We present a general strategy to simulate a D + 1-dimensional quantum system using a D-dimensional one. We analyze in detail a feasible implementation of our scheme using optical lattice technology. The simplest nontrivial realization of a fourth dimension corresponds to the creation of a bivolume geometry. We also propose single-and many-particle experimental signatures to detect the effects of the extra dimension.Introduction. There is long-standing interest in Physics for the possible existence and effects of extra dimensions. This interest was brought by the seminal papers by Kaluza and Klein [1] aimed at unifying interactions using the presence of a fourth spacial dimension. Later on, extra dimensions became a sine qua non element for the construction of string theory [2]. Separately, enormous progress has been made in recent years to achieve real quantum simulations, that is, to simulate quantum mechanical models using other well controlled quantum systems [3]. It is now reasonable to investigate to what extent present technology can be used to faithfully simulate a quantum theory living in extra dimensions.Let us briefly recall recent progress on quantum simulation of condensed matter models using cold atoms [4,5]. By confining atoms to an optical lattice, the Hubbard model may be realized [6], and the superfluid-to-Mott-insulator transition observed [7]. Furthermore, several schemes to couple neutral cold atoms to artificial Abelian [8] and non-Abelian [9, 10] magnetic and electric fields have been put forth [11][12][13][14]. This opens the door to creating strongly correlated quantum-Hall states with cold atoms [15]. Although cold atoms are non-relativistic, it is possible to simulate relativistic effects by looking at the low-energy behaviour of some special lattice models -lattice models where the band structure presents Dirac cones, e.g. honeycomb lattices [16,17] or lattices dressed with internal degrees of freedom [18]. Hence, it is not far-fetched that that cold atoms may provide some insight into particle physics models that are not completely understood, such as quantum chromodynamics (see for instance [19]) or, as presented here, in the analysis of extra dimensions.General strategy. The basic idea to achieve a quantum simulation of an extra dimension consists in engineering the connectivity of the system partly on real dimensions, and partly on the use of different species for the degrees of freedom (for an alternative approach cf. [20]). Let us illustrate this construction in the case of the simplest quantum mechanical model of a free particle on a hypercubic D + 1 spacial lattice. The Hamiltonian for this system is

We argue that the Fermi-Hubbard Hamiltonian describing the physics of ultracold atoms on optical lattices in the presence of artificial non-Abelian gauge fields, is exactly equivalent to the gauge theory Hamiltonian describing Dirac fermions in the lattice. We show that it is possible to couple the Dirac fermions to an "artificial" gravitational field, i.e. to consider the Dirac physics in a curved spacetime. We identify the special class of spacetime metrics that admit a simple realization in terms of a Fermi-Hubbard model subjected to an artificial SU (2) field, corresponding to position dependent hopping matrices. As an example, we discuss in more detail the physics of the 2+1D Rindler metric, its possible experimental realization and detection.

We propose several designs to simulate quantum many-body systems in manifolds with a nontrivial topology. The key idea is to create a synthetic lattice combining real-space and internal degrees of freedom via a suitable use of induced hoppings. The simplest example is the conversion of an open spin-ladder into a closed spin-chain with arbitrary boundary conditions. Further exploitation of the idea leads to the conversion of open chains with internal degrees of freedom into artificial tori and Möbius strips of different kinds. We show that in synthetic lattices the Hubbard model on sharp and scalable manifolds with non-Euclidean topologies may be realized. We provide a few examples of the effect that a change of topology can have on quantum systems amenable to simulation, both at the single-particle and at the many-body level.

The role of noise in the transport properties of quantum excitations is a topic of great importance in many fields, from organic semiconductors for technological applications to light-harvesting complexes in photosynthesis. In this paper we study a semi-classical model where a tight-binding Hamiltonian is fully coupled to an underlying spatially extended nonlinear chain of atoms. We show that the transport properties of a quantum excitation are subtly modulated by (i) the specific type (local versus non-local) of exciton-phonon coupling and by (ii) nonlinear effects of the underlying lattice. We report a non-monotonic dependence of the exciton diffusion coefficient on temperature, in agreement with earlier predictions, as a direct consequence of the lattice-induced fluctuations in the hopping rates due to long-wavelength vibrational modes. A standard measure of transport efficiency confirms that both nonlinearity in the underlying lattice and off-diagonal exciton-phonon coupling promote transport efficiency at high temperatures, preventing the Zeno-like quench observed in other models lacking an explicit noise-providing dynamical system. ( ) ( ) and J J u u . 5

A scheme for measuring complex temperature partition functions of Ising models is introduced. In the context of ordered qubit registers this scheme finds a natural translation in terms of global operations, and single particle measurements on the edge of the array. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimation error, valid with high confidence, are provided through a central-limit theorem, which validity extends beyond the present context. It holds for example for estimations of the Jones polynomial. Interestingly, the kind of state preparations and measurements involved in this application can in principle be made "instantaneous", i.e. independent of the system size or the parameters being simulated. Second, the scheme allows to accurately estimate some non-trivial invariants of links. A third result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models on a square lattice. We provide conditions under which estimating such partition functions allows one to reconstruct scattering amplitudes of quantum circuits making the problem BQP-hard. Using this mapping, we show that fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we show that the ability to accurately measure corner magnetizations on thermal states of two-dimensional Ising models with magnetic field leads to fully polynomial random approximation schemes (FPRAS) for the partition function. Each of these results corresponds to a section of the text that can be essentially read independently.