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Register a new userComparative study of semiclassical approaches to quantum dynamics
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Quantum states can be described equivalently by density matrices, Wigner functions or quantum tomograms. We analyze the accuracy and performance of three related semiclassical approaches to quantum dynamics, in particular with respect to their numerical implementation. As test cases, we consider the time evolution of Gaussian wave packets in different one-dimensional geometries, whereby tunneling, resonance and anharmonicity effects are taken into account. The results and methods are benchmarked against an exact quantum mechanical treatment of the system, which is based on a highly efficient Chebyshev expansion technique of the time evolution operator.
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We study the costs and benefits of different quantum approaches to finding approximate solutions of constrained combinatorial optimization problems with a focus on Maximum Independent Set. In the Lagrange multiplier approach we analyze the dependence of the output on graph density and circuit depth. The Quantum Alternating Ansatz Approach is then analyzed and we examine the dependence on different choices of initial states. The Quantum Alternating Ansatz Approach, although powerful, is expensive in terms of quantum resources. A new algorithm based on a Dynamic Quantum Variational Ansatz (DQVA) is proposed that dynamically changes to ensure the maximum utilization of a fixed allocation of quantum resources. Our analysis and the new proposed algorithm can also be generalized to other related constrained combinatorial optimization problems.
The family of trajectories-based approximations employed in computational quantum physics and chemistry is very diverse. For instance, Bohmian and Hellers frozen Gaussian semiclassical trajectories seem to have nothing in common. Based on a hydrodynamic analogy to quantum mechanics, we furnish the unified gauge theory of all such models. In the light of this theory, currently known methods are just a tip of the iceberg, and there exists an infinite family of yet unexplored trajectory-based approaches. Specifically, we show that each definition for a semiclassical trajectory corresponds to a specific hydrodynamic analogy, where a quantum system is mapped to an effective probability fluid in the phase space. We derive the continuity equation for the effective fluid representing dynamics of an arbitrary open bosonic many-body system. We show that unlike in conventional fluid, the flux of the effective fluid is defined up to Skodjes gauge [R. T. Skodje et. al. Phys. Rev. A 40, 2894 (1989)]. We prove that the Wigner, Husimi and Bohmian representations of quantum mechanics are particular cases of our generic hydrodynamic analogy, and all the differences among them reduce to the gauge choice. Infinitely many gauges are possible, each leading to a distinct quantum hydrodynamic analogy and a definition for semiclassical trajectories. We propose a scheme for identifying practically useful gauges and apply it to improve a semiclassical initial value representation employed in quantum many-body simulations.
Measurements on classical systems are usually idealized and assumed to have infinite precision. In practice, however, any measurement has a finite resolution. We investigate the theory of non-ideal measurements in classical mechanics using a measurement probe with finite resolution. We use the von Neumann interaction model to represent the interaction between system and probe. We find that in reality classical systems are affected by measurement in a similar manner as quantum systems. In particular, we derive classical equivalents of Luders rule, the collapse postulate, and the Lindblad equation.
We present a systematic comparison of different methods of fidelity estimation of a linear optical quantum controlled-Z gate implemented by two-photon interference on a partially polarizing beam splitter. We have utilized a linear fidelity estimator based on the Monte Carlo sampling technique as well as a non-linear estimator based on maximum likelihood reconstruction of a full quantum process matrix. In addition, we have also evaluated lower bound on quantum gate fidelity determined by average quantum state fidelities for two mutually unbiased bases. In order to probe various regimes of operation of the gate we have introduced a tunable delay line between the two photons. This allowed us to move from high-fidelity operation to a regime where the photons become distinguishable and the success probability of the scheme significantly depends on input state. We discuss in detail possible systematic effects that could influence the gate fidelity estimation.
We numerically study the dynamics and stationary states of a spin ensemble strongly coupled to a single-mode resonator subjected to loss and external driving. Employing a generalized cumulant expansion approach we analyze finite-size corrections to a semiclassical description of amplitude bistability, which is a paradigm example of a driven-dissipative phase transition. Our theoretical model allows us to include inhomogeneous broadening of the spin ensemble and to capture in which way the quantum corrections approach the semiclassical limit for increasing ensemble size $N$. We set up a criterion for the validity of the Maxwell-Bloch equations and show that close to the critical point of amplitude bistability even very large spin ensembles consisting of up to $10^4$ spins feature significant deviations from the semiclassical theory.
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