Sampling diverse near-optimal solutions via algorithmic quantum annealing.

Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open problems is the lack of ergodicity, or mode collapse, for typical stochastic solvers based on Monte Carlo techniques leading to poor generalization or lack of robustness to uncertainties. Currently, there is no universal metric to quantify such performance deficiencies across various solvers. Here, we introduce a new diversity measure for quantifying the number of independent approximate solutions for NP-hard optimization problems. Among others, it allows benchmarking solver performance by a required time-to-diversity (TTD), a generalization of often used time-to-solution (TTS). We illustrate this metric by comparing the sampling power of various quantum annealing strategies. In particular, we show that the inhomogeneous quantum annealing schedules can redistribute and suppress the emergence of topological defects by controlling space-time separated critical fronts, leading to an advantage over standard quantum annealing schedules with respect to both TTS and TTD for finding rare solutions. Using path-integral Monte Carlo simulations for up to 1600 qubits, we demonstrate that nonequilibrium driving of quantum fluctuations, guided by efficient approximate tensor network contractions, can significantly reduce the fraction of hard instances for random frustrated 2D spin glasses with local fields. Specifically, we observe that by creating a class of algorithmic quantum phase transitions, the diversity of solutions can be enhanced by up to 40% with the fraction of hard-to-sample instances reducing by more than 25%.

Quantum phase transition dynamics in the two-dimensional transverse-field Ising model.

The quantum Kibble-Zurek mechanism (QKZM) predicts universal dynamical behavior near the quantum phase transitions (QPTs). It is now well understood for the one-dimensional quantum matter. Higher-dimensional systems, however, remain a challenge, complicated by the fundamentally different character of the associated QPTs and their underlying conformal field theories. In this work, we take the first steps toward theoretical exploration of the QKZM in two dimensions for interacting quantum matter. We study the dynamical crossing of the QPT in the paradigmatic Ising model by a joint effort of modern state-of-the-art numerical methods, including artificial neural networks and tensor networks. As a central result, we quantify universal QKZM behavior close to the QPT. We also note that, upon traversing further into the ferromagnetic regime, deviations from the QKZM prediction appear. We explain the observed behavior by proposing an extended QKZM taking into account spectral information as well as phase ordering. Our work provides a testing platform for higher-dimensional quantum simulators.

Coherent Many-Body Oscillations Induced by a Superposition of Broken Symmetry States in the Wake of a Quantum Phase Transition.

It is now widely accepted that quenches through the critical region of quantum phase transitions result in post-transition states populated with topological defects-analogs of the classical topological defects. However, consequences of the very nonclassical fact that the state after a quench is a superposition of distinct, broken-symmetry vacua with different numbers and locations of defects have remained largely unexplored. We identify coherent quantum oscillations induced by such superpositions in observables complementary to the one involved in symmetry breaking. These oscillations satisfy Kibble-Zurek dynamical scaling laws with the quench rate, with an instantaneous oscillation frequency set primarily by the gap of the system. In addition to the obvious fundamental significance of a superposition of different broken symmetry states, quantum coherent oscillations can be used to verify unitarity and test for imperfections of the experimental implementations of quantum simulators.

Performance of reservoir discretizations in quantum transport simulations.

Quantum transport simulations often use explicit, yet finite, electronic reservoirs. These should converge to the correct continuum limit, albeit with a trade-off between discretization and computational cost. Here, we study this interplay for extended reservoir simulations, where relaxation maintains a bias or temperature drop across the system. Our analysis begins in the non-interacting limit, where we parameterize different discretizations to compare them on an even footing. For many-body systems, we develop a method to estimate the relaxation that best approximates the continuum by controlling virtual transitions in Kramers turnover for the current. While some discretizations are more efficient for calculating currents, there is little benefit with regard to the overall state of the system. Any gains become marginal for many-body, tensor network simulations, where the relative performance of discretizations varies when sweeping other numerical controls. These results indicate that typical reservoir discretizations have little impact on numerical costs for certain computational tools. The choice of a relaxation parameter is nonetheless crucial, and the method we develop provides a reliable estimate of the optimal relaxation for finite reservoirs.

Approximate optimization, sampling, and spin-glass droplet discovery with tensor networks.

We devise a deterministic algorithm to efficiently sample high-quality solutions of certain spin-glass systems that encode hard optimization problems. We employ tensor networks to represent the Gibbs distribution of all possible configurations. Using approximate tensor-network contractions, we are able to efficiently map the low-energy spectrum of some quasi-two-dimensional Hamiltonians. We exploit the local nature of the problems to compute spin-glass droplets geometries, which provides a new form of compression of the low-energy spectrum. It naturally extends to sampling, which otherwise, for exact contraction, is #P-complete. In particular, for one of the hardest known problem-classes devised on chimera graphs known as deceptive cluster loops and for up to 2048 spins, we find on the order of 10^{10} degenerate ground states in a single run of our algorithm, computing better solutions than have been reported on some hard instances. Our gradient-free approach could provide new insight into the structure of disordered spin-glass complexes, with ramifications both for machine learning and noisy intermediate-scale quantum devices.

Open System Tensor Networks and Kramers' Crossover for Quantum Transport.

Tensor networks are a powerful tool for many-body ground states with limited entanglement. These methods can nonetheless fail for certain time-dependent processes-such as quantum transport or quenches-where entanglement growth is linear in time. Matrix-product-state decompositions of the resulting out-of-equilibrium states require a bond dimension that grows exponentially, imposing a hard limit on simulation timescales. However, in the case of transport, if the reservoir modes of a closed system are arranged according to their scattering structure, the entanglement growth can be made logarithmic. Here, we apply this ansatz to open systems via extended reservoirs that have explicit relaxation. This enables transport calculations that can access steady states, time dynamics and noise, and periodic driving (e.g., Floquet states). We demonstrate the approach by calculating the transport characteristics of an open, interacting system. These results open a path to scalable and numerically systematic many-body transport calculations with tensor networks.

Breaking the Entanglement Barrier: Tensor Network Simulation of Quantum Transport.

The recognition that large classes of quantum many-body systems have limited entanglement in the ground and low-lying excited states led to dramatic advances in their numerical simulation via so-called tensor networks. However, global dynamics elevates many particles into excited states, and can lead to macroscopic entanglement and the failure of tensor networks. Here, we show that for quantum transport-one of the most important cases of this failure-the fundamental issue is the canonical basis in which the scenario is cast: When particles flow through an interface, they scatter, generating a "bit" of entanglement between spatial regions with each event. The frequency basis naturally captures that-in the long-time limit and in the absence of inelastic scattering-particles tend to flow from a state with one frequency to a state of identical frequency. Recognizing this natural structure yields a striking-potentially exponential in some cases-increase in simulation efficiency, greatly extending the attainable spatial and time scales, and broadening the scope of tensor network simulation to hitherto inaccessible classes of nonequilibrium many-body problems.

Single-Chain Magnet Based on Cobalt(II) Thiocyanate as XXZ Spin Chain.

Invited for the cover of this issue is the group of Michal Rams at Jagiellonian University (Kraków, Poland) and colleagues at Christian-Albrechts-Universität zu Kiel, Friedrich-Schiller-Universität Jena, and Helmholtz-Zentrum Berlin. The image represents a 1D coordination polymer with Co(II) spins that are flipped by photons during an EPR experiment. Read the full text of the article at 10.1002/chem.201903924.

Single-Chain Magnet Based on Cobalt(II) Thiocyanate as XXZ Spin Chain.

The cobalt(II) in [Co(NCS)2 (4-methoxypyridine)2 ]n are linked by pairs of thiocyanate anions into linear chains. In contrast to a previous structure determination, two crystallographically independent cobalt(II) centers have been found to be present. In the antiferromagnetic state, below the critical temperature (Tc =3.94 K) and critical field (Hc =290 Oe), slow relaxations of the ferromagnetic chains are observed. They originate mainly from defects in the magnetic structure, which has been elucidated by micromagnetic Monte Carlo simulations and ac measurements using pristine and defect samples. The energy barriers of the relaxations are Δτ1 =44.9(5) K and Δτ2 =26.0(7) K for long and short spin chains, respectively. The spin excitation energy, measured by using frequency-domain EPR spectroscopy, is 19.1 cm-1 and shifts 0.1 cm-1 due to the magnetic ordering. Ab initio calculations revealed easy-axis anisotropy for both CoII centers, and also an exchange anisotropy Jxx /Jzz of 0.21. The XXZ anisotropic Heisenberg model (solved by using the density renormalization matrix group technique) was used to reconcile the specific heat, susceptibility, and EPR data.

Symmetry Breaking Bias and the Dynamics of a Quantum Phase Transition.

The Kibble-Zurek mechanism predicts the formation of topological defects and other excitations that quantify how much a quantum system driven across a quantum critical point fails to be adiabatic. We point out that, thanks to the divergent linear susceptibility at the critical point, even a tiny symmetry breaking bias can restore the adiabaticity. The minimal required bias scales like τ_{Q}^{-ßδ/(1+zν)}, where ß, δ, z, ν are the critical exponents and τ_{Q} is a quench time. We test this prediction by DMRG simulations of the quantum Ising chain. It is directly applicable to the recent emulation of quantum phase transition dynamics in the Ising chain with ultracold Rydberg atoms.

Reversible and Topotactic Solvent Removal in a Magnetic Ni(NCS)2 Coordination Polymer.

Reaction of Ni(NCS)2 with 4-(Boc-amino)pyridine in acetonitrile leads to the formation of a new coordination polymer with the composition Ni(NCS)2(4-(Boc-amino)pyridine)2·MeCN (1-MeCN). In the crystal structure the Ni(II) cations are linked by the anionic ligands into chains that are further connected into layers by intermolecular N-H···O hydrogen bonding. These layers are stacked and channels are formed, in which acetonitrile molecules are located. Solvent removal leads to the ansolvate 1, which shows microporosity as proven by sorption measurements. Single crystal X-ray investigations reveal that the solvent removal leads to a change in symmetry from primitive to C-centered, which is reversible and which proceeds via a topotactic reaction leaving the network intact. The magnetic properties of 1-MeCN and 1 are governed by the ferromagnetic exchange between spins of Ni(II) forming chains. The susceptibility and specific heat for such a quantum Heisenberg chain of S = 1 spins with zero-field splitting are calculated using the DMRG method and compared with the experimental results.

Anomalous behavior of the energy gap in the one-dimensional quantum XY model.

We reexamine the well-studied one-dimensional spin-1/2 XY model to reveal its nontrivial energy spectrum, in particular the energy gap between the ground state and the first excited state. In the case of the isotropic XY model, the XX model, the gap behaves very irregularly as a function of the system size at a second order transition point. This is in stark contrast to the usual power-law decay of the gap and is reminiscent of the similar behavior at the first order phase transition in the infinite-range quantum XY model. The gap also shows nontrivial oscillatory behavior for the phase transitions in the anisotropic model in the incommensurate phase. We observe a close relation between this anomalous behavior of the gap and the correlation functions. These results, those for the isotropic case in particular, are important from the viewpoint of quantum annealing where the efficiency of computation is strongly affected by the size dependence of the energy gap.

Assisted finite-rate adiabatic passage across a quantum critical point: exact solution for the quantum Ising model.

The dynamics of a quantum phase transition is inextricably woven with the formation of excitations, as a result of critical slowing down in the neighborhood of the critical point. We design a transitionless quantum driving through a quantum critical point, allowing one to access the ground state of the broken-symmetry phase by a finite-rate quench of the control parameter. The method is illustrated in the one-dimensional quantum Ising model in a transverse field. Driving through the critical point is assisted by an auxiliary Hamiltonian, for which the interplay between the range of the interaction and the modes where excitations are suppressed is elucidated.

A quantum phase transition in a quantum external field: superposing two magnetic phases.

We study an Ising chain undergoing a quantum phase transition in a quantum magnetic field. Such a field can be emulated by coupling the chain to a central spin initially in a superposition state. We show that--by adiabatically driving such a system--one can prepare a quantum superposition of any two ground states of the Ising chain. In particular, one can end up with the Ising chain in a superposition of ferromagnetic and paramagnetic phases--a scenario with no analogue in prior studies of quantum phase transitions. Remarkably, the resulting magnetization of the chain encodes the position of the critical point and universal critical exponents, as well as the ground state fidelity.

Quantum fidelity in the thermodynamic limit.

We study quantum fidelity, the overlap between two ground states of a many-body system, focusing on the thermodynamic regime. We show how a drop in fidelity near a critical point encodes universal information about a quantum phase transition. Our general scaling results are illustrated in the quantum Ising chain for which a remarkably simple expression for fidelity is found.

Multiscale entanglement renormalization ansatz in two dimensions: quantum Ising model.

We propose a symmetric version of the multiscale entanglement renormalization ansatz in two spatial dimensions (2D) and use this ansatz to find an unknown ground state of a 2D quantum system. Results in the simple 2D quantum Ising model on the 8x8 square lattice are found to be very accurate even with the smallest nontrivial truncation parameter.