Quantum critical dynamics in a 5,000-qubit programmable spin glass.

Experiments on disordered alloys1-3 suggest that spin glasses can be brought into low-energy states faster by annealing quantum fluctuations than by conventional thermal annealing. Owing to the importance of spin glasses as a paradigmatic computational testbed, reproducing this phenomenon in a programmable system has remained a central challenge in quantum optimization4-13. Here we achieve this goal by realizing quantum-critical spin-glass dynamics on thousands of qubits with a superconducting quantum annealer. We first demonstrate quantitative agreement between quantum annealing and time evolution of the Schrödinger equation in small spin glasses. We then measure dynamics in three-dimensional spin glasses on thousands of qubits, for which classical simulation of many-body quantum dynamics is intractable. We extract critical exponents that clearly distinguish quantum annealing from the slower stochastic dynamics of analogous Monte Carlo algorithms, providing both theoretical and experimental support for large-scale quantum simulation and a scaling advantage in energy optimization.

Observation of topological phenomena in a programmable lattice of 1,800 qubits.

The work of Berezinskii, Kosterlitz and Thouless in the 1970s1,2 revealed exotic phases of matter governed by the topological properties of low-dimensional materials such as thin films of superfluids and superconductors. A hallmark of this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom-typified by the classical XY model-owing to thermal fluctuations. In the two-dimensional Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently, a Kosterlitz-Thouless phase transition has been predicted in the quantum system-the two-dimensional transverse-field Ising model-by theory and simulation3-5. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting niobium flux qubits whose pairwise couplings are arranged in a fully frustrated square-octagonal lattice. Essential to the critical behaviour, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We describe and use a simple approach to statistical estimation with an annealing-based quantum processor that performs Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials.

Kagome qubit ice.

Topological phases of spin liquids with constrained disorder can host a kinetics of fractionalized excitations. However, spin-liquid phases with distinct kinetic regimes have proven difficult to observe experimentally. Here we present a realization of kagome spin ice in the superconducting qubits of a quantum annealer, and use it to demonstrate a field-induced kinetic crossover between spin-liquid phases. Employing fine control over local magnetic fields, we show evidence of both the Ice-I phase and an unconventional field-induced Ice-II phase. In the latter, a charge-ordered yet spin-disordered topological phase, the kinetics proceeds via pair creation and annihilation of strongly correlated, charge conserving, fractionalized excitations. As these kinetic regimes have resisted characterization in other artificial spin ice realizations, our results demonstrate the utility of quantum-driven kinetics in advancing the study of topological phases of spin liquids.

Scaling advantage over path-integral Monte Carlo in quantum simulation of geometrically frustrated magnets.

The promise of quantum computing lies in harnessing programmable quantum devices for practical applications such as efficient simulation of quantum materials and condensed matter systems. One important task is the simulation of geometrically frustrated magnets in which topological phenomena can emerge from competition between quantum and thermal fluctuations. Here we report on experimental observations of equilibration in such simulations, measured on up to 1440 qubits with microsecond resolution. By initializing the system in a state with topological obstruction, we observe quantum annealing (QA) equilibration timescales in excess of one microsecond. Measurements indicate a dynamical advantage in the quantum simulation compared with spatially local update dynamics of path-integral Monte Carlo (PIMC). The advantage increases with both system size and inverse temperature, exceeding a million-fold speedup over an efficient CPU implementation. PIMC is a leading classical method for such simulations, and a scaling advantage of this type was recently shown to be impossible in certain restricted settings. This is therefore an important piece of experimental evidence that PIMC does not simulate QA dynamics even for sign-problem-free Hamiltonians, and that near-term quantum devices can be used to accelerate computational tasks of practical relevance.

Computational hardness of spin-glass problems with tile-planted solutions.

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 multidimensional tuning parameter space. Our results show that the problems have a wide range of tunable hardness. Moreover, we observe multiple transitions in the hardness phase space, which we further corroborate using simulated annealing and simulated quantum annealing. By investigating thermodynamic properties of these planted systems, we demonstrate that the harder samples undergo magnetic ordering transitions which are also ultimately responsible for the observed hardness transitions on changing the sample composition.

Wishart planted ensemble: A tunably rugged pairwise Ising model with a first-order phase transition.

We propose the Wishart planted ensemble, a class of zero-field Ising models with tunable algorithmic hardness and specifiable (or planted) ground state. The problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a sign-inverted variant of the Hopfield model. The Hamiltonian contains only 2-spin interactions, with the coupler matrix following a type of Wishart distribution. The class exhibits a classical first-order phase transition in temperature. For some parameter settings the model has a locally stable paramagnetic state, a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape. We analytically probe the ensemble thermodynamic properties by deriving the Thouless-Anderson-Palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis; extensive Monte Carlo simulations confirm our predictions of the first-order transition temperature. The class exhibits a wide variation in algorithmic hardness as a generation parameter is varied, with a pronounced easy-hard-easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes. By deriving the ensemble-averaged energy distribution and taking into account finite-precision representation, we propose an analytical expression for the location of the hardness peak and show that at fixed precision, the number of constraints in the integer program must increase with system size to yield truly hard problems. The Wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms.

Computing with noise: phase transitions in boolean formulas.

Computing circuits composed of noisy logical gates and their ability to represent arbitrary boolean functions with a given level of error are investigated within a statistical mechanics setting. Existing bounds on their performance are straightforwardly retrieved, generalized, and identified as the corresponding typical-case phase transitions. Results on error rates, function depth, and sensitivity, and their dependence on the gate-type and noise model used are also obtained.

Phase diagram of the 1-in-3 satisfiability problem.

We study typical case properties of the 1-in-3 satisfiability problem, the Boolean satisfaction problem, where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 satisfiability and exact 3-cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region and develop a one-step replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.

Mean-field method with correlations determined by linear response.

We introduce a mean-field approximation based on the reconciliation of maximum entropy and linear response for correlations in the cluster variation method. Within a general formalism that includes previous mean-field methods, we derive formulas improving on, e.g., the Bethe approximation and the Sessak-Monasson result at high temperature. Applying the method to direct and inverse Ising problems, we find improvements over standard implementations.

Noisy random Boolean formulae: a statistical physics perspective.

Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field.

Equilibrium properties of disordered spin models with two-scale interactions.

Methods for understanding classical disordered spin systems with interactions conforming to some idealized graphical structure are well developed. The equilibrium properties of the Sherrington-Kirkpatrick model, which has a densely connected structure, have become well understood. Many features generalize to sparse Erdös-Rényi graph structures above the percolation threshold and to Bethe lattices when appropriate boundary conditions apply. In this paper, we consider spin states subject to a combination of sparse strong interactions with weak dense interactions, which we term a composite model. The equilibrium properties are examined through the replica method, with exact analysis of the high-temperature paramagnetic, spin-glass, and ferromagnetic phases by perturbative schemes. We present results of replica symmetric variational approximations, where perturbative approaches fail at lower temperature. Results demonstrate re-entrant behaviors from spin glass to ferromagnetic phases as temperature is lowered, including transitions from replica symmetry broken to replica symmetric phases. The nature of high-temperature transitions is found to be sensitive to the connectivity profile in the sparse subgraph, with regular connectivity a discontinuous transition from the paramagnetic to ferromagnetic phases is apparent.