess.

Combinatorial optimization models a vast range of industrial processes aiming at improving their efficiency. In general, solving this type of problem exactly is computationally intractable. Therefore, practitioners rely on heuristic solution approaches. Variational quantum algorithms are optimization heuristics that can be demonstrated with available quantum hardware. In this case study, we apply four variational quantum heuristics running on IBM’s superconducting quantum processors to the job shop scheduling problem. Our problem optimizes a steel manufacturing process. A comparison on 5 qubits shows that the recent filtering variational quantum eigensolver (F-VQE) converges faster and samples the global optimum more frequently than the quantum approximate optimization algorithm (QAOA), the standard variational quantum eigensolver (VQE), and variational quantum imaginary time evolution (VarQITE). Furthermore, F-VQE readily solves problem sizes of up to 23 qubits on hardware without error mitigation post processing.