Adi Makmal

Quantum Speedup for Active Learning Agents

Can quantum mechanics help us build intelligent learning agents? A defining signature of intelligent behavior is the capacity to learn from experience. However, a major bottleneck for agents to learn in reallife situations is the size and complexity of the corresponding task environment. Even in a moderately realistic environment, it may simply take too long to rationally respond to a given situation. If the environment is impatient, allowing only a certain time for a response, an agent may then be unable to cope with the situation and to learn at all. Here, we show that quantum physics can help and provide a quadratic speedup for active learning as a genuine problem of artificial intelligence. This result will be particularly relevant for applications involving complex task environments.

A self-interaction-free local hybrid functional: Accurate binding energies vis-\`a-vis accurate ionization potentials from Kohn-Sham eigenvalues

We present and test a new approximation for the exchange-correlation (xc) energy of Kohn-Sham density functional theory. It combines exact exchange with a compatible non-local correlation functional. The functional is by construction free of one-electron self-interaction, respects constraints derived from uniform coordinate scaling, and has the correct asymptotic behavior of the xc energy density. It contains one parameter that is not determined ab initio. We investigate whether it is possible to construct a functional that yields accurate binding energies and affords other advantages, specifically Kohn-Sham eigenvalues that reliably reflect ionization potentials. Tests for a set of atoms and small molecules show that within our local-hybrid form accurate binding energies can be achieved by proper optimization of the free parameter in our functional, along with an improvement in dissociation energy curves and in Kohn-Sham eigenvalues. However, the correspondence of the latter to experimental ionization potentials is not yet satisfactory, and if we choose to optimize their prediction, a rather different value of the functionals parameter is obtained. We put this finding in a larger context by discussing similar observations for other functionals and possible directions for further functional development that our findings suggest.

Fully Numerical All-Electron Solutions of the Optimized Effective Potential Equation for Diatomic Molecules.

We present an approach for fully numerical, all-electron solutions of the optimized effective potential equation within Kohn-Sham density functional theory for diatomic molecules. The approach is based on a real-space, prolate-spheroidal coordinate grid for solving the all-electron Kohn-Sham equations and an iterative scheme for solving the optimized effective potential equation. The accuracy of this method is demonstrated by comparison with previously reported calculations. New fully numerical benchmark results for selected diatomic molecules are provided.

Projective Simulation for Classical Learning Agents: A Comprehensive Investigation

We study the model of projective simulation (PS), a novel approach to artificial intelligence based on stochastic processing of episodic memory which was recently introduced. 2) Here we provide a detailed analysis of the model and examine its performance, including its achievable efficiency, its learning times and the way both properties scale with the problems’ dimension. In addition, we situate the PS agent in different learning scenarios, and study its learning abilities. A variety of new scenarios are being considered, thereby demonstrating the model’s flexibility. Furthermore, to put the PS scheme in context, we compare its performance with those of Q-learning and learning classifier systems, two popular models in the field of reinforcement learning. It is shown that PS is a competitive artificial intelligence model of unique properties and strengths.

Projective simulation with generalization

The ability to generalize is an important feature of any intelligent agent. Not only because it may allow the agent to cope with large amounts of data, but also because in some environments, an agent with no generalization capabilities cannot learn. In this work we outline several criteria for generalization, and present a dynamic and autonomous machinery that enables projective simulation agents to meaningfully generalize. Projective simulation, a novel, physical approach to artificial intelligence, was recently shown to perform well in standard reinforcement learning problems, with applications in advanced robotics as well as quantum experiments. Both the basic projective simulation model and the presented generalization machinery are based on very simple principles. This allows us to provide a full analytical analysis of the agent’s performance and to illustrate the benefit the agent gains by generalizing. Specifically, we show that already in basic (but extreme) environments, learning without generalization may be impossible, and demonstrate how the presented generalization machinery enables the projective simulation agent to learn.

Quantum walks on embedded hypercubes

It has been proved by Kempe that discrete quantum walks on the hypercube (HC) hit exponentially faster than the classical analog. The same was also observed numerically by Krovi and Brun for a slightly different property, namely, the expected hitting time. Yet, to what extent this striking result survives in more general graphs is to date an open question. Here, we tackle this question by studying the expected hitting time for quantum walks on HCs that are embedded into larger symmetric structures. By performing numerical simulations of the discrete quantum walk and deriving a general expression for the classical hitting time, we observe an exponentially increasing gap between the expected classical and quantum hitting times, not only for walks on the bare HC, but also for a large family of embedded HCs. This suggests that the quantum speedup is stable with respect to such embeddings.

Projective simulation applied to the grid-world and the mountain-car problem

We study the model of projective simulation (PS) which is a novel approach to artificial intelligence (AI). Recently it wasshown that the PS agent performs well in a number of simple task environments, also when compared to standard models ofreinforcement learning (RL). In this paper we study the performance of the PS agent further in more complicated scenarios. Tothat end we chose two well-studied benchmarking problems, namely the “grid-world” and the “mountain-car” problem, whichchallenge the model with large and continuous input space. We compare the performance of the PS agent model with those ofexisting models and show that the PS agent exhibits competitive performance also in such scenarios.

Dissociation of diatomic molecules and the exact-exchange Kohn-Sham potential: The case of LiF

We examine the role of the exact-exchange (EXX) Kohn-Sham potential in curing the problem of fractional molecular dissociation. This is achieved by performing EXX calculations for the illustrative case of the LiF molecule. We show that by choosing the lowest-energy electronic configuration for each interatomic distance, a qualitatively correct binding energy curve, reflecting integer dissociation, is obtained. Surprisingly, for LiF this comes at the cost of violating the Aufbau principle, a phenomenon we discuss at length. Furthermore, we numerically confirm that in the EXX potential of the diatomic molecule, one of the atomic potentials is shifted by a constant while the other one is not, depending on where the highest occupied molecular orbital is localized. This changes the relative positions of the energies of each atom and enforces the integer configuration by preventing spurious charge transfer. The size of the constant shift becomes increasingly unstable numerically the larger the interatomic separation is, reflecting the increasing absence of coupling between the atoms.

Quantum walks on embedded hypercubes: Nonsymmetric and nonlocal cases

The expected hitting time of discrete quantum walks on a hypercube (HC) is numerically known to be exponentially shorter than that of their classical analogs in terms of the scaling with the HC dimension. Recent numerical analyses illustrated that this scaling exists not only on the bare HC, but also when the HC graph is symmetrically and locally embedded into larger graphs. The present work investigates the necessity of symmetry and locality for the speed-up by considering embeddings that are nonsymmetric or nonlocal. We provide numerical evidence that the exponential speed-up survives also in these cases. Furthermore, our numerical simulations demonstrate that removing a single edge from the HC also does not destroy the exponential speed-up. In the nonlocal embedding of the HC we encounter dark states, which we analyze. We provide a general and detailed presentation of the mapping that reduces the exponentially large Hilbert space of the quantum walk to an effective subspace of polynomial scaling. This mapping is our essential tool to numerically study quantum walks in such high-dimensional structures.