We study first order quantum phase transitions in mean-field spin glasses. We solve the quantum Random Energy Model using elementary methods and show that at the transition the eigenstate suddenly projects onto the unperturbed ground state and that the gap between the lowest states is exponentially small in the system size. We argue that this is a generic feature of all `Random First Order' models, which includes benchmarks such as random satisfiability. We introduce a two-time instanton to calculate this gap in general, and discuss the consequences for quantum annealing.
Probing the lowest energy configuration of a complex system by quantum annealing was recently found to be more effective than its classical, thermal counterpart. Comparing classical and quantum Monte Carlo annealing protocols on the random two-dimensional Ising model we confirm the superiority of quantum annealing relative to classical annealing. We also propose a theory of quantum annealing, based on a cascade of Landau-Zener tunneling events. For both classical and quantum annealing, the residual energy after annealing is inversely proportional to a power of the logarithm of the annealing ti…
The strongest evidence for superiority of quantum annealing on spin glass problems has come from comparing simulated quantum annealing using quantum Monte Carlo (QMC) methods to simulated classical annealing [G. Santoro et al., Science 295, 2427(2002)]. Motivated by experiments on programmable quantum annealing devices we revisit the question of when quantum speedup may be expected for Ising spin glass problems. We find that even though a better scaling compared to simulated classical annealing can be achieved for QMC simulations, this advantage is due to time discretization and measurements w…
A technique inspired on quantum annealing is proposed in order to obtain the classical ground state of a spin-glass by tracking the full wavefunction of a given system within the subspace of matrix product states (MPS), using the density matrix renormalization group (DMRG). The technique is exemplified within the problem of obtention of the classical ground state of an Ising spin glass on ladder geometries. Its performance is evaluated and related to the entanglement entropy.
We discuss an Ising spin glass where each $S=1/2$ spin is coupled antiferromagnetically to three other spins (3-regular graphs). Inducing quantum fluctuations by a time-dependent transverse field, we use out-of-equilibrium quantum Monte Carlo simulations to study dynamic scaling at the quantum glass transition. Comparing the dynamic exponent and other critical exponents with those of the classical (temperature-driven) transition, we conclude that quantum annealing is less efficient than classical simulated annealing in bringing the system into the glass phase. Quantum computing based on the qu…
Quantum annealing promises to solve complex combinatorial optimization problems faster than current transistor-based computer technologies. Although to date only one commercially-available quantum annealer is procurable, one can already start to map out the application scope of these novel optimization machines. These mid-scale programmable analog special-purpose devices could, potentially, revolutionize optimization. However, their disruptive application domain remains to be found. While the commercial analog quantum optimization machine by D-Wave Systems Inc. already exceeds 1000 qubits, her…
We derive a number of exact relations between equilibrium and nonequilibrium quantities for spin glasses in external fields using the Jarzynski equality and gauge symmetry. For randomly-distributed longitudinal fields, a lower bound is established for the work done on the system in nonequilibrium processes, and identities are proven to relate equilibrium and nonequilibrium quantities. In the case of uniform transverse fields, identities are proven between physical quantities and exponentiated work done to the system at different parts of the phase diagram with the context of quantum annealing …
We present an implementation of Quantum Annealing (QA) via lattice Green's function Monte Carlo (GFMC), focusing on its application to the Ising spin-glass in transverse field. In particular, we study whether or not such method is more effective than the Path-Integral Monte Carlo (PIMC) based QA, as well as classical simulated annealing (CA), previously tested on the same optimization problem. We identify the issue of importance sampling, i.e., the necessity of possessing reasonably good (variational) trial wavefunctions, as the key point of the algorithm. We have considered two possible class…
We apply a generalized Kibble-Zurek out-of-equilibrium scaling ansatz to simulated annealing when approaching the spin-glass transition at temperature $T=0$ of the two-dimensional Ising model with random $J= \pm 1$ couplings. Analyzing the spin-glass order parameter and the excess energy as functions of the system size and the annealing velocity in Monte Carlo simulations with Metropolis dynamics, we find scaling where the energy relaxes slower than the spin-glass order parameter, i.e., there are two different dynamic exponents. The values of the exponents relating the relaxation time scales t…
We study the problem to infer the ground state of a spin-glass Hamiltonian using data from another Hamiltonian with interactions disturbed by noise from the original Hamiltonian, motivated by the ground-state inference in quantum annealing on a noisy device. It is shown that the average Hamming distance between the inferred spin configuration and the true ground state is minimized when the temperature of the noisy system is kept at a finite value, and not at zero temperature. We present a spin-glass generalization of a well-established result that the ground state of a purely ferromagnetic Ham…
Recently, a programmable quantum annealing machine has been built that minimizes the cost function of hard optimization problems by adiabatically quenching quantum fluctuations. Tests performed by different research teams have shown that, indeed, the machine seems to exploit quantum effects. However experiments on a class of random-bond instances have not yet demonstrated an advantage over classical optimization algorithms on traditional computer hardware. Here we present evidence as to why this might be the case. These engineered quantum annealing machines effectively operate coupled to a dec…
Exploiting quantum properties to outperform classical ways of information-processing is an outstanding goal of modern physics. A promising route is quantum simulation, which aims at implementing relevant and computationally hard problems in controllable quantum systems. Here we demonstrate that in a trapped ion setup, with present day technology, it is possible to realize a spin model of the Mattis type that exhibits spin glass phases. Remarkably, our method produces the glassy behavior without the need for any disorder potential, just by controlling the detuning of the spin-phonon coupling. A…
There has been considerable progress in the design and construction of quantum annealing devices. However, a conclusive detection of quantum speedup over traditional silicon-based machines remains elusive, despite multiple careful studies. In this work we outline strategies to design hard tunable benchmark instances based on insights from the study of spin glasses - the archetypal random benchmark problem for novel algorithms and optimization devices. We propose to complement head-to-head scaling studies that compare quantum annealing machines to state-of-the-art classical codes with an approa…
We study the role of Hamiltonian complexity in the performance of quantum annealers. We consider two general classes of annealing Hamiltonians: stoquastic ones, which can be simulated efficiently using the quantum Monte Carlo algorithm, and nonstoquastic ones, which cannot be treated efficiently. We implement the latter by adding antiferromagnetically coupled two-spin driver terms to the traditionally studied transverse-field Ising model, and compare their performance to that of similar stoquastic Hamiltonians with ferromagnetically coupled additional terms. We focus on a model of long-range I…
An important challenge in superconducting quantum computing is the need to physically couple many devices using quasi-two-dimensional fabrication processes. Recent advances in the design and fabrication of quantum annealing processors have enabled an increase in pairwise connectivity among thousands of qubits. One benefit of this is the ability to minor-embed optimization problems using fewer physical qubits for each logical spin. Here we demonstrate the benefit of this progress in the problem of minimizing the energy of three-dimensional spin glasses. Comparing the previous generation D-Wave …
Recently, Heim, Ronnow, Isakov and Troyer [Science 348 (2015) 215] have reported that Monte Carlo simulations for the Ising spin glass model on the square lattice in the physically relevant continuous-imaginary-time limit do not show superiority of quantum annealing (QA) using transverse field against classical annealing (CA). Although the QA schedule that they had used has been using conventionally, however the QA schedule mathematically has no guarantee that the used schedule is the best QA schedule for performance of optimization. We propose a new QA schedule for studying transverse-field-b…
To gain better insight into the complexity theory of quantum annealing, we propose and solve a class of spin systems which contain bottlenecks of the kind expected to dominate the runtime of quantum annealing as it tries to solve difficult optimization problems. We uncover a noise amplification effect at these bottlenecks, whereby tunneling rates caused by flux-qubit noise scale in proportion to the number of qubits $N$ in the limit that $N\to \infty$. By solving the incoherent annealing dynamics exactly, we find a wide range of regimes where the probability that a quantum annealer remains in …
Here we discuss the annealing behavior of an infinite-range $\pm J$ Ising spin glass in presence of a transverse field using a zero-temperature quantum Monte Carlo. Within the simulation scheme, we demonstrate that quantum annealing not only helps finding the ground state of a classical spin glass, but can also help simulating the ground state of a quantum spin glass, in particularly, when the transverse field is low, much more efficiently.
We review here the recent success in quantum annealing, i.e., optimization of the cost or energy functions of complex systems utilizing quantum fluctuations. The concept is introduced in successive steps through the studies of mapping of such computationally hard problems to the classical spin glass problems. The quantum spin glass problems arise with the introduction of quantum fluctuations, and the annealing behavior of the systems as these fluctuations are reduced slowly to zero. This provides a general framework for realizing analog quantum computation.
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, $H_I$. Here, we point to the existence of an analytical solution for such a problem for an arbitrary $H_I$ 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 monotoni…
We investigate the computational hardness of spin-glass instances on a square lattice, generated via a recently introduced tunable and scalable approach for planting solutions. The method relies on partitioning the problem graph into edge-disjoint subgraphs, and planting frustrated, elementary subproblems that share a common local ground state, which guarantees that the ground state of the entire problem is known a priori. Using population annealing Monte Carlo, we compare the typical hardness of problem classes over a large region of the multi-dimensional tuning parameter space. Our results s…
We analyze the performance of quantum annealing as formulated by Lechner, Hauke, and Zoller (LHZ), by which a Hamiltonian with all-to-all two-body interactions is reduced to a corresponding Hamiltonian with local many-body interactions. Mean-field analyses show that problematic first-order quantum phase transitions that exist in the original LHZ formulation can be avoided, and thus an exponential speedup is achieved, if we drive the coefficient of the many-body term, which represents the constraint, non-linearly as a function of time. This result applies not only to a simple ferromagnetic mode…
Recent developments in quantum annealing techniques have been indicating potential advantage of quantum annealing for solving NP-hard optimization problems. In this article we briefly indicate and discuss the beneficial features of quantum annealing techniques and compare them with those of simulated annealing techniques. We then briefly discuss the quantum annealing studies of some model spin glass and kinetically constrained systems.
Quantum annealing is a generic solver of classical optimization problems that makes full use of quantum fluctuations. We consider work statistics given by a repetition of quantum annealing processes by employing the Jarzynski equality proposed in nonequilibrium statistical physics. In particular, we analyze the distribution of the work performed by a transverse field. A special symmetry, gauge symmetry, leads to a non-trivial relationship between quantum annealing toward different targets in the theory of spin glasses. We believe that our results will be a step toward an alternative realizatio…
Quantum annealing is analogous to simulated annealing with a tunneling mechanism substituting for thermal activation. Its performance has been tested in numerical simulation with mixed conclusions. There is a class of optimization problems for which the efficiency can be studied analytically using techniques based on the statistical mechanics of spin glasses.
Recently, Lechner, Hauke and Zoller [Science Advances, 1(9)e1500838, (2015)] have proposed a quantum annealing architecture, in which a classical spin glass with all-to-all connectivity is simulated by a spin glass with geometrically local interactions. We interpret this architecture as a classical error-correcting code, which is highly robust against weakly correlated bit-flip noise, and we analyze the code's performance using a belief-propagation decoding algorithm. Our observations may also apply to more general encoding schemes and noise models.
There are many factors that influence the design of quantum annealing processing units. Here we address the issue of improving quantum annealing processing unit designs from the point of view of the critical behavior of spin glasses. It has been argued [Phys. Rev. X 4, 021008 (2014)] that among the most difficult Ising spin-glass ground-state problems are those related to lattices which exhibit a finite-temperature spin-glass transition. Here, we show that adding small-world couplers between qubits (spins) to the native quasi-planar quantum processing unit graph results in a topology where a d…
We introduce a novel Simulated Quantum Annealing (SQA) algorithm which employs a multispin quantum fluctuation operator. At variance with the usual transverse field, short-range two-spin flip interactions are included in the driver Hamiltonian. A Quantum Monte Carlo algorithm, capable of efficiently simulating large disordered systems, is described and tested. A first application to SQA, on a random square lattice Ising spin glass reveals that the multi-spin driver Hamiltonian improves upon the usual transverse field. This work paves the way for more systematic investigations using multi-spin …
The annealing schedule is optimized for a parameter in the Lechner-Hauke-Zoller (LHZ) scheme for quantum annealing designed for the all-to-all-interacting Ising model representing generic combinatorial optimization problems. We adapt the variational approach proposed by Matsuura et al. (arXiv:2003.09913) to the annealing schedule of a term representing a constraint for variables intrinsic to the LHZ scheme with the annealing schedule of other terms kept intact. Numerical results for a simple ferromagnetic model and the spin-glass problem show that nonmonotonic annealing schedules optimize the …
Classical and quantum annealing are two heuristic optimization methods that search for an optimal solution by slowly decreasing thermal or quantum fluctuations. Optimizing annealing schedules is important both for performance and fair comparisons between classical annealing, quantum annealing, and other algorithms. Here we present a heuristic approach for the optimization of annealing schedules for quantum annealing and apply it to 3D Ising spin glass problems. We find that if both classical and quantum annealing schedules are similarly optimized, classical annealing outperforms quantum anneal…
We propose a modified quantum annealing protocol, i. e., pulsed quantum annealing} (PQA), in order to increase the success probability by a pulse application during the quantum annealing process. It is well known that the success probability of the conventional quantum annealing is reduced due to the Landau-Zener transitions. By applying a pulse to the system, we modulate the success probability and increase it, compared to the conventional quantum annealing, by optimizing the pulse parameters. We demonstrate our findings for a single qubit both numerically and analytically. The analytical mod…
We show two fascinating topics lying between quantum information processing and statistical mechanics. First, we introduce an elaborated technique, the surface code, to prepare the particular quantum state with robustness against decoherence. Second, we show another interesting technique to employ quantum nature, quantum annealing. Through both of the topics, we would shed light on the birth of the interdisciplinary field between quantum mechanics and statistical mechanics.
We study various annealing dynamics, both classical and quantum, for simple mean-field models and explain how to describe their behavior in the thermodynamic limit in terms of differential equations. In particular we emphasize the differences between quantum annealing (i.e. evolution with Schrödinger equation) and simulated quantum annealing (i.e. annealing of a Quantum Monte Carlo simulation).
For many optimization algorithms the time-to-solution depends not only on the problem size but also on the specific problem instance and may vary by many orders of magnitude. It is then necessary to investigate the full distribution and especially its tail. Here we analyze the distributions of annealing times for simulated annealing and simulated quantum annealing (by path integral quantum Monte Carlo) for random Ising spin glass instances. We find power-law distributions with very heavy tails, corresponding to extremely hard instances, but far broader distributions - and thus worse performanc…
We investigate quantum annealing with antiferromagnetic transverse interactions for the generalized Hopfield model with $k$-body interactions. The goal is to study the effectiveness of antiferromagnetic interactions, which were shown to help us avoid problematic first-order quantum phase transitions in pure ferromagnetic systems, in random systems. We estimate the efficiency of quantum annealing by analyzing phase diagrams for two cases where the number of embedded patterns is finite or extensively large. The phase diagrams of the model with finite patterns show that there exist annealing path…
The Sherrington-Kirkpatrick model with random $\pm1$ couplings is programmed on the D-Wave Two annealer featuring 509 qubits interacting on a Chimera-type graph. The performance of the optimizer compares and correlates to simulated annealing. When considering the effect of the static noise, which degrades the performance of the annealer, one can estimate an improvement on the comparative scaling of the two methods in favor of the D-Wave machine. The optimal choice of parameters of the embedding on the Chimera graph is shown to be associated to the emergence of the spin-glass critical temperatu…
Recently, it was demonstrated both theoretically and experimentally on the D-Wave quantum annealer that transverse-field quantum annealing does not find all ground states with equal probability. In particular, it was proposed that more complex driver Hamiltonians beyond transverse fields might mitigate this shortcoming. Here, we investigate the mechanisms of (un)fair sampling in quantum annealing. While higher-order terms can improve the sampling for selected small problems, we present multiple counterexamples where driver Hamiltonians that go beyond transverse fields do not remove the samplin…
Quantum annealing has the potential to provide a speedup over classical algorithms in solving optimization problems. Just as for any other quantum device, suppressing Hamiltonian control errors will be necessary before quantum annealers can achieve speedups. Such analog control errors are known to lead to $J$-chaos, wherein the probability of obtaining the optimal solution, encoded as the ground state of the intended Hamiltonian, varies widely depending on the control error. Here, we show that $J$-chaos causes a catastrophic failure of quantum annealing, in that the scaling of the time-to-solu…
We propose a method to interpolate dynamics of von Neumann and classical master equations with an arbitrary mixing parameter to investigate the thermal effects in quantum dynamics. The two dynamics are mixed by intervening to continuously modify their solutions, thus coupling them indirectly instead of directly introducing a coupling term. This maintains the quantum system in a pure state even after the introduction of thermal effects and obtains not only a density matrix but also a state vector representation. Further, we demonstrate that the dynamics of a two-level system can be rewritten as…
A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of $N$ qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by $O(\log N)$ bottlenecks inside the spin glass phase, …
We study the performance of the D-Wave 2X quantum annealing machine on systems with well-controlled ground-state degeneracy. While obtaining the ground state of a spin-glass benchmark instance represents a difficult task, the gold standard for any optimization algorithm or machine is to sample all solutions that minimize the Hamiltonian with more or less equal probability. Our results show that while naive transverse-field quantum annealing on the D-Wave 2X device can find the ground-state energy of the problems, it is not well suited in identifying all degenerate ground-state configurations a…
We study the problem to infer the original ground state of a spin-glass Hamiltonian out of the information from the Hamiltonian with interactions deviated from the original ones. Our motivation comes from quantum annealing on a real device in which the values of interactions are degraded by noise. We show numerically for quasi-one-dimensional systems that the Hamming distance between the original ground state and the inferred spin state is minimized when we stop the process of quantum annealing before the amplitude of the transverse field reaches zero in contrast to the conventional prescripti…
A novel quantum-classical hybrid scheme is proposed to efficiently solve large-scale combinatorial optimization problems. The key concept is to introduce a Hamiltonian dynamics of the classical flux variables associated with the quantum spins of the transverse-field Ising model. Molecular dynamics of the classical fluxes can be used as a powerful preconditioner to sort out the frozen and ambivalent spins for quantum annealers. The performance and accuracy of our smooth hybridization in comparison to the standard classical algorithms (the tabu search and the simulated annealing) are demonstrate…
A shorter processing time is desirable for quantum computation to minimize the effects of noise. We propose a simple procedure to variationally determine a set of parameters in the transverse-field Ising model for quantum annealing appended with a field along the $y$ axis. The method consists of greedy optimization of the signs of coefficients of the $y$-field term based on the outputs of short annealing processes. We test the idea in the ferromagnetic system with all-to-all couplings and spin-glass problems, and find that the method outperforms the traditional form of quantum annealing and si…
The debate around the potential superiority of quantum annealers over their classical counterparts has been ongoing since the inception of the field by Kadowaki and Nishimori close to two decades ago. Recent technological breakthroughs in the field, which have led to the manufacture of experimental prototypes of quantum annealing optimizers with sizes approaching the practical regime, have reignited this discussion. However, the demonstration of quantum annealing speedups remains to this day an elusive albeit coveted goal. Here, we examine the power of quantum annealers to provide a different …
