Quantum Spin Glass Research Articles

Information scrambling and butterfly velocity in quantum spin glass chains

Information scrambling and butterfly velocity in quantum spin glass chains

Exact Numerical Solution of the Fully Connected Classical and Quantum Heisenberg Spin Glass.

We present the mean field solution of the quantum and classical Heisenberg spin glasses, using the combination of a high precision numerical solution of the Parisi full replica symmetry breaking equations and a continuous time quantum MonteCarlo algorithm. We characterize the spin glass order and its low-energy excitations down to zero temperature. The Heisenberg spin glass has a rougher energy landscape than its Ising analog, and exhibits a very slow temperature evolution of its dynamical properties. We extend our analysis to the doped, metallic Heisenberg spin glass, which displays unexpectedly slow spin dynamics, reflecting the proximity to the melting quantum critical point and its associated Sachdev-Ye-Kitaev Planckian dynamics.

NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians

Motivated by the ground state structure of quantum models with all-to-all interactions such as mean-field quantum spin glass models and the Sachdev-Ye-Kitaev (SYK) model, we propose a tensor network architecture which can accomodate volume law entanglement and a large ground state degeneracy. We call this architecture the non-local renormalization ansatz (NoRA) because it can be viewed as a generalization of MERA, DMERA, and branching MERA networks with the constraints of spatial locality removed. We argue that the architecture is potentially expressive enough to capture the entanglement and complexity of the ground space of the SYK model, thus making it a suitable variational ansatz, but we leave a detailed study of SYK to future work. We further explore the architecture in the special case in which the tensors are random Clifford gates. Here the architecture can be viewed as the encoding map of a random stabilizer code. We introduce a family of codes inspired by the SYK model which can be chosen to have constant rate and linear distance at the cost of some high weight stabilizers. We also comment on potential similarities between this code family and the approximate code formed from the SYK ground space.

Backscattering silicon spectrometer (BASIS): sixteen years in advanced materials characterization.

Quasielastic neutron scattering (QENS) is an experimental technique that can measure parameters of mobility, such as diffusion jump rate and jump length, as well as localized relaxations of chemical species (molecules, ions, and segments) at atomic and nanometer length scales. Due to the high penetrative power of neutrons and their sensitivity to neutron scattering cross-section of chemical species, QENS can effectively probe mobility inside most bulk materials. This review focuses on QENS experiments performed using a neutron backscattering silicon spectrometer (BASIS) to explore the dynamics in various materials and understand their structure-property relationship. BASIS is a time-of-flight near-backscattering inverted geometry spectrometer with very high energy resolution (approximately 0.0035 meV of full width at half maximum), allowing measurements of dynamics on nano to picosecond timescales. The science areas studied with BASIS are diverse, with a focus on soft matter topics, including traditional biological and polymer science experiments, as well as measurements of fluids ranging from simple hydrocarbons and aqueous solutions to relatively complex room-temperature ionic liquids and deep-eutectic solvents, either in the bulk state or confined. Additionally, hydrogen confined in various materials is routinely measured on BASIS. Other topics successfully investigated at BASIS include quantum fluids, spin glasses, and magnetism. BASIS has been in the user program since 2007 at the Spallation Neutron Source of the Oak Ridge National Laboratory, an Office of Science User Facility supported by the U.S. Department of Energy. Over the past sixteen years, BASIS has contributed to various scientific disciplines, exploring the structure and dynamics of many chemical species and their fabrication for practical applications. A comprehensive review of BASIS contributions and capabilities would be an asset to the materials science community, providing insights into employing the neutron backscattering technique for advanced materials characterization.

Statistical aspects of a quantum Hopfield model

In this paper, we study the eigenstate properties of a quantum Hopfield model by the exact diagonalization method. The local permutational symmetry in this model organizes the spins into clusters, which can each be considered a large quantum spin interacting with others. It is shown that such a quantum Hopfield model, even though without dissipation, is interesting in its own right as an example of quantum frustrated magnetism and quantum spin glass. It exhibits three distinct phases: a low-energy spin-glass phase at a low transverse field, a thermal paramagnetic phase at a high transverse field, and a nonthermal high-energy paramagnetic phase. The dynamics of the revival probability starting from a memory pattern in such a closed quantum many-body model has also been studied.

The Restricted Boltzmann Machine Ansatz through Adiabiatic Routes

We discuss the expressive power of the quantum restricted Boltzmann machine Ansatz in order to represent faithfully the operation process of a quantum spin-glass system undergoing an adiabatic quantum computation process. We address both the representability and the trainability problems associated to the restricted Boltzmann machine Ansatz when we try to describe the ground state of a quantum Ising spin-glass in a transverse field. In our approach, we obtain the ground state of the system using exact diagonalization and find the optimal restricted Boltzman machine. Our preliminary results show that due to the complex spin-glass landscape, for some topologies of the restricted Bolztmann machine, the parameters of the representation are hard to be found.

Reentrant phase transitions involving glassy and superfluid orders in the random hopping Bose–Hubbard model

We study a system of strongly correlated bosons with off-diagonal disorder, i.e., randomness in the kinetic energy, and find a family of reentrant phase transitions that occur as a function of the on-site interaction. We model the system using the paradigmatic Bose–Hubbard Hamiltonian with a random hopping term and solve it employing the replica trick and Trotter–Suzuki expansion known from quantum spin-glasses. From subsequent numerical calculations, we find three distinct phase boundaries at which the reentrant transitions occur: between glass and disordered phase, between superglass and superfluid ones, and between superfluid and disordered phases. All three happen at temperatures slightly above critical temperatures of corresponding non-interacting systems. When the emerging and disappearing order is glassy, this corresponds to the interplay of the thermal energy and the spread of hoppings. When superfluidity is involved, thermal fluctuations must slightly overcome the mean hopping in turn for the reentrance to occur.

Variational Ansatz for the Ground State of the Quantum Sherrington-Kirkpatrick Model.

We present an Ansatz for the ground states of the quantum Sherrington-Kirkpatrick model, a paradigmatic model for quantum spin glasses. Our Ansatz, based on the concept of generalized coherent states, very well captures the fundamental aspects of the model, including the ground state energy and the position of the spin glass phase transition. It further enables us to study some previously unexplored features, such as the nonvanishing longitudinal field regime and the entanglement structure of the ground states. We find that the ground state entanglement can be captured by a simple ensemble of weighted graph states with normally distributed phase gates, leading to a volume law entanglement, contrasting with predictions based on entanglement monogamy.

Lattice connectivity and entanglement in quantum spin-glasses

I have studied the role of lattice connectivity and coupling weights distribution on the entanglement of quantum spin-glasses. It's found in this work that the connectivity of the lattice weakly influence the degree of entanglement in the spin-glass compared to the distribution of the coupling constants between the spins. This suggest important implications for machine learning models such as Boltzmann machines and the study of complex quantum systems.

Spectral form factor of a quantum spin glass

It is widely expected that systems which fully thermalize are chaotic in the sense of exhibiting random-matrix statistics of their energy level spacings, whereas integrable systems exhibit Poissonian statistics. In this paper, we investigate a third class: spin glasses. These systems are partially chaotic but do not achieve full thermalization due to large free energy barriers. We examine the level spacing statistics of a canonical infinite-range quantum spin glass, the quantum p-spherical model, using an analytic path integral approach. We find statistics consistent with a direct sum of independent random matrices, and show that the number of such matrices is equal to the number of distinct metastable configurations — the exponential of the spin glass “complexity” as obtained from the quantum Thouless-Anderson-Palmer equations. We also consider the statistical properties of the complexity itself and identify a set of contributions to the path integral which suggest a Poissonian distribution for the number of metastable configurations. Our results show that level spacing statistics can probe the ergodicity-breaking in quantum spin glasses and provide a way to generalize the notion of spin glass complexity beyond models with a semi-classical limit.

Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, HI. Here, we point to the existence of an analytical solution for such a problem for an arbitrary HI beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotonic power-law suppression of nonadiabatic excitations with time T of QA, without any signature of a transition to a glass phase, which is usually characterized by a logarithmic energy relaxation. This behavior suggests that the energy relaxation can differ in classical and quantum spin glasses strongly, when it is assisted by external time-dependent fields. In specific cases of HI, the solution also shows a considerable quantum speedup in computations.

Dynamic parallel spin stripes from the 1/8 anomaly to the end of superconductivity in La1.6−xNd0.4SrxCuO4

We have carried out neutron spectroscopic measurements on single crystals of ${\mathrm{La}}_{1.6\ensuremath{-}x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ from $0.12\ensuremath{\le}x\ensuremath{\le}0.26$ using time-of-flight techniques. These measurements allow us to follow the evolution of parallel spin stripe fluctuations with energies less than $\ensuremath{\sim}33$ meV, from $x=0.12$ to 0.26. Samples at these hole-doping levels are known to display static (on the neutron-scattering time scale) parallel spin stripes at low temperature, with onset temperatures and intensities which decrease rapidly with increasing $x$. Nonetheless, we report remarkably similar dynamic spectral weight for the corresponding dynamic parallel spin stripes, between 5 and 33 meV, from the 1/8 anomaly near $x=0.12$, to optimal doping near $x=0.19$ to the quantum critical point for the pseudogap phase near $x=0.24$, and finally to the approximate end of superconductivity near $x=0.26$. This observed dynamic magnetic spectral weight is structured in energy with a peak near 17 meV at all dopings studied. Earlier neutron and resonant x-ray scattering measurements on related cuprate superconductors have reported both a disappearance with increasing doping of magnetic fluctuations at ($\ensuremath{\pi}$, $\ensuremath{\pi}$) wave vectors characterizing parallel spin stripe structures and persistant paramagnon scattering away from this wave vector, respectively. Our results for ${\mathrm{La}}_{1.6\ensuremath{-}x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ from 0.12 $\ensuremath{\le}x\ensuremath{\le}0.26$ clearly show persistent parallel spin stripe fluctuations at and around at ($\ensuremath{\pi}$, $\ensuremath{\pi}$), and across the full range of doping studied. These results are also compared to recent theory. Together with a rapidly declining $x$ dependence to the static parallel spin stripe order, the persistent parallel spin stripe fluctuations show a remarkable similarity to the expectations of a quantum spin glass, random t-J model, recently introduced to describe strong local correlations in cuprates.

Noise-tolerant quantum speedups in quantum annealing without fine tuning

Quantum annealing is a powerful alternative model of quantum computing, which can succeed in the presence of environmental noise even without error correction. However, despite great effort, no conclusive demonstration of a quantum speedup (relative to state of the art classical algorithms) has been shown for these systems, and rigorous theoretical proofs of a quantum advantage (such as the adiabatic formulation of Grover’s search problem) generally rely on exponential precision in at least some aspects of the system, an unphysical resource guaranteed to be scrambled by experimental uncertainties and random noise. In this work, we propose a new variant of quantum annealing, called RFQA, which can maintain a scalable quantum speedup in the face of noise and modest control precision. Specifically, we consider a modification of flux qubit-based quantum annealing which includes low-frequency oscillations in the directions of the transverse field terms as the system evolves. We show that this method produces a quantum speedup for finding ground states in the Grover problem and quantum random energy model, and thus should be widely applicable to other hard optimization problems which can be formulated as quantum spin glasses. Further, we explore three realistic noise channels and show that the speedup from RFQA is resilient to 1/f-like local potential fluctuations and local heating from interaction with a sufficiently low temperature bath. Another noise channel, bath-assisted quantum cooling transitions, actually accelerates the algorithm and may outweigh the negative effects of the others. We also detail how RFQA may be implemented experimentally with current technology.

The quantum random energy model as a limit of p-spin interactions

We consider the free energy of a mean-field quantum spin glass described by a [Formula: see text]-spin interaction and a transversal magnetic field. Recent rigorous results for the case [Formula: see text], i.e. the quantum random energy model (QREM), are reviewed. We show that the free energy of the [Formula: see text]-spin model converges in a joint thermodynamic and [Formula: see text] limit to the free energy of the QREM.

Dynamical correlations and a quantum glass phase in a random hopping Bose–Hubbard model

We investigate a system of interacting bosons with random intersite tunnelling amplitudes. We describe these by introducing Gaussian-distributed hopping integrals into the standard Bose–Hubbard model. This system has been recently shown to exhibit a quantum phase transition to a glassy state. The latter is characterized by a quenched disorder of boson wave-function phases. In this aspect, the system resembles quantum spin-glass systems that attracted much attention. By exploiting this analogy, we employ the well-established methodology originated by Sherrington and Kirkpatrick, which bases on the replica trick and the Trotter–Suzuki expansion. This treatment transforms the original quantum problem into an effective classical one with an additional time-like dimension. Here, we focus on autocorrelation functions of canonical variables of the effective system in the time-like domain. Deep in the disordered phase, we find a highly dynamical nature of correlations in agreement with the expected short memory of the system. This behaviour weakens while approaching and passing the phase boundary, where in the glassy phase asymptotically non-vanishing correlations are encountered. Thus, the state features infinite memory, which is consistent with the quenched nature of glassy disorder with random but frozen boson phases.

Erratum: Quantum simulation of long-range XY quantum spin glass with strong area-law violation using trapped ions [Phys. Rev. A 99 , 052342 (2019)

Ground states of local Hamiltonians are known to obey the entanglement entropy area law. While area law violation of a mild kind (logarithmic) is commonly encountered, strong area-law violation (more than logarithmic) is rare. In this paper, we study the long range quantum spin glass in one dimension whose couplings are disordered and fall off with distance as a power-law. We show that this system exhibits more than logarithmic area law violation in its ground state. Strikingly this feature is found to be true even in the short range regime in sharp contrast to the spinless long range disordered fermionic model. This necessitates the study of large systems for the quantum $XY$ spin glass model which is challenging since these numerical methods depend on the validity of the area law. This situation lends itself naturally for the exploration of a quantum simulation approach. We present a proof-of-principle implementation of this non-trivially interacting spin model using trapped ions and provide a detailed study of experimentally realistic parameters.

On the Question of Ergodicity in Quantum Spin Glass Phase and Its Role in Quantum Annealing

We first review, following our earlier studies, the critical behavior of the quantum Sherrington–Kirkpatrick (SK) model at finite as well as at zero temperatures. Through the analysis of the Binder cumulant we determined the entire phase diagram of the model and from the scaling analysis of the numerical data we obtained the correlation length exponent. For both the critical Binder cumulant and the correlation length exponent, we observed a crossover from classical- to quantum-fluctuation-dominated values at a finite temperature. We studied the behavior of the order parameter distribution of the model in the glass phase (at finite and zero temperatures). Along with a classical-fluctuation-dominated nonergodic region (where the replica symmetry is broken), we also found a quantum-fluctuation-dominated low-temperature ergodic region in the spin glass phase. In this quantum-fluctuation-dominated region, the order parameter distribution has a narrow peak around its most probable value, eventually becoming a delta function in the infinite-system-size limit (indicating replica symmetry restoration or ergodicity in the system). We also found that the annealing time (to reach a very low energy level of the classical SK model) becomes practically system-size-independent when the annealing paths pass through this ergodic region. In contrast, when such paths pass through the nonergodic region, the convergence time grows rapidly with the system size. We present a new study of the autocorrelation of the spins in both ergodic and nonergodic regions. We found a significant increase in the relaxation time (and also a change in the relaxation behavior) in the classical-fluctuation-dominated (nonergodic) region compared with that in the quantum-fluctuation-dominated (ergodic) region of the spin glass phase.

Quantum simulation of long-range XY quantum spin glass with strong area-law violation using trapped ions

Ground states of local Hamiltonians are known to obey the entanglement entropy area law. While area-law violation of a mild kind (logarithmic) is commonly encountered, strong area-law violation (more than logarithmic) is rare. In this paper, we study the long-range quantum spin glass in one dimension whose couplings are disordered and fall off with distance as a power law. We show that this system exhibits more than logarithmic area-law violation in its ground state. Strikingly this feature is found to be true even in the short-range regime, in sharp contrast to the spinless long-range disordered fermionic model. This necessitates the study of large systems for the quantum $XY$ spin glass model, which is challenging since these numerical methods depend on the validity of the area law. This situation lends itself naturally to the exploration of a quantum simulation approach. We present a proof-of-principle implementation of this nontrivially interacting spin model using trapped ions and provide a detailed study of experimentally realistic parameters.

Emergent and broken symmetries of atomic self-organization arising from Gouy phase shifts in multimode cavity QED

Optical cavities can induce photon-mediated interactions among intracavity-trapped atoms. Multimode cavities provide the ability to tune the form of these interactions, e.g., by inducing a nonlocal, sign-changing term to the interaction. By accounting for the Gouy phase shifts of the modes in a nearly degenerate, confocal, Fabry-Perot cavity, we provide a theoretical description of this interaction, along with additional experimental confirmation to complement that presented in the companion paper, Ref. [1]. Furthermore, we show that this interaction should be written in terms of a complex order parameter, allowing for a U(1)-symmetry to emerge. This symmetry corresponds to the phase of the atomic density wave arising from self-organization when the cavity is transversely pumped above a critical threshold power. We theoretically and experimentally show how this phase depends on the position of the Bose-Einstein condensate (BEC) within the cavity and discuss mechanisms that break the U(1)-symmetry and lock this phase. We then consider alternative Fabry-Perot multimode cavity geometries (i.e., beyond the confocal) and schemes with more than one pump laser and show that these provide additional capabilities for tuning the cavity-meditated interaction among atoms, including the ability to restore the U(1)-symmetry despite the presence of symmetry-breaking effects. These photon-mediated interactions may be exploited for realizing quantum liquid crystalline states and spin glasses using multimode optical cavities.

Glassy Phase of Optimal Quantum Control.

We study the problem of preparing a quantum many-body system from an initial to a target state by optimizing the fidelity over the family of bang-bang protocols. We present compelling numerical evidence for a universal spin-glasslike transition controlled by the protocol time duration. The glassy critical point is marked by a proliferation of protocols with close-to-optimal fidelity and with a true optimum that appears exponentially difficult to locate. Using a machine learning (ML) inspired framework based on the manifold learning algorithm t-distributed stochastic neighbor embedding, we are able to visualize the geometry of the high-dimensional control landscape in an effective low-dimensional representation. Across the transition, the control landscape features an exponential number of clusters separated by extensive barriers, which bears a strong resemblance with replica symmetry breaking in spin glasses and random satisfiability problems. We further show that the quantum control landscape maps onto a disorder-free classical Ising model with frustrated nonlocal, multibody interactions. Our work highlights an intricate but unexpected connection between optimal quantum control and spin glass physics, and shows how tools from ML can be used to visualize and understand glassy optimization landscapes.