Ising Coupling Research Articles

Single-step implementation of high-fidelity n -bit Toffoli gates

The family of $n$-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary $n$-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to $n$ other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely strongly coupling $n+1$ qubits and a driving frequency that scales with $n$, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and on superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-\textsc{not}-gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.

Phase diagram of a three-dimensional dipolar Ising model with textured Ising axes

We study from tempered Monte Carlo simulations the magnetic phase diagram of a textured dipolar Ising model on a face centered cubic lattice. The Ising coupling of the model follow the dipole–dipole interaction. The Ising axes are distributed with a uniaxial symmetry along the direction with a Gaussian probability density of the polar angles. This distribution provides a quenched disorder realization of the dipolar Ising model making a continuous link between the parallel axes dipoles and the random axes dipole models. As expected the phase diagram presents three distinctive phases: paramagnetic, ferromagnetic and spin-glass. A quasi long range ferromagnetic and a reentrant spin-glass phases are obtained in the vicinity of the ferromagnetic spin-glass line. This model provides a way to predict the magnetic phases of magnetic nanoparticles supracrystals in terms of the texturation of the easy axes distribution in the strong anisotropy limit.

Breakdown of intermediate one-half magnetization plateau of spin-1/2 Ising-Heisenberg and Heisenberg branched chains at triple and Kosterlitz-Thouless critical points.

The spin-1/2 Ising-Heisenberg branched chain composed of regularly alternating Ising spins and Heisenberg dimers involving an additional side branching is rigorously solved in a magnetic field by the transfer-matrix approach. The ground-state phase diagram, the magnetization process and the concurrence measuring a degree of bipartite entanglement within the Heisenberg dimers are examined in detail. Three different ground states were found depending on a mutual interplay between the magnetic field and two different coupling constants: the modulated quantum antiferromagnetic phase, the quantum ferrimagnetic phase, and the classical ferromagnetic phase. Two former quantum ground states are manifested in zero-temperature magnetization curves as intermediate plateaus at zero and one-half of the saturation magnetization, whereas the one-half plateau disappears at a triple point induced by a strong-enough ferromagnetic Ising coupling. The ground-state phase diagram and zero-temperature magnetization curves of the analogous spin-1/2 Heisenberg branched chain were investigated using density-matrix renormalization group calculations. The latter fully quantum Heisenberg model involves, besides two gapful phases manifested as zero and one-half magnetization plateaus, gapless quantum spin-liquid phase. The intermediate one-half plateau of the spin-1/2 Heisenberg branched chain vanishes at Kosterlitz-Thouless quantum critical point between gapful and gapless quantum ground states unlike the triple point of the spin-1/2 Ising-Heisenberg branched chain.

Subspace controllability of bipartite symmetric spin networks

We consider a class of spin networks where each spin in a certain set interacts, via Ising coupling, with a set of central spins, and the control acts simultaneously on all the spins. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability in every case.

The Effects of an Impurity in an Ising‐XXZ Diamond Chain on Thermal Entanglement, on Quantum Coherence, and on Quantum Teleportation

AbstractThe effects of an impurity plaquette on the thermal quantum correlations measurement by the concurrence, on quantum coherence quantified by the recently proposed l1‐norm of coherence and on quantum teleportation in a Ising‐ diamond chain are discussed. Such an impurity is formed by the interaction between the interstitial Heisenberg dimers and the nearest‐neighbor Ising coupling between the nodal and interstitial spins. All the interaction parameters are different from those of the rest of the chain. By tailoring them, quantum entanglement and quantum coherence can be controlled and tuned. Therefore, the quantum resources—thermal entanglement and quantum coherence—of the model exhibit a clear performance improvement in comparison to the original model without impurities. It is demonstrated that quantum teleportation can be tuned by its inclusion. Thermal teleportation is modified in a significant way as well, and a strong increase in the average fidelity is observed. The exact solution is furnished by the use of the transfer‐matrix method.

Ising Model Parameter Estimation with Confidence Evaluation Using the Exchange Monte Carlo Method

We propose a method to evaluate confidence of estimation of both the Ising coupling constant and observation noise intensity from the observed spin states. The Ising model has a variety of applicat...

Nonlocal string order parameter in the S=12 Kitaev-Heisenberg ladder

We study the spin-$\frac{1}{2}$ Kitaev-Heisenberg (KJ) model in a two-leg ladder. Without a Heisenberg interaction, the Kitaev phase in the ladder model has Majorana fermions with local Z$_2$ gauge fields, and is usually described as a disordered phase without any order parameter. Here we prove the existence of a non-local string order parameter (SOP) in the Kitaev phase which survives with a finite Heisenberg interaction. The SOP is obtained by relating the Kitaev ladder, through a non-local unitary transformation, to a one-dimensional $XY$ chain with an Ising coupling to a dangling spin at every site. This differentiates the Kitaev phases from other nearby phases including a rung singlet. Two phases with non-zero SOP corresponding to ferromagnetic and antiferromagnetic Kitaev interactions are identified. The full phase diagram of the KJ ladder is determined using exact diagonalization and density matrix renormalization group methods, which shows a striking similarity to the KJ model on a two-dimensional honeycomb lattice.

FK–Ising coupling applied to near-critical planar models

We consider the Ising model at its critical temperature with external magnetic field ha15∕8 on aZ2. We give a purely probabilistic proof, using FK methods rather than reflection positivity, that for a=1, the correlation length is ≥const.h−8∕15 as h↓0. We extend to the a↓0 continuum limit the FK–Ising coupling for all h>0, and obtain tail estimates for the largest renormalized cluster area in a finite domain as well as an upper bound with exponent 1∕8 for the one-arm event. Finally, we show that for a=1, the average magnetization, M(h), in Z2 satisfies M(h)∕h1∕15→ some B∈(0,∞) as h↓0.

Tripartite quantum correlations in the renormalized space of Heisenberg-Ising spin−1∕2 chain

The dynamics of tripartite entanglement and tripartite quantum discord in the superposition of the two degenerate ground states of the Heisenberg-Ising spin −1∕2 chain with the increasing size of the system are studied through quantum renormalization group (QRG) technique. It is shown that although, both the correlations exhibit quantum phase transition in the thermodynamics limit, however, quantum discord is more sensitive to the size of the system. Unlike the bipartite case, the tripartite correlations are maximum when the Ising coupling between the spins dominates the Heisenberg coupling. Interestingly enough, it is found that the regions of entangled phase and product phase in the tripartite case are swapped with respect to the ones in the bipartite case.

Temporal Order in Periodically Driven Spins in Star-Shaped Clusters.

We experimentally study the response of star-shaped clusters of initially unentangled N=4, 10, and 37 nuclear spin-1/2 moments to an inexact π-pulse sequence and show that an Ising coupling between the center and the satellite spins results in robust period-2 magnetization oscillations. The period is stable against bath effects, but the amplitude decays with a timescale that depends on the inexactness of the pulse. Simulations reveal a semiclassical picture in which the rigidity of the period is due to a randomizing effect of the Larmor precession under the magnetization of surrounding spins. The timescales with stable periodicity increase with net initial magnetization, even in the presence of perturbations, indicating a robust temporal ordered phase for large systems with finite magnetization per spin.

The Ising model coupled to 2d orders

In this article we make first steps in coupling matter to causal set theory in the path integral. We explore the case of the Ising model coupled to the 2d discrete Einstein Hilbert action, restricted to the 2d orders. We probe the phase diagram in terms of the Wick rotation parameter β and the Ising coupling j and find that the matter and the causal sets together give rise to an interesting phase structure. The couplings give rise to five different phases. The causal sets take on random or crystalline characteristics as described in Surya (2012 Class. Quantum Grav. 29 132001) and the Ising model can be correlated or uncorrelated on the random orders and correlated, uncorrelated or anti-correlated on the crystalline orders. We find that at least one new phase transition arises, in which the Ising spins push the causal set into the crystalline phase.

Long range of indirect exchange interaction on the edges of MoS2 flakes

We study the Ruderman–Kittel–Kasuya–Yosida interaction between two magnetic impurities connected to the edges of zigzag-terminated MoS2 flakes. When the impurities lie on the edges of the flake, the effective exchange interaction exhibits sizable noncollinear Dzyaloshinskii–Moriya character that competes with a strong Ising coupling. We analyze the characteristic decay exponent for doping levels inside the band gap of the infinite layer, corresponding to edge states of the flake at the Fermi level. The characteristic exponents show sub-two-dimensional behavior for these band fillings, with decays much slower than quadratic. The Ising interaction has effectively one-dimensional long range, while the noncollinear component that grows for short impurity separation becomes comparable in magnitude. The resulting tunable exchange interaction on these systems opens the way for the study of interesting phases of impurity arrays with long-range stable helical order.

Mapping of the Bak, Tang, and Wiesenfeld sandpile model on a two-dimensional Ising-correlated percolation lattice to the two-dimensional self-avoiding random walk.

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.

The influence of further-neighbor spin-spin interaction on a ground state of 2D coupled spin-electron model in a magnetic field

An exhaustive ground-state analysis of extended two-dimensional (2D) correlated spin-electron model consisting of the Ising spins localized on nodal lattice sites and mobile electrons delocalized over pairs of decorating sites is performed within the framework of rigorous analytical calculations. The investigated model, defined on an arbitrary 2D doubly decorated lattice, takes into account the kinetic energy of mobile electrons, the nearest-neighbor Ising coupling between the localized spins and mobile electrons, the further-neighbor Ising coupling between the localized spins and the Zeeman energy. The ground-state phase diagrams are examined for a wide range of model parameters for both ferromagnetic as well as antiferromagnetic interaction between the nodal Ising spins and non-zero value of external magnetic field. It is found that non-zero values of further-neighbor interaction leads to a formation of new quantum states as a consequence of competition between all considered interaction terms. Moreover, the new quantum states are accompanied with different magnetic features and thus, several kinds of field-driven phase transitions are observed.

Phase transitions and thermal entanglement of the distorted Ising–Heisenberg spin chain: topology of multiple-spin exchange interactions in spin ladders

We consider a symmetric spin-1/2 Ising-XXZ double sawtooth spin ladder obtained from distorting a spin chain, with the XXZ interaction between the interstitial Heisenberg dimers (which are connected to the spins based on the legs via an Ising-type interaction), the Ising coupling between nearest-neighbor spins of the legs and rungs spins, respectively, and additional cyclic four-spin exchange (ring exchange) in the square plaquette of each block. The presented analysis supplemented by results of the exact solution of the model with infinite periodic boundary implies a rich ground state phase diagram. As well as the quantum phase transitions, the characteristics of some of the thermodynamic parameters such as heat capacity, magnetization and magnetic susceptibility are investigated. We prove here that among the considered thermodynamic and thermal parameters, solely heat capacity is sensitive versus the changes of the cyclic four-spin exchange interaction. By using the heat capacity function, we obtain a singularity relation between the cyclic four-spin exchange interaction and the exchange coupling between pair spins on each rung of the spin ladder. All thermal and thermodynamic quantities under consideration should be investigated by regarding those points which satisfy the singularity relation. The thermal entanglement within the Heisenberg spin dimers is investigated by using the concurrence, which is calculated from a relevant reduced density operator in the thermodynamic limit.

Magnetization processes and existence of reentrant phase transitions in coupled spin-electron model on doubly decorated planar lattices

An alternative model for a description of magnetization processes in coupled 2D spin-electron systems has been introduced and rigorously examined using the generalized decoration-iteration transformation and the corner transfer matrix renormalization group method. The model consists of localized Ising spins placed on nodal lattice sites and mobile electrons delocalized over the pairs of decorating sites. It takes into account a hopping term for mobile electrons, the Ising coupling between mobile electrons and localized spins as well as the Zeeman term acting on both types of particles. The ground-state and finite-temperature phase diagrams were established and comprehensively analyzed. It was found that the ground-state phase diagrams are very rich depending on the electron hopping and applied magnetic field. The diversity of magnetization curves can be related to intermediate magnetization plateaus, which may be continuously tuned through the density of mobile electrons. In addition, the existence of several types of reentrant phase transitions driven either by temperature or magnetic field was proven.

Thermal entanglement in a spin-1/2 Ising-XYZ distorted diamond chain with the second-neighbor interaction between nodal Ising spins

We consider a spin-1/2 Ising-XYZ distorted diamond chain with the XYZ interaction between the interstitial Heisenberg dimers, the nearest-neighbor Ising coupling between the nodal and interstitial spins, respectively, and the second-neighbor Ising coupling between the nodal spins. The ground-state phase diagram of the spin-1/2 Ising-XYZ distorted diamond chain exhibits several intriguing phases due to the XY anisotropy and the second-neighbor interaction, whereas the model can be exactly solved using the transfer-matrix technique. The quantum entanglement within the Heisenberg spin dimers is studied through a bipartite measure concurrence, which is calculated from a relevant reduced density operator. The concurrence may either show a standard thermal dependence with a monotonous decline with increasing temperature or a more peculiar thermal dependence accompanied with reentrant behavior of the concurrence. Based in the present model, it is conjectured that the bipartite entanglement between the interstitial Heisenberg spin pairs in the natural mineral azurite is quite insensitive to the applied magnetic field and it persists up to approximately 30 K.

Random quantum discord in a mixed three-spin Ising-XY model with added Dzyaloshinskii-Moriya (DM) interaction

In the present work, we first verify pairwise quantum discord (QD) between half-spins of a mixed three-spin (1/2,1,1/2) system with Ising-XY model for which spins (1,1/2) have Ising interaction and spins (1/2,1/2) have both XY and Dzyaloshinskii-Moriya (DM) interactions. A single-ion anisotropy property ζ is assumed for integer spin. This system is considered at the front of an external homogeneous magnetic field B in thermal equilibrium. Then, we investigate the pairwise entanglement between half-spins by using entanglement of formation (EoF) as functions of temperature T, magnetic field B and other introduced parameters. In this case, all parameters are independent of the temperature. We compare the QD and the EoF verified for the spins (1/2,1/2) and express some interesting temperature, magnetic and quantum correlation properties. Also, we study the anisotropy influences on the quantum correlation between such spins in detail, and we find that this parameter can be used as a quantum discord controller. In the second step, we investigate QD for the half-spins in a stochastic manner, i.e., when all parameters of the tripartite system are constant (independent of the temperature) except the anisotropy, which is considered as a random variable dependent on the temperature. Finally, we assume that all parameters are constant except the Ising coupling strength, which can change randomly with temperature. Ultimately, we conclude that in the experimental conditions under which some parameters may have stochastic behavior, the anisotropy can also be a good candidate to control QD.

Rise and Fall of Reentrant Phase Transitions in a Coupled Spin-Electron Model on a Doubly Decorated Honeycomb Lattice

Phase diagrams and spontaneous magnetization are rigorously calculated for a coupled spin-electron model on a doubly decorated honeycomb lattice, which accounts for a quantum-mechanical hopping of the mobile electrons on decorating sites, the nearest-neighbor Ising coupling between mobile electrons and localized spins, as well as, the further-neighbor Ising coupling between the localized spins placed on nodal sites. The spontaneously ordered ferromagnetic phase, spontaneously ordered antiferromagnetic phase and disordered paramagnetic phase emerge in a phase diagram depending on an electron filling of the decorating sites, a relative size of the hopping term and both considered coupling constants. It is evidenced that a nature and size of the further-neighbor Ising coupling between the localized spins basically influences rise and fall of reentrant transitions close to a phase boundary between the paramagnetic phase and both spontaneously ordered phases.

Spin-Liquid-to-Spin-Liquid Transition in Kitaev Magnets Driven by Fractionalization.

While phase transitions between magnetic analogs of the three states of matter-a long-range ordered state, paramagnet, and spin liquid-are extensively studied, the possibility of "liquid-liquid" transitions, namely, between different spin liquids, remains elusive. By introducing the additional Ising coupling into the honeycomb Kitaev model with bond asymmetry, we discover that the Kitaev spin liquid turns into a spin-nematic quantum paramagnet before a magnetic order is established by the Ising coupling. The quantum phase transition between the two liquid states accompanies a topological change driven by fractionalized excitations, the Z_{2} gauge fluxes, and is of first order. At finite temperatures, this yields a persisting first-order transition line that terminates at a critical point located deep inside the regime where quantum spins are fractionalized. It is suggested that similar transitions may occur in other perturbed Kitaev magnets with bond asymmetry.