Abstract
The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of its dynamics remains beyond the reach of current numerical methods. In this article, we show that general quantum computations can be encoded into the physics of wave packets propagating through a planar graph, with scattering interactions governed by the fermionic Hubbard model. Therefore, simulating the model on planar graphs is as hard as simulating quantum computation. We give two different arguments, demonstrating that the simulation is difficult both for wave packets prepared as excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the t-J model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. In that sense, the construction provides a quantum mechanical analog of the Fredkin–Toffoli billiard ball computer.
Highlights
The aim of quantum Hamiltonian complexity theory is to categorize the basic questions of physics by how difficult they are to resolve computationally [1]
Childs et al demonstrated that a wide class of quantum systems, including the Bose– Hubbard model, is universal for quantum computation, in the sense that it is possible to encode arbitrary quantum computations into their dynamics if their interactions are arranged between the vertices of a particular computation-dependent planar graph [12]
Since the Bose–Hubbard model can be simulated on a quantum computer and can simulate arbitrary quantum computations, the complexity of simulating the model is precisely the power of quantum computation
Summary
Content from this work Abstract may be used under the The Hubbard model may be the simplest model of particles interacting on a lattice, but simulation of terms of the Creative. General quantum computations can be encoded into the physics of wave packets propagating through. Any further distribution of this work must maintain a planar graph, with scattering interactions governed by the fermionic Hubbard model. Attribution to the author(s) and the title of simulating the model on planar graphs is as hard as simulating quantum computation. Excitations of the fermionic vacuum, and for hole wave packets at filling fraction one-half in the limit of strong coupling. In the latter case, which is described by the t-J model, there is only reflection and no transmission in the scattering events, as would be the case for classical hard spheres. The construction provides a quantum mechanical analog of the Fredkin–Toffoli billiard ball computer
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