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 quantum 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 j∈v j∈p turbation (and we assume ∆ J in order for it to where i ∈ v denotes the set of edges touching a ver- be a small deviation, otherwise the model undergoes tex of the lattice and i ∈ 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 = {|ψii}i=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) |ψ i = |ψ i , ∀v; σ(j) |ψ i = |ψ i , ∀p. (2) x i i z i i of excitations are always created in pairs, being the j∈v j∈p 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 geometry needed to perform a toric code simulation has   1 1 1 1 already been constructed with a typical separation of 1  1 1 −1 −1  the atoms which can be tuned to be in the interval S =   . (3) 2  1 −1 1 −1  (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]. Intuitively, translation invariance guar- gates [10]. The more interesting two qubit gates can antees that the possible anyonic excitations can move be performed by two different methods: on the one freely without paying any energetic penalty. The hand Rydberg systems interacting through van der temperature induced creation of pair of anyonic ex- Waals interactions in the blockade regime can be de- citations and their subsequent annihilation after the scribed by density-density interaction hamiltonians (1) (2) −6 curve resulting from the union of their trajectories of the form HI = V (R)σz σz , with V (R) ∼ R performing a non contractible makes available a dy- being R the distance between both atoms[11]. On namical connection of topologically different sectors. the other hand, spin exchange or XY hamiltonians (1) (2) Despite of this fact, simulations can allow us to H = V (R)σ− σ+ + h.c., with V (R) scaling now as observe this kind of anyonic statistics features if they R−3, can be achieved by using two different Rydberg are done fast enough, for example [7], where just a states, interacting directly via resonant dipole-dipole minimal implementation was done, that is, just a pla- interaction [12]. The experimental results reported quette was simulated by encoding the model in the strongly suggest the feasibility of the toric code to be multi-partite entangled state of polarized photons. simulated with current technology in ultracold Ryd- berg atoms. 3 Rydberg atoms 4 Simulation of the Toric code Rydberg atoms are excited atoms (usually alkali with optical tweezers atoms like Rb and Cs) with one or more electrons that have a big principal quantum number n [8]. They Stator approach have several interesting properties like their long range strong interactions, since their dipole dipole In order to obtain a four body interaction between interactions scales as n4 or radiative lifetimes that the four qubits in a certain plaquette or between the scale as n3. This features can be exploded for quan- four qubits that touch a certain vertex, we will fol- tum simulation purposes, since it allows to efficiently low the stator approach which we describe now. Let perform long-range two qubit gates making it a po- us focus in obtaining the four σx effective interaction tential quantum simulator for quantum many body between the links that touch a certain vertex. Focus-

3 ing on a certain vertex, we will assume that we are described above but considering now the unitary op- able to perform arbitrary local operations on each of erator the links and on each of the ancillary systems located (P )† at the vertex, what can be implemented in Rydberg UE0 = UEVy (π/4). (9) atoms as we discussed before. Our objective will be In this way, we manage to construct the full interac- to perform evolution under the hamiltonian tions we need to run a simulation of the toric code.

4 Since the hamiltonian is a sum of commuting parts, Y (i) we do not even need to trotterize nor our method Hv = −J σx , (4) i=1 relies on perturbative expansions, what makes this suitable for exploring regimes of strong coupling and for the four qubits on the links. In order to do this, long times. we can take advantage of the system located at the vertex and design a sequence which consists of: an unitary gate that entangles the physical qubits with Running the simulation for the whole the ancilla by means of a certain unitary gate UAP , lattice a subsequent evolution of the ancilla under a local In order to simulate the full toric code model, we hamiltonian and then the inverse of the original uni- need to implement the four body interactions for tary gate that now disentangles the physical and an- all the plaquettes and vertex. Since all the terms cilla system. In other words, we want H , U such A E of the Hamiltonian commute among themselves, the that full evolution for a generic time can be achieved by performing the evolution term by term one after the † −iHAt −iHv t without needing to perform some kind of trotteriza- UEe UE(|iniA |ψiP ) = |iniA e |ψiP , (5) tion. Since plaquettes or vertex with different parity being |iniA the state of the ancilla qubit at the be- do not share any of the links among themselves, we ginning of the simulation and |ψi the generic state of can perform the following protocol to simulate the the physical qubits. full toric code: Considering the operators acting on a certain ver- 1. Run the simulation for all the vertex with a fixed tex and the corresponding links touching the vertex parity and implement the sequence to obtain given by evolution under the four σx term of each of those vertex. (L) P4 (i) (A) −iσiφ −iφ i=1 σx σx Vi (φ) = e ,Dx,x(φ) = e , (6) 2. Do the same for the vertex with the the other where i = x, y, z and A, P stands for ancilla and all parity. the physical qubits at the same time, we can perform 3. Run the simulation for the plaquettes with fixed the evolution under the hamiltonian 4 by choosing parity and implement the sequence to obtain the following UE evolution under the four σz gates. 4. Do the same for the plaquettes with the opposite (A)† (P )† A UE =V (π/4)V (π/2)Dx,x(π/4)V (π/4) parity. y y y (7) P P (A) (A) Vy (π/2)Vx (π/4)Vz (π/4)Vy (π/4), We need to do even and odd plaquettes (vertex) at different times because otherwise, the entanglement and choosing of the same edge with two plaquettes, two vertex or H = −Jσ(A). (8) A x a vertex and a plaquette at the same time will spoil In order to simulate the four σz term on a certain pla- the disentangling proccess at the end and the physical quette, we just need to perform the same operations system will remain entangled with the ancillas.

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