Stroboscopic Generation of Topological Protection
Trapped neutral atoms offer a powerful route to robust simulation of complex quantum systems. We present here a stroboscopic scheme for realization of a Hamiltonian with -body interactions on a set of neutral atoms trapped in an addressable optical lattice, using only 1- and 2-body physical operations together with a dissipative mechanism that allows thermalization to finite temperature or cooling to the ground state. We demonstrate this scheme with application to the toric code Hamiltonian, ground states of which can be used to robustly store quantum information when coupled to a low temperature reservoir.
Present Address: ] Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA
Introduction.– Among the most exciting aspects of quantum simulation is the possibility of generating and studying exotic quantum phases such as those possessing topological order that can be used to robustly store and process quantum information. The Hamiltonians governing these phases frequently require more-than-2-body interactions that are hard or even impossible to realize naturally. This difficulty has spurred much theoretical and experimental effort in the artificial engineering of Hamiltonians, particularly for trapped neutral atoms Lewenstein et al. (2007). Many proposals have been made for the generation of 2-body Hamiltonians using static emulation schemes and some experimental realizations have appeared Greiner et al. (2002); Spielman et al. (2008). Specific proposals have appeared for generating -body interactions Büchler et al. (2007), but have focused on static emulation.
We present here an alternative, dynamic emulation approach to systematic generation of -body interactions that is based on sequences of control pulses which individually realize - and -body operations on internal atomic levels. We show that this stroboscopic realization of the Hamiltonian can be implemented simultaneously with a dissipative thermalization protocol to stabilize the system from the effects of imperfect quantum operations and environmental noise. In the zero temperature limit, this can be view as replacing algorithmic error correction in an equivalent quantum circuit model with a dissipative procedure to remove errors Lloyd et al. (). The resource requirements for this thermalization protocol are different from those of algorithmic error correction, and may be more accessible to experiment in the foreseeable future. We illustrate the approach here with stroboscopic generation of the 4-body toric code Hamiltonian, which constitutes one of the simplest exactly solvable models with a ground state topological phase Kitaev (2003):
where denotes a Pauli operator on the links of a square lattice and denote the vertex/plaquette of the lattice. The ground state of this model possesses topological order, and therefore has anyonic quasiparticle excitations and, on a lattice with periodic boundary conditions, an emergent topological degeneracy. Quantum information can be encoded in this ground state degeneracy and manipulated with controlled creation and braiding of anyons Kitaev (2003); Nayak et al. (2008). In a finite sized system Nussinov and Ortiz (2008); Castelnovo and Chamon (2007), the topological order of the ground state and gap to excited states protects against decoherence and loss of quantum information due to noise provided the system is coupled to a low temperature bath. Our analysis below will provide a scheme for generating both and an effective low temperature bath, realizing the topological protection characteristic of the toric code.
The physical context for our analysis is a set of individual Cs atoms trapped at the sites of an addressable simple cubic optical lattice Nelson et al. (2007). A lattice spacing of m Nelson et al. (2007) allows essentially perfect addressability Beals et al. (2008). The orbital degrees of freedom are frozen on the time scales relevant to our analysis and we need consider only internal atomic degrees of freedom. Two hyperfine levels (e.g., ) define a 2-level pseudospin system. We realize in the interaction representation defined by the pseudospin energies. Auxiliary internal levels are used to realize 1-spin and 2-spin quantum operations, using optical frequency Raman pulses to generate arbitrary single-spin operations and excitation of one atom to a Rydberg state, e.g., the state, to generate controlled-phase gates, CPHASE Jaksch et al. (2000). To achieve thermalization or cooling, the Hamiltonian is supplemented by coupling the primary system spins to an ancillary set of pseudospins that will be dissipatively controlled to simulate a thermal reservoir. Since the pseudospins are localized at the sites of a cubic lattice, one can choose to either realize on a single plane using a surface code Bravyi and Kitaev () or in a three-dimensional cubic array with toroidal boundary conditions realized by SWAP operations (see Fig. 1).
Effective Hamiltonian Evolution.– Given the ability to perform both Rydberg-induced CPHASE gates between atoms in neighboring sites and arbitrary 1-body rotations, , on individual atoms, where is a variable phase angle, sequences of these operations can be chosen to generate effective -body interactions through high-order terms in the Magnus expansion Klarsfeld and Oteo (1989), allowing stroboscopic simulation of a broad class of Hamiltonians. Consider the operator sequence, , where the are the 1- or 2-body gates described above. Effective interactions are found through:
Consider now simulation of the 4-body interactions in . We use the notation and define , , and , where is the identity operator. For simplicity, it is assumed that each takes a time to execute. We construct the operator sequence,
where . This sequence acts over a time to generate the following effective Hamiltonian at a single vertex, :
where denotes the application of the (up to) four-body operator to the spins meeting at a vertex or surrounding a plaquette ; we choose ; and with . By repeating the operator sequence a second time with sign reversals and , we cancel the fourth order terms in , giving that acts for a time to generate the effective Hamiltonian
The sequence is specifically designed to cancel the lowest-order () perturbation terms without affecting the gap. The remaining term is a 2-body perturbation to . Repeating this sequence with appropriate sign reversals will cancel these higher order terms. However, the ground state subspace of is robust to these remaining perturbations (see below). A shorter operator sequence may then be preferable to reduce gate errors. The plaquette operator, , can be generated by cyclic permutation of the Pauli operators in the above expressions for and .
Simulation of then requires application of the pulse sequence to all vertices and plaquettes conforming to a two dimensional square lattice with periodic boundary conditions. Vertex and plaquette terms may be applied serially as: . Because only the perturbation terms fail to commute, the truncation error in the above expression occurs at orders larger than . For 18 pseudospins, representing a system with toroidal boundary conditions, a completely serial implementation yields a stroboscopic cycle time of 720s using the estimate ns Weiss (2009) and the minimal count of one CPHASE and four 1-spin gates to realize all Zhang et al. (2003). This may be reduced by implementing some operators in parallel.
Simulated Thermalization.– The pseudospin subspace of the system defined by the internal states of the trapped atoms will interact with the external environment through the controlled quantum operations in the above pulse sequences and uncontrolled noisy interactions. Noise in the optical lattice system will not drive the simulation subspace to a state that is thermal under the simulated Hamiltonian Brown (2007). Additionally, noise in the above sequence of control gates will add entropy and effectively heat the system. The entropy production resulting from imperfect gate operation , where is the error per gate Young et al. (). Quantum circuit models are usually supplemented by error correction schemes to effectively remove entropy from the system. We take a different approach here, constructing an effective system-reservoir interaction to control system entropy and relax the system to the ground state or a thermal state.
To maintain the simulated system at a thermal steady state we add an interaction of the system pseudospins with a set of ancillary pseudospins. In the optical lattice system, these ancillary pseudospins, which may be a 2 species of atom, are trapped in an offset, intercalated optical lattice, such that each ancillary atom is adjacent to a system atom. Consider a Hamiltonian with local -body interactions of the form: , where is an -body operator involving a neighborhood of pseudospins, , including pseudospin , with eigenvalues , labels the type of interaction, and is a constant. Additionally we define the pseudospin flip operator such that when . When all , as is the case for Eq. (1), we can define with . These areare -body interactions; creates a pair of excitations about and translates an excitation about . The energy gap for creation of a pair of excitations is .
A route to guaranteeing the thermal equilibration of this system is for it to evolve under the Lindblad master equation , where is the density matrix and is the superoperator , with the Lindblad operators. With given by
the Lindblad master equation describes equilibration with a bath of temperature . The unique stationary state of the system is then the thermal state under with temperature . and are relaxation rates, and their values dictate the thermalization time. For simplicity we have set . To generate evolution under such a master equation, we introduce a set of non-interacting ancillary pseudospins that independently undergo strong dissipation. Each local neighborhood of the system interacts locally with a single thermal ancillary pseudospin and a single maximally mixed ancillary pseudospin for each : The master equation of the system and ancilla pseudospins combined is of the above Lindblad form with Lindblad operators:
where and define the relaxation rates of the individual ancillary pseudospins. With this choice of it can be shown Young et al. () that for the system pseudospins evolve under a renormalized Lindblad master equation with given by Eq. (4) and , thus leaving the thermal state of as the unique stationary state.
For , the effective system-reservoir interaction cools the system towards the ground state, and the Lindblad operators can be reduced to the -body terms Diehl et al. (2008); Kraus et al. (2008). In this limit the ancillary pseudospins become an effective low temperature bath with a cooling rate and heating rate determined by gate errors and any environmental noise. Competition between these rates leads to a minimum reachable temperature for the system , where , with EPG and the error rate and frequency of application of , respectively.
The Lindblad master equation, with operators given by Eq. (4), generates a unitary system-reservior interaction but nonunitary reservoir relaxation. Stroboscopic simulation of is performed in a manner analogous to the simulation described above. Phase angles are chosen in the 1- and 2-body gates to generate an effective static interaction strength over the time between applications of , such that . Nonunitary evolution of the reservoir is generated by encoding the reservoir as two levels of a -system. The pseudospin states are the ground state and the meta-stable state . State is chosen to have fast spontaneous emission to , with rate . This spontaneous emission is the decoherence mechanism required to generate the nonunitary Lindblad evolution. The ancillary pseudospin levels can be placed in a thermal state via the following procedure: i) -pulse on the transition. ii) Wait for decay to ground state, . iii) -pulse on transition. iv) -pulse on the transition. v) Wait for decay, which now yields the final pseudospin state, , corresponding to an effective temperature . The above stroboscopic procedure generates in Eq. (4). The procedure can be simplified in the limit of cooling towards zero temperature by eliminating steps iii-v, when it becomes similar to the optical pumping scheme employed in measurement of qubit states for trapped ions Wineland et al. (1980). This thermalization procedure is then repeated and interleaved with the stroboscopic application of .
Thermalization of the Toric Code.– is in the local form of , with two types of excitations, electric charges and magnetic vortices, () that reside on vertices and plaquettes, respectively, of the square lattice. The excitation operators are defined with , , and . Each link must interact with four ancillary pseudospins in the limit or to allow thermalization to the ground state or the thermal state of , respectively.
The stroboscopic generation of outlined above introduces truncation perturbations in the perturbative expansion, e.g, the second term in Eq. (3), which are distinct from extrinsic errors due to experimental noise and gate inaccuracies. If sufficiently large, such truncation perturbations could drive the system away from the desired ground state phase . We now show on a finite sized system accessible to current experiments Nelson et al. (2007), that the intrinsic perturbations can be kept sufficiently small. Fig. 2(a) plots the gap of as a function of the strength of the perturbation for a 33 planar lattice with toroidal boundary conditions (Fig. 1) . The Zeeman field is added here to fully split the ground state degeneracy and ensure robust characterization of the eigenstates of even in the presence of small additional perturbations.
We define the ground state fidelity as , where the are the degenerate ground states of and are the nearly degenerate ground states of . Fig. 2(b) shows the ground state fidelity as a function of . This fidelity determines the robustness of topological operations that will be performed via string operators Kitaev (2003) to measure or perform gates on the system. We see that for the features of the topological phase persist, including the approximate four-fold degeneracy of the ground state and a finite gap to excitations. This corresponds to a maximum value of , which constrains the gate operations in the pulse sequences, Eq. (2). This robustness should increase with increasing lattice size, and is consistent with known stability of to perturbations Trebst et al. (2007).
Increasing increases and therefore the gap of the ; however it also increases , which reduce the gap of and topological protection for large . We also note that where is the number of sequential gates used to simulate . For larger lattices, some degree of parallelization is thus desirable to ensure that the gap does not decrease with the lattice size. Choosing , the gap achieved by a completely serial implementation is , where is the number of system atoms used. With the cooling sequence serially interleaved, and are achievable Young et al. (). For a minimal system of 18 system atoms, this allows for an effective temperature to be reached with an error rate of or less Young et al. ().
Sources of Errors.– This scheme is designed to be robust against errors within the pseudospin subspace. The dominant source of residual error in the implementation discussed here is leakage from the Rydberg levels due to spontaneous emission and black body radiation. The latter may be effectively suppressed by working at low temperatures Hessels et al. (1998), and spontaneous emission is minimized by utilizing states with larger . With , we estimate that spointaneous emission errors can be reduced to per gate, allowing for up to stroboscopic cycles.
Discussion.– We have developed a formalism for the stroboscopic generation of -body Hamiltonians using - and -body quantum operations together with a dissipative thermalization scheme. We have applied this here to the toric code Hamiltonian in the context of addressable optical lattice experiments Nelson et al. (2007). Our method applies to a wide range of lattice spin models Young et al. () as well as to other experimental setups Porto et al. (2003). The dynamic generation both of a Hamiltonian possessing a topologically ordered ground state and of an effective thermalization mechanism offers the possibility of robust simulation of the ground state and of the creation and braiding of anyonic excitations han (). These are essential components required for the topologically protected storage and manipulation of quantum information.
We thank D. Weiss for useful discussions. This material is based upon work supported by DARPA under Award No. 3854-UCB-AFOSR-0041. During the preparation of this manuscript related results discussing ground state preparation were reported Weimer et al. ().
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