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Quantum Simulation of Topological Models with Ultracold Atoms

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Quantum Simulation of Topological Models with Ultracold Atoms Quantum simulation of topological models with ultracold atoms Gerardo Garc´ıaMoreno Supervisor: Alejandro G´onzalezTudela December 17, 2019 1 Introduction with this in order to exploit all the potential of quan- tum technologies. A proposal for circumventing this It is clear that we are living the so-called second quan- difficulty is topological quantum computation, a pro- tum revolution, or in other words, the era of quantum posal based on storing the quantum information on technologies. Concepts like quantum supremacy [1], non-local properties of the quantum system. In this the ability of quantum computers to perform tasks way, because of the local nature of the noise induced more efficiently than classical computers, are, nowa- by the environment, the information is stored in a days, trending topics in the community. Moreover, robust way. The archetypical example of quantum technologically, it is expected that the growth in the system with topological order that could be used for efficiency of classical computers will stop within the this purpose is the toric code introduced by Kitaev following years due to fundamental limitations com- [3]. The experimental challenge of such model, how- ing from physical principles. Thus, it seems clear ever, stems from the design of the four-body inter- that, in order to solve certain computational prob- actions required which does not typically appear in lems, quantum technologies will be needed. In par- natural systems. ticular, one of these computational problems hard to calculate classically is the simulation of quantum Nanophotonic crystals are dielectric media with a many-body physics. For such purpose, Feynman en- periodic refractive index, what makes the photons visioned [2] quantum simulators, that are, quantum see an effective lattice that modifies their dispersion computers that are able to solve just one specific type relation. They are very useful for quantum simula- of problem or, put in it in other words, quantum sys- tion purposes because of their parameters high tun- tems used to study the properties of other physical ability and the possibility of combining them with systems. The archetypical example of such a problem ultracold atom physics to create a quantum simu- is high temperature superconductivity since there is lator. We initially focused on this kind of systems a microscopic model that is assumed to capture some as a possibility to simulate topological models. Our of the main properties of the phenomenon. However, objective was, initially, to combine ultracold atom the inability to simulate it on a classical computer physics and nanophotonic physics like superlattice leaves open the question of whether there are more photonic crystals with the stator approach we will suitable models for describing this physical property describe later to be able to simulate physical systems or other mechanisms responsible for it. displaying this kind of topological features. At the One of the main problems of quantum technolo- end, we found that Rydberg atom physics combined gies is the phenomenon of quantum decoherence, the with holographic techniques [4] and the stator ap- breakdown of quantum correlations induced by the proach were suitable for simulating the toric code, presence of a noisy environment. Thus, we must deal allowing to even implement periodic boundary con- 1 ditions. vacuum we understand a certain ground state of the system, can be identified with a certain topological sector of the theory. This is because, first, no local 2 Toric code measurement can distinguish between this different vacua but all of these states turn out to be mutu- The toric code [3] is a two-dimensional lattice with ally orthogonal. Second, the degeneracy is robust in spin 1=2 degrees of freedom located on each of the the sense of being preserved under local perturba- edges described by the following Hamiltonian tions [5], where the splitting in energies is estimated X Y X Y to be exp(−∆L),being L the smallest dimension of H = −J σ(j) − J σ(j); (1) x z the lattice and ∆ the typical energy scale of the per- v p j2v j2p turbation (and we assume ∆ J in order for it to where i 2 v denotes the set of edges touching a ver- be a small deviation, otherwise the model undergoes tex of the lattice and i 2 p the set of edges sorround- a phase transition). ing a plaquette on the lattice, and a sum over the We can now analyze excitations above the ground whole set of plaquettes and vertex of the lattice is state. The lowest energy excitations correspond to a performed. J is a constant with dimensions of en- failure of one of the equations (2) to hold at a given ergy. The hamiltonian is a sum of terms such that vertex or at a certain plaquette. We can have a ver- each of the terms commutes with all the other ones tex where the eigenvalue of the vertex operator takes and with the property that each of the terms squares the eigenvalue −1 instead of +1 or a plaquette where to the identity and thus its eigenvalues are ±1. In or- the plaquette operator takes also −1 instead of +1 der to find the ground state manifold of such system and paying a penalty on energy of +4J above the dim(M) M = fj iigi=1 , we can look for the states that ground state energy. We can interpret the first case have eigenvalue +1 for all the individual operators as the creation of a single electric quasiparticle which we denote as e and the second as the creation of a Y Y magnetic particle which we denote by m. This kind σ(j) j i = j i ; 8v; σ(j) j i = j i ; 8p: (2) x i i z i i of excitations are always created in pairs, being the j2v j2p vertex electric and plaquette magnetic defects cre- Such subspace of the Hilbert space is called the ated by applying σx or σz operator to a certain link stabilizer space of the code and it turns out that respectively. We can create a pair of such particles, dim(M) = 4g, being g the genus of the manifold perform a closed loop and annihilate them. The in- on which the lattice is placed on. To prove this one teresting feature of this model is that, although do- needs to notice that within the set of closed paths ing this operation for an electric charge in a loop that that can be defined on the lattice and its dual, some contains no magnetic charge provides us with a trivial of these closed paths have the specific property that operator, doing it in a loop that contains inside it a they are non-contractible. There is a mapping from magnetic charge provides us with a relative phase be- the space of closed non-contractible spaces and the tween the states of π. In other words, we can think of ground state manifold. For the case of g = 0, which another kind of excitations that results from the non- corresponds to the open boundary conditions we find trivial braiding of e and m which is the f excitation out that the ground state is not degenerate and it made up of a magnetic bound to an electric charge turns out to be unique. However, for g = 1 corre- which corresponds to putting an additional 8J energy sponding to periodic boundary conditions (the man- above the ground state. All the excitations consist of ifold on which we put the lattice being a torus in putting additional quasiparticles on a certain ground this case) we find that the ground state is fourfold state and thus the energy spectrum can be written degenerate. This degeneracy of the ground state is as E = E0 + 4JN, being E0 the ground state energy a characteristic property of topological quantum or- and N the number of electric and magnetic parti- dered systems, since each of the `vacua', where by cles we have in our system. The non trivial braid- 2 ing of this quasiparticles follows a so-called abelian physics systems [9]. anyonic statistics and can be summarized by the fol- Programmable holographic optical tweezers can be lowing modular S-matrix which contains the factor used to create three-dimensional arrays of traps, of- that multiplies a certain state after a quasiparticle i fering a high tunability of the lattice geometry. In performs a loop around a quasiparticle j where we fact, it has been shown that arbitrarily designed ar- list the particles in the order I; e; m; f being I the rays of up to about 120 traps can be generated with abscence of particles this kind of approach [4]. In particular, the geom- etry needed to perform a toric code simulation has 0 1 1 1 1 1 already been constructed with a typical separation of 1 B 1 1 −1 −1 C the atoms which can be tuned to be in the interval S = B C : (3) 2 @ 1 −1 1 −1 A (3−40 µm), which is a suitable range for implement- 1 −1 −1 1 ing fast two qubit gates because interaction energies However, it turns out that in order to observe long- between Rydberg states at those distances are typi- live fault tolerant topological order at finite temper- cally in the megahertz range. ature we need to perform some feedback operations Single qubit gates acting on single individual qubits on the system. This is because in 2D local and trans- encoded in atomic hyperfine states can be performed lation invariant stabilizer Hamiltonian systems do combining microwaves with a gradient field with high not have thermal stability (generically in dimension fidelity and are not as challenging as the two qubit D ≤ 3) [6].
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