Superconducting quantum simulator for topological order and the toric code

Abstract

Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations and prospects for protecting stored quantum information against errors. Here we show that the Hamiltonian of the central model of this class of systems, the Toric Code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via Superconducting Quantum Interference Devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along non-contractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single qubit operations allow to create and propagate elementary excitations of the Toric Code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.Topological phases of quantum matter [1,2] are of scientific interest for their intriguing quantum correlation properties, exotic excitations with fractional and non-Abelian quantum statistics, and their prospects for physically protecting quantum information against errors [3].The model which takes the central role in the discussion of topological order is the Toric Code [4-6], which is an example of a Z 2 lattice gauge theory. It consists of a two-dimensional spin lattice with quasi local fourbody interactions between the spins and features 4 g degenerate, topologically ordered ground states for a lattice on a surface of genus g. The topological order of these 4 g ground states shows up in their correlation properties. The topologically ordered states are locally indistinguishable (the reduced density matrices of a single spin are identical for all of them) and only show differences for global properties (correlations along non-contractible loops). Moreover local perturbations cannot transform these states into one another and can therefore only excite higher energy states that are separated by a finite energy gap.In equilibrium, the Toric Code ground states are therefore protected against local perturbations that are small compared to the gap, which renders them ideal candidates for storing quantum information [4]. Yet for a selfcorrecting quantum memory, the mobility of excitations needs to be suppressed as well, which so far has only been shown to be achievable in four or more lattice dimensions [7,8]...

Floquet engineering in superconducting circuits: From arbitrary spin-spin interactions to the Kitaev honeycomb model

“…The Abelian phase is realized when |J α | > |J β | + |J γ |, where α, β, γ ∈ {xx, yy, zz}. In this phase the model can be mapped onto the toric code model [7]. Using our proposed circuits, one can also investigate the non-abelian phase of the model.…”

“…For longitudinal modulation, a small oscillating flux is threaded on top of the dc flux through the loop of the qubit's SQUID to modulate its effective Josephson energy, c.f. [7]. As a bimodal modulation we refer to a quasiperiodic Hamiltonian,…”

“…For explicit resonances [6], one can transform the system to the rotating frame of the resonant interaction to obtain an effective high-frequency regime. Yet, the applicability of such an approach is not clear where implicit resonances happen in higher order of perturbation [7]. In this work, we develop a Floquet engineering approach in the resonant modulation regime and apply it to coupled superconducting qubits.…”

Circuit QED with qutrits: Coupling three or more atoms via virtual-photon exchange

“…I of the main text, there exist one other type of effective interactions between the four atoms in the fourth-order perturbation terms, namely, the four-spin ring exchange interaction σ [26,27,55]. Noted that the total excitation number is conserved in this case.…”

Left-handed superlattice metamaterials for circuit QED

“…One of the most promising applications of quantum technologies is the simulation of physical phenomena too complex to be dealt with by other techniques. Amongst the many possible physical implementations of such simulations, superconducting circuits take a central role, with experiments being able to engineer a large number of different many-body Hamiltonians [1][2][3][4][5][6] and reaching regimes that are otherwise challenging for other platforms, e.g. the strong and ultrastrong coupling regime [7][8][9][10][11][12][13][14].…”