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arXiv:2006.11190 · Physical Review A

Solving optimization problems with Rydberg analog quantum computers: Realistic requirements for quantum advantage using noisy simulation and classical benchmarks

Michel Fabrice Serret, Bertrand Marchand, Thomas Ayral

Published Nov 23, 2020• 32 citations
arXivHTMLPDF

Abstract

Platforms of Rydberg atoms have been proposed as promising candidates to solve some combinatorial optimization problems. Here we compute quantitative requirements on the system sizes and noise levels that these platforms must fulfill to reach quantum advantage in approximately solving the Unit-Disk Maximum Independent Set problem. Using noisy simulations of Rydberg platforms of up to 26 atoms interacting through realistic van der Waals interactions, we compute the average approximation ratio that can be attained with a simple quantum annealing-based heuristic within a fixed temporal computational budget. Based on estimates of the correlation lengths measured in the engineered quantum state, we extrapolate the results to large atom numbers and compare them to a simple classical approximation heuristic. We find that approximation ratios of at least $\ensuremath{\approx}0.84$ are within reach for near-future noise levels. Not taking into account further classical and quantum algorithmic improvements, we estimate that quantum advantage could be reached by attaining a number of controlled atoms of $\ensuremath{\sim}8000$ for a time budget of 2 s, and $\ensuremath{\sim}1000--1200$ for a time budget of 0.2 s, provided the coherence levels of the system can be improved by a factor 10 while maintaining a constant repetition rate.

DOI
10.1103/physreva.102.052617

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Platforms of Rydberg atoms have been proposed as promising candidates to solve some combinatorial optimization problems. Here we compute quantitative requirements on the system sizes and noise levels that these platforms must fulfill to reach quantum advantage in approximately solving the Unit-Disk Maximum Independent Set problem. Using noisy simulations of Rydberg platforms of up to 26 atoms interacting through realistic van der Waals interactions, we compute the average approximation ratio that can be attained with a simple quantum annealing-based heuristic within a fixed temporal computational budget. Based on estimates of the correlation lengths measured in the engineered quantum state, we extrapolate the results to large atom numbers and compare them to a simple classical approximation heuristic. We find that approximation ratios of at least \ensuremath≈0.84\ensuremath{\approx}0.84\ensuremath≈0.84 are within reach for near-future noise levels. Not taking into account further classical and quantum algorithmic improvements, we estimate that quantum advantage could be reached by attaining a number of controlled atoms of \ensuremath∼8000\ensuremath{\sim}8000\ensuremath∼8000 for a time budget of 2 s, and \ensuremath∼1000−−1200\ensuremath{\sim}1000--1200\ensuremath∼1000−−1200 for a time budget of 0.2 s, provided the coherence levels of the system can be improved by a factor 10 while maintaining a constant repetition rate.
Analog quantum computing with Rydberg atoms is seen as an avenue to solve hard graph optimization problems, because they naturally encode the maximum independent set (MIS) problem on unit disk (UD) graphs, a problem that admits rather efficient approximation schemes on classical computers. Going beyond UD-MIS to address generic graphs requires embedding schemes, typically with chains of ancilla atoms, and an interpretation algorithm to map results back to the original problem. However, interpreting approximate solutions obtained with realistic quantum computers proves to be a difficult problem. As a case study, we evaluate the ability of two interpretation strategies to correct errors in the recently introduced crossing lattice embedding. We find that one strategy, based on finding the closest embedding solution, leads to very high qualities, albeit at an exponential cost. The second strategy, based on ignoring defective regions of the embedding graph, is polynomial in the graph size, but it leads to a degradation of the solution quality which is prohibitive under realistic assumptions on the defect generation. Moreover, more favorable defect scalings lead to a contradiction with well-known approximability conjectures. Therefore, it is unlikely that a scalable and generic improvement in solution quality can be achieved with Rydberg platforms---thus moving the focus to heuristic algorithms.
Programmable quantum systems based on Rydberg atom arrays have recently been used for hardwareefficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] with hundreds of qubits.In particular, the maximum independent set problem on so-called unit-disk graphs, was shown to be efficiently encodable in such a quantum system.Here, we extend the classes of problems that can be efficiently encoded in Rydberg arrays by constructing explicit mappings from a wide class of problems to maximum-weighted independent set problems on unit-disk graphs, with at most a quadratic overhead in the number of qubits.We analyze several examples, including maximum-weighted independent set on graphs with arbitrary connectivity, quadratic unconstrained binary optimization problems with arbitrary or restricted connectivity, and integer factorization.Numerical simulations on small system sizes indicate that the adiabatic time scale for solving the mapped problems is strongly correlated with that of the original problems.Our work provides a blueprint for using Rydberg atom arrays to solve a wide range of combinatorial optimization problems with arbitrary connectivity, beyond the restrictions imposed by the hardware geometry.
Quantum information science has emerged as one of the most promising fields in contemporary research, relying on both software and hardware innovations. This thesis looks for both algorithms with quantum features that provide advantages over classical algorithms, and better hardware platforms for experiments and quantum computing. The work spans theoretical studies in both algorithm and hardware design, including hybrid quantum-classical algorithms and the development of quantum information processing platforms. The algorithmic part has focused on the performance of a hybrid quantum algorithm - the Grover Quantum Approximate Optimization Algorithm (Grover QAOA) - designed for problems with multiple solutions. In practice, we find its potential for speedup in solution search and its ability to find all solutions. Furthermore, we propose a simplified protocol that reduces the classical complexity of optimizing the algorithm’s parameters, enhancing its practicality for future applications. Our implementation of Grover QAOA for multiple combinatorial optimization problems on trapped-ion quantum computers demonstrates that the algorithm can fulfill its fair-sampling advantage even on noisy devices. In the hardware part, we mainly explore how the motion of cold atoms can either be used to engineer interactions or lead to previously overlooked decoherence. The first hardware platform we discuss is trapped ions, where we focus on implementing individual addressing to natively simulate new types of many-body systems. Our proposal leverages the exceptional controllability of trapped ions to explore dynamical models such as topological pumping. The second hardware platform we study is Rydberg atom lattices, where we investigate the decoherence processes introduced by atomic motion during dynamics. Using the numerical tool of discrete truncated Wigner approximation, we simulate the coupled dynamics of electronic and motional degrees of freedom, demonstrating that atom motion induced by strong van der Waals interactions in Rydberg atoms can lead to significant decoherence in analog simulation experiments. We have also explored specialized topics involving other quantum hardware platforms. One area of study is the reduction of frequency crowding in superconducting circuit quantum chips. By properly designing the frequencies for each transmon qubit, we can improve the manufacturing yield of collision-free processors. Another area focuses on the SU(N) Fermi-Hubbard models on alkaline-earth-metal optical lattice platforms. We have obtained the phase diagram of unit-filling models with imbalanced spin flavors. This work aids future experiments in searching for potential ground states of the unit-filling SU(N) model.
Recent progress in quantum computing and quantum simulation of many-body systems with arrays of neutral atoms using Rydberg excitation has provided unforeseen opportunities towards computational advantage in solving various optimization problems. The problem of a maximum-weight independent set of unit-disk graphs is an example of an NP-hard optimization problem. It involves finding the largest set of vertices with the maximum sum of their weights for a graph which has edges connecting all pairs of vertices within a unit distance. This problem can be solved using quantum annealing with an array of interacting Rydberg atoms. For a particular graph, a spatial arrangement of atoms represents vertices of the graph, while the detuning from resonance at Rydberg excitation defines the weights of these vertices. The edges of the graph can be drawn according to the unit-disk criterion. Maximum-weight independent sets can be obtained by applying a variational quantum adiabatic algorithm. We consider driving the quantum system of interacting atoms to the many-body ground state using a nonlinear quasiadiabatic profile for sweeping the Rydberg detuning. We also propose using a quantum wire, which is a set of auxiliary atoms of a different chemical element, to mediate strong coupling between the remote vertices of the graph. We investigate this effect for different lengths of the quantum wire. We also investigate the quantum phases of matter realizing commensurate and incommensurate phases in one- and two-dimensional spatial arrangements of the atomic array.
Programmable quantum systems utilizing Rydberg atoms have recently been shown to efficiently encode combinatorial optimization problems. The relationships among the variables in these optimization problems can be transformed into a maximum weighted independent set problem within the Rydberg atom system, requiring at most a quadratic overhead in terms of the number of qubits. Although experiments with current Rydberg quantum systems have demonstrated linear speedup compared to classical algorithms, the overhead in both the number of qubits and time remains substantial. To further reduce the overhead of encoding these optimization problems in Rydberg atom systems, we propose a simplification strategy and design a new heuristic algorithm that incorporates a topological sort approach to optimize the encoding process. We conduct extensive experiments on small graphs with dozens of qubits. The numerical results demonstrate the effectiveness of our algorithm in both runtime and optimization efficiency. Specifically, our algorithm reduces the number of required encoding qubits by 12.0% compared to existing methods. Additionally, our approach reduces the time complexity to a quadratic overhead, enabling experiments on larger graphs. The results further indicate that our algorithm provides significant time advantages over existing solutions.
A novel class of hybrid quantum-classical algorithms based on the variational approach have recently emerged from separate proposals addressing, for example, quantum chemistry and combinatorial problems. These algorithms provide an approximate solution to the problem at hand by encoding it in the state of a quantum computer. The operations used to prepare the state are not a priori fixed but, quite the opposite, are subjected to a classical optimization procedure that modifies the quantum gates and improves the quality of the approximate solution. While the quantum hardware determines the size of the problem and what states are achievable (limited, respectively, by the number of qubits and by the kind and number of possible quantum gates), it is the classical optimization procedure that determines the way in which the quantum states are explored and whether the best available solution is actually reached. In addition, the quantities required in the optimization, for example the objective function itself, have to be estimated with finite precision in any experimental implementation. While it is desirable to have very precise estimates, this comes at the cost of repeating the state preparation and measurement multiple times. Here we analyze the competing requirements of high precision and low number of repetitions and study how the overall performance of the variational algorithm is affected by the precision level and the choice of the optimization method. Finally, this study introduces quasi-Newton optimization methods in the general context of hybrid variational algorithms and presents quantitative results for the Quantum Approximate Optimization Algorithm.