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International Conference on Learning Representations

Fidelity-based Deep Adiabatic Scheduling

Eli Ovits, Lior Wolf

Published May 3, 2021• 0 citations
OpenAlex

Abstract

Adiabatic quantum computation is a form of computation that acts by slowly interpolating a quantum system between an easy to prepare initial state and a final state that represents a solution to a given computational problem. The choice of the interpolation schedule is critical to the performance: if at a certain time point, the evolution is too rapid, the system has a high probability to transfer to a higher energy state, which does not represent a solution to the problem. On the other hand, an evolution that is too slow leads to a loss of computation time and increases the probability of failure due to decoherence. In this work, we train deep neural models to produce optimal schedules that are conditioned on the problem at hand. We consider two types of problem representation: the Hamiltonian form, and the Quadratic Unconstrained Binary Optimization (QUBO) form. A novel loss function that scores schedules according to their approximated success probability is introduced. We benchmark our approach on random QUBO problems, Grover search, 3-SAT, and MAX-CUT problems and show that our approach outperforms, by a sizable margin, the linear schedules as well as alternative approaches that were very recently proposed.

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Adiabatic quantum computation is a form of computation that acts by slowly interpolating a quantum system between an easy to prepare initial state and a final state that represents a solution to a given computational problem. The choice of the interpolation schedule is critical to the performance: if at a certain time point, the evolution is too rapid, the system has a high probability to transfer to a higher energy state, which does not represent a solution to the problem. On the other hand, an evolution that is too slow leads to a loss of computation time and increases the probability of failure due to decoherence. In this work, we train deep neural models to produce optimal schedules that are conditioned on the problem at hand. We consider two types of problem representation: the Hamiltonian form, and the Quadratic Unconstrained Binary Optimization (QUBO) form. A novel loss function that scores schedules according to their approximated success probability is introduced. We benchmark our approach on random QUBO problems, Grover search, 3-SAT, and MAX-CUT problems and show that our approach outperforms, by a sizable margin, the linear schedules as well as alternative approaches that were very recently proposed.
Abstract Adiabatic quantum computing and optimization have garnered much attention recently as possible models for achieving a quantum advantage over classical approaches to optimization and other special purpose computations. Both techniques are probabilistic in nature and the minimum gap between the ground state and first excited state of the system during evolution is a major factor in determining the success probability. In this work we investigate a strategy for increasing the minimum gap and success probability by introducing intermediate Hamiltonians that modify the evolution path between initial and final Hamiltonians. We focus on an optimization problem relevant to recent hardware implementations and present numerical evidence for the existence of a purely local intermediate Hamiltonian that achieve the optimum performance in terms of pushing the minimum gap to one of the end points of the evolution. As a part of this study we develop a convex optimization formulation of the search for optimal adiabatic schedules that makes this computation more tractable, and which may be of independent interest. We further study the effectiveness of random intermediate Hamiltonians on the minimum gap and success probability, and empirically find that random Hamiltonians have a significant probability of increasing the success probability, but only by a modest amount.
Abstract Several fundamental questions demand space exploration, posing tremendous computational tasks to address modelling and data processing. Quantum computing has been proposed to address deep space-related problems from quantum gravity, nuclear processes and many-body field theory simulation to artificial intelligence and related space mission planning. Adiabatic quantum computers have been proposed for solving optimization problems, such as space mission scheduling. The problem of assigning jobs to resources in a scientific mission, while satisfying certain constraints, can be tackled with the help of D-Wave’s quantum annealer, which exploits the adiabatic theorem to find the minimum energy solutions of a given system, encoded as the Hamiltonian of a quadratic unconstrained binary optimization (QUBO) problem. In its simplest em-bodiement, the scheduling can be implemented as a graph coloring (GC) problem, where the goal is to assign different colors (out of a given set) to connected nodes of an input graph, while respecting given constraints. The feasibility of our methods and the performance of quantum annealers are studied on randomly generated GC instances. A second framework studied is the job shop scheduling problem (JSP) that encodes more real-like scheduling problems of interest to agencies and industries. By implementing the problems with a QUBO formulation, we are able to find the optimal solution to three JSP instances present in the literature. The biggest one is solved with a quantum computation approach here for the first time.
Quantum algorithm design plays a crucial role in exploiting the computational advantage of quantum devices. Here we develop a deep-reinforcement-learning based approach for quantum adiabatic algorithm design. Our approach is generically applicable to a class of problems with solution hard-to-find but easy-to-verify, e.g., searching and NP-complete problems. We benchmark this approach in Grover-search and 3-SAT problems, and find that the adiabatic algorithm obtained by our RL approach leads to significant improvement in the resultant success probability. In application to Grover search, our RL design automatically produces an adiabatic quantum algorithm that has the quadratic speedup. We find for all our studied cases that quantitatively the RL-designed algorithm has a better performance compared to the analytically constructed nonlinear Hamiltonian path when the encoding Hamiltonian is solvable, and that this RL-design approach remains applicable even when the nonlinear Hamiltonian path is not analytically available. In 3-SAT we find RL design has fascinating transferability---the adiabatic algorithm obtained by training on a specific choice of clause number leads to better performance consistently over the linear algorithm on different clause numbers. These findings suggest the applicability of reinforcement learning for automated quantum adiabatic algorithm design. Further considering the established complexity equivalence of circuit and adiabatic quantum algorithms, we expect the RL-designed adiabatic algorithm to inspire novel circuit algorithms as well. Our approach is potentially applicable to different quantum hardware from trapped ions and optical lattices to superconducting-qubit devices.
Quantum annealing is a practical approach to execute the native instruction set of the adiabatic quantum computation model. The key of running adiabatic algorithms is to maintain a high success probability of evolving the system into the ground state of a problem-encoded Hamiltonian at the end of an annealing schedule. This is typically done by executing the adiabatic algorithm slowly to enforce adiabacity. However, properly optimized annealing schedule can accelerate the computational process. Inspired by the recent success of DeepMind's AlphaZero algorithm that can efficiently explore and find a good winning strategy from a large combinatorial search with a neural-network-assisted Monte Carlo Tree Search (MCTS), we adopt MCTS and propose a neural-network-enabled version, termed QuantumZero (QZero), to automate the design of an optimal annealing schedule in a hybrid quantum-classical framework. The flexibility of having neural networks allows us to apply transfer-learning technique to boost QZero's performance. We find both MCTS and QZero to perform very well in finding excellent annealing schedules even when the annealing time is short in the 3-SAT examples we consider in this study. We also find MCTS and QZero to be more efficient than many other leading reinforcement leanring algorithms for the task of desining annealing schedules. In particular, if there is a need to solve a large set of similar problems using a quantum annealer, QZero is the method of choice when the neural networks are first pre-trained with examples solved in the past.
Adiabatic quantum algorithms must evolve slowly enough to suppress non-adiabatic transitions while remaining fast enough to be practical. In open systems, this trade-off is reshaped by decoherence. For Hamiltonians subject to dephasing Lindbladians, Avron et al. [Phys. Rev. A 82, 040304(R) (2010)] showed that a unique timetable exists that maximizes the fidelity with a target state. This optimal schedule is characterized by a constant tunneling rate along the adiabatic path. In this work, we revisit their analysis and apply it to the adiabatic Grover search framework, obtaining closed-form expressions for the optimal evolution schedule, the minimum runtime, and the resulting achievable fidelity. Moreover, by invoking an energy-time uncertainty argument, we identify a critical dephasing threshold beyond which further noise-assisted acceleration is prohibited, showing that the apparent advantage of strong dephasing is a Zeno-type stabilization mechanism rather than a conventional algorithmic speedup over unitary Grover search.
Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring the properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time.