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Programmable Hamiltonian engineering with quadratic quantum Fourier transform
Quantum Fourier transform (QFT) is a widely used building block for quantum algorithms, whose scalable implementation is challenging in experiments. Here, we propose a protocol of quadratic QFT (QQFT) considering cold atoms confined in an optical lattice. This QQFT is equivalent to QFT in the single-particle subspace and generalizes to a many-body setting through the second quantization formalism. We show this QQFT protocol can be implemented using a programmable laser potential with the digital-micromirror-device techniques recently developed in experiments. The QQFT protocol enables programmable Hamiltonian engineering and allows quantum simulations of Hamiltonian models, which are difficult to realize with conventional approaches. The flexibility of our approach is demonstrated by performing quantum simulations of one-dimensional Poincaré crystal physics and two-dimensional topological flat bands, where the QQFT protocol effectively generates the required long-range tunnelings despite the locality of the cold atom system. We find the discrete Poincaré symmetry and topological properties in the two examples, respectively, have robustness against a certain degree of noise that is potentially existent in the experimental realization. We expect this approach would open wide opportunities for optical lattice-based programmable quantum simulations.
Relativistic quantum field theory of stochastic dynamics in the Hilbert space
In this paper, we develop an action formulation of stochastic dynamics in the Hilbert space. In this formulation, the quantum theory of random unitary evolution is easily reconciled with special relativity. We generalize the Wiener process into 1+3-dimensional spacetime, and then define a scalar random field which keeps invariant under Lorentz transformations. By adding to the action of quantum field theory a coupling term between random and quantum fields, we obtain a random-number action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentz-invariant equation of motion for the state vector or density matrix. We derive the path integral formula of S matrix, based on which we develop a diagrammatic technique for doing the calculation. We find the diagrammatic rules for both the stochastic free field theory and stochastic phi-4 theory. The Lorentz invariance of the random S matrix is strictly proved by using the diagrammatic technique. We then develop a diagrammatic technique for calculating the density matrix of final quantum states after scattering. In the absence of interaction, we obtain the exact expressions of both S matrix and density matrix. In the presence of interaction, we prove a simple relation between the density matrices of stochastic and conventional phi-4 theory. Our formalism leads to an ultraviolet divergence that has a similar origin as that in quantum field theory. The divergence can be canceled by renormalizing the coupling strength to random field. We prove that the stochastic quantum field theory is renormalizable even in the presence of interaction. In the models with a linear coupling between random and quantum fields, the random field excites particles out of the vacuum, driving the Universe towards an infinite-temperature state. The number of excited particles follows the Poisson distribution. The collision between particles is not affected by the random field. But the signals of colliding particles are gradually covered by the background excitations caused by random field.
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