Creation, manipulation, and detection of Abelian and non-Abelian anyons in optical lattices

Separability Transitions in Topological States Induced by Local Decoherence

We study states with intrinsic topological order subjected to local decoherence from the perspective of separability, i.e., whether a decohered mixed state can be expressed as an ensemble of short-range entangled pure states. We focus on toric codes and the X-cube fracton state and provide evidence for the existence of decoherence-induced separability transitions that precisely coincide with the threshold for the feasibility of active error correction. A key insight is that local decoherence acting on the "parent" cluster states of these models results in a Gibbs state. As an example, for the 2D (3D) toric code subjected to bit-flip errors, we show that the decohered density matrix can be written as a convex sum of short-range entangled states for p>p_{c}, where p_{c} is related to the paramagnetic-ferromagnetic transition in the 2D (3D) random bond Ising model along the Nishimori line.

Unveiling the Non-Abelian Statistics of D(S_{3}) Anyons Using a Classical Photonic Simulator

Simulators can realize novel phenomena by separating them from the complexities of a full physical implementation. Here, we put forward a scheme that can simulate the exotic statistics of D(S_{3}) non-Abelian anyons with minimal resources. The qudit lattice representation of this planar code supports local encoding of D(S_{3}) anyons. As a proof-of-principle demonstration, we employ a classical photonic simulator to encode a single qutrit and manipulate it to perform the fusion and braiding properties of non-Abelian D(S_{3}) anyons. The photonic technology allows us to perform the required nonunitary operations with much higher fidelity than what can be achieved with current quantum computers. Our approach can be directly generalized to larger systems or to different anyonic models, thus enabling advances in the exploration of quantum error correction and fundamental physics alike.

Preparation of Matrix Product States with Log-Depth Quantum Circuits

We consider the preparation of matrix product states (MPS) on quantum devices via quantum circuits of local gates. We first prove that faithfully preparing translation-invariant normal MPS of N sites requires a circuit depth T=Ω(logN). We then introduce an algorithm based on the renormalization-group transformation to prepare normal MPS with an error ε in depth T=O[log(N/ε)], which is optimal. We also show that measurement and feedback leads to an exponential speedup of the algorithm to T=O[loglog(N/ε)]. Measurements also allow one to prepare arbitrary translation-invariant MPS, including long-range non-normal ones, in the same depth. Finally, the algorithm naturally extends to inhomogeneous MPS.

Adjustable rotation of multiple vortices produced by diode-pumped Nd:YVO4 lasers using intracavity second harmonic generation

The criteria for achieving adjustable rotation of optical vortices are analyzed and used to design a diode-pumped solid-state laser that incorporates intracavity second harmonic generation within a concave-flat cavity to produce frequency-doubled Hermite-Gaussian (FDHG) modes. These FDHG modes are subsequently employed to generate various structured lights containing 2, 4, and 6 nested vortices using an external cylindrical mode converter. Through theoretical exploration, we propose that increasing the radius of curvature of the concave mirror and extending the cavity length can enhance the rotational angles of multiple vortices by expanding the adjustable range of phase shift for FDHG modes. Moreover, theoretical analyses assess vortex rotation concerning the positions of a nonlinear medium, successfully validating the experimental observations and elucidating the phase structures of the transformed beams.

Nishimori's Cat: Stable Long-Range Entanglement from Finite-Depth Unitaries and Weak Measurements

In the field of monitored quantum circuits, it has remained an open question whether finite-time protocols for preparing long-range entangled states lead to phases of matter that are stable to gate imperfections, that can convert projective into weak measurements. Here, we show that in certain cases, long-range entanglement persists in the presence of weak measurements, and gives rise to novel forms of quantum criticality. We demonstrate this explicitly for preparing the two-dimensional Greenberger-Horne-Zeilinger cat state and the three-dimensional toric code as minimal instances. In contrast to random monitored circuits, our circuit of gates and measurements is deterministic; the only randomness is in the measurement outcomes. We show how the randomness in these weak measurements allows us to track the solvable Nishimori line of the random-bond Ising model, rigorously establishing the stability of the glassy long-range entangled states in two and three spatial dimensions. Away from this exactly solvable construction, we use hybrid tensor network and Monte Carlo simulations to obtain a nonzero Edwards-Anderson order parameter as an indicator of long-range entanglement in the two-dimensional scenario. We argue that our protocol admits a natural implementation in existing quantum computing architectures, requiring only a depth-3 circuit on IBM's heavy-hexagon transmon chips.

Shortest Route to Non-Abelian Topological Order on a Quantum Processor

A highly coveted goal is to realize emergent non-Abelian gauge theories and their anyonic excitations, which encode decoherence-free quantum information. While measurements in quantum devices provide new hope for scalably preparing such long-range entangled states, existing protocols using the experimentally established ingredients of a finite-depth circuit and a single round of measurement produce only Abelian states. Surprisingly, we show there exists a broad family of non-Abelian states-namely those with a Lagrangian subgroup-which can be created using these same minimal ingredients, bypassing the need for new resources such as feed forward. To illustrate that this provides realistic protocols, we show how D_{4} non-Abelian topological order can be realized, e.g., on Google's quantum processors using a depth-11 circuit and a single layer of measurements. Our work opens the way toward the realization and manipulation of non-Abelian topological orders, and highlights counterintuitive features of the complexity of non-Abelian phases.

Quantum Circuits Assisted by Local Operations and Classical Communication: Transformations and Phases of Matter

We introduce deterministic state-transformation protocols between many-body quantum states that can be implemented by low-depth quantum circuits followed by local operations and classical communication. We show that this gives rise to a classification of phases in which topologically ordered states or other paradigmatic entangled states become trivial. We also investigate how the set of unitary operations is enhanced by local operations and classical communication in this scenario, allowing one to perform certain large-depth quantum circuits in terms of low-depth ones.

One-dimensional mixtures of several ultracold atoms: a review

Recent theoretical and experimental progress on studying one-dimensional systems of bosonic, fermionic, and Bose-Fermi mixtures of a few ultracold atoms confined in traps is reviewed in the broad context of mesoscopic quantum physics. We pay special attention to limiting cases of very strong or very weak interactions and transitions between them. For bosonic mixtures, we describe the developments in systems of three and four atoms as well as different extensions to larger numbers of particles. We also briefly review progress in the case of spinor Bose gases of a few atoms. For fermionic mixtures, we discuss a special role of spin and present a detailed discussion of the two- and three-atom cases. We discuss the advantages and disadvantages of different computation methods applied to systems with intermediate interactions. In the case of very strong repulsion, close to the infinite limit, we discuss approaches based on effective spin chain descriptions. We also report on recent studies on higher-spin mixtures and inter-component attractive forces. For both statistics, we pay particular attention to impurity problems and mass imbalance cases. Finally, we describe the recent advances on trapped Bose-Fermi mixtures, which allow for a theoretical combination of previous concepts, well illustrating the importance of quantum statistics and inter-particle interactions. Lastly, we report on fundamental questions related to the subject which we believe will inspire further theoretical developments and experimental verification.

Digital Quantum Simulation of Z_{2} Lattice Gauge Theories with Dynamical Fermionic Matter

We propose a scheme for digital quantum simulation of lattice gauge theories with dynamical fermions. Using a layered optical lattice with ancilla atoms that can move and interact with the other atoms (simulating the physical degrees of freedom), we obtain a stroboscopic dynamics which yields the four-body plaquette interactions, arising in models with (2+1) and higher dimensions, without the use of perturbation theory. As an example we show how to simulate a Z_{2} model in (2+1) dimensions.

Digital Quantum Simulation of Z_{2} Lattice Gauge Theories with Dynamical Fermionic Matter

We propose a scheme for digital quantum simulation of lattice gauge theories with dynamical fermions. Using a layered optical lattice with ancilla atoms that can move and interact with the other atoms (simulating the physical degrees of freedom), we obtain a stroboscopic dynamics which yields the four-body plaquette interactions, arising in models with (2+1) and higher dimensions, without the use of perturbation theory. As an example we show how to simulate a Z_{2} model in (2+1) dimensions.

Anyon Hubbard Model in One-Dimensional Optical Lattices

Raman-assisted hopping may be used to realize the anyon Hubbard model in one-dimensional optical lattices. We propose a feasible scenario that significantly improves the proposal of T. Keilmann et al. [Nat. Commun. 2, 361 (2011)], allowing as well for an exact realization of the two-body hard-core constraint, and for controllable effective interactions without the need of Feshbach resonances. We show that the combination of anyonic statistics and two-body hard-core constraint leads to a rich ground-state physics, including Mott insulators with attractive interactions, pair superfluids, dimer phases, and multicritical points. Moreover, the anyonic statistics results in a novel two-component superfluid of holon and doublon dimers, characterized by a large but finite compressibility and a multipeaked momentum distribution, which may be easily revealed experimentally.

Dynamic creation of a topologically-ordered Hamiltonian using spin-pulse control in the Heisenberg model

Hamiltonian engineering is an important approach for quantum information processing, when appropriate materials do not exist in nature or are unstable. So far there is no stable material for the Kitaev spin Hamiltonian with anisotropic interactions on a honeycomb lattice, which plays a crucial role in the realization of both Abelian and non-Abelian anyons. Here, we show two methods to dynamically realize the Kitaev spin Hamiltonian from the conventional Heisenberg spin Hamiltonian using pulse-control techniques based on the Baker-Campbell-Hausdorff (BCH) formula. In the first method, the Heisenberg interaction is changed into Ising interactions in the first process of the pulse sequence. In the next process of the first method, we transform them to a desirable anisotropic Kitaev spin Hamiltonian. In the second more efficient method, we show that if we carefully design two-dimensional pulses that vary depending on the qubit location, we can obtain the desired Hamiltonian in only one step of applying the BCH formula. As an example, we apply our methods to spin qubits based on quantum dots, in which the effects of both the spin-orbit interaction and the hyperfine interaction are estimated.

Nonequilibrium dynamics of one-dimensional hard-core anyons following a quench: complete relaxation of one-body observables

We demonstrate the role of interactions in driving the relaxation of an isolated integrable quantum system following a sudden quench. We consider a family of integrable hard-core lattice anyon models that continuously interpolates between noninteracting spinless fermions and strongly interacting hard-core bosons. A generalized Jordan-Wigner transformation maps the entire family to noninteracting fermions. We find that, aside from the singular free-fermion limit, the entire single-particle density matrix and, therefore, all one-body observables relax to the predictions of the generalized Gibbs ensemble (GGE). This demonstrates that, in the presence of interactions, correlations between particles in the many-body wave function provide the effective dissipation required to drive the relaxation of all one-body observables to the GGE. This relaxation does not depend on translational invariance or the tracing out of any spatial domain of the system.

Order parameter for symmetry-protected phases in one dimension

We introduce an order parameter for symmetry-protected phases in one dimension which allows us to directly identify those phases. The order parameter consists of stringlike operators and swaps, but differs from conventional string order operators in that it only depends on the symmetry but not on the state. We verify our framework through numerical simulations for the SO(3) invariant spin-1 bilinear-biquadratic model which exhibits a dimerized and a Haldane phase, and find that the order parameter not only works very well for the dimerized and the Haldane phase, but it also returns a distinct signature for gapless phases. Finally, we discuss possible ways to measure the order parameter in experiments with cold atoms.

Physical implementation of protected qubits

We review the general notion of topological protection of quantum states in spin models and its relation with the ideas of quantum error correction. We show that topological protection can be viewed as a Hamiltonian realization of error correction: for a quantum code for which the minimal number of errors that remain undetected is N, the corresponding Hamiltonian model of the effects of the environment noise appears only in the Nth order of the perturbation theory.We discuss the simplest model Hamiltonians that realize topological protection and their implementation in superconducting arrays. We focus on two dual realizations: in one the protected state is stored in the parity of the Cooper pair number, in the other, in the parity of the flux number. In both cases the superconducting arrays allow a number of fault-tolerant operations that should make the universal quantum computation possible.

Interface between topological and superconducting qubits

We propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled-phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit.

Non-Abelian anions from degenerate landau levels of ultracold atoms in artificial gauge potentials

We show that non-Abelian potentials acting on ultracold gases with two hyperfine levels can give rise to ground states with non-Abelian excitations. We consider a realistic gauge potential for which the Landau levels can be exactly determined: The non-Abelian part of the vector potential makes the Landau levels nondegenerate. In the presence of strong repulsive interactions, deformed Laughlin ground states occur in general. However, at the degeneracy points of the Landau levels, non-Abelian quantum Hall states appear: These ground states, including deformed Moore-Read states (characterized by Ising anyons as quasiholes), are studied for both fermionic and bosonic gases.

Fault-tolerant Greenberger-Horne-Zeilinger paradox based on non-Abelian anyons

We propose a scheme to test the Greenberger-Horne-Zeilinger paradox based on braidings of non-Abelian anyons, which are exotic quasiparticle excitations of topological states of matter. Because topological ordered states are robust against local perturbations, this scheme is in some sense "fault-tolerant" and might close the detection inefficiency loophole problem in previous experimental tests of the Greenberger-Horne-Zeilinger paradox. In turn, the construction of the Greenberger-Horne-Zeilinger paradox reveals the nonlocal property of non-Abelian anyons. Our results indicate that the non-Abelian fractional statistics is a pure quantum effect and cannot be described by local realistic theories. Finally, we present a possible experimental implementation of the scheme based on the anyonic interferometry technologies.