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Efficient algorithms to solve atom reconfiguration problems. II. Assignment-rerouting-ordering algorithm

Remy El Sabeh, Jessica Bohm, Zhiqian Ding, Stephanie Maaz, Naomi Nishimura, Izzat El Hajj, Amer E. Mouawad, and Alexandre Cooper
Phys. Rev. A 108, 023108 – Published 4 August 2023
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  • INTRODUCTION
  • ATOM RECONFIGURATION PROBLEMS
  • THE ASSIGNMENT-REROUTING-ORDERING…
  • QUANTIFYING PERFORMANCE IN THE ABSENCE…
  • QUANTIFYING PERFORMANCE IN THE PRESENCE…
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Programmable arrays of optical traps enable the assembly of configurations of single atoms to perform controlled experiments on quantum many-body systems. Finding the sequence of control operations to transform an arbitrary configuration of atoms into a predetermined one requires solving an atom reconfiguration problem quickly and efficiently. A typical approach to solving atom reconfiguration problems is to use an assignment algorithm to determine which atoms to move to which traps. This approach results in control protocols that exactly minimize the number of displacement operations; however, this approach does not optimize for the number of displaced atoms or the number of times each atom is displaced, resulting in unnecessary control operations that increase the execution time and failure rate of the control protocol. In this work we propose the assignment-rerouting-ordering (ARO) algorithm to improve the performance of assignment-based algorithms in solving atom reconfiguration problems. The ARO algorithm uses an assignment subroutine to minimize the total distance traveled by all atoms, a rerouting subroutine to reduce the number of displaced atoms, and an ordering subroutine to guarantee that each atom is displaced at most once. The ordering subroutine relies on the existence of a partial ordering of moves that can be obtained using a polynomial-time algorithm that we introduce within the formal framework of graph theory. We numerically quantify the performance of the ARO algorithm in the presence and in the absence of loss and show that it outperforms the exact, approximation, and heuristic algorithms that we use as benchmarks. Our results are useful for assembling large configurations of atoms with high success probability and fast preparation time, as well as for designing and benchmarking novel atom reconfiguration algorithms.

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    • Received 13 October 2022
    • Revised 1 May 2023
    • Accepted 10 July 2023

    DOI:https://doi.org/10.1103/PhysRevA.108.023108

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Atom & ion trapping & guidingCooling & trappingMolecule trapping & guidingOptical lattices & trapsQuantum simulation
    1. Techniques
    Graph theory
    Atomic, Molecular & OpticalGeneral Physics

    Authors & Affiliations

    Remy El Sabeh1, Jessica Bohm2, Zhiqian Ding2, Stephanie Maaz3, Naomi Nishimura3, Izzat El Hajj1, Amer E. Mouawad1,3,4, and Alexandre Cooper2,*

    • 1Department of Computer Science, American University of Beirut, Beirut 1107 2020, Lebanon
    • 2Institute for Quantum Computing, University of Waterloo, Waterloo, Ontarion N2L 3G1, Canada
    • 3David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
    • 4AG Theoretische Informatik, University of Bremen, 28359 Bremen, Germany

    • *alexandre.cooper@uwaterloo.ca

    See Also

    Efficient algorithms to solve atom reconfiguration problems. I. Redistribution-reconfiguration algorithm

    Barry Cimring, Remy El Sabeh, Marc Bacvanski, Stephanie Maaz, Izzat El Hajj, Naomi Nishimura, Amer E. Mouawad, and Alexandre Cooper
    Phys. Rev. A 108, 023107 (2023)

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    Vol. 108, Iss. 2 — August 2023

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    • Figure 1
      Figure 1

      Assignment-rerouting-ordering algorithm. The atom reconfiguration problem consists of finding a sequence of moves to transform an arbitrary configuration of atoms (black dots) contained in a static array of optical traps (circles) into a target configuration of atoms (shaded green disks). First, the ARO algorithm uses the assignment subroutine to find a sequence of moves that minimizes the total distance traveled by all atoms or, equivalently, the total number of displacement operations. Second, the rerouting subroutine attempts to update the path of each move to reduce the number of atoms displaced without increasing the total displacement distance, here choosing the path P1′ over the path P1. Third, the ordering subroutine finds a sequence of moves that prevents an atom from obstructing the path of another atom, here choosing to execute the move associated with P2 before executing the move associated with P3.

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    • Figure 2
      Figure 2

      Justification for the cycle-breaking procedure. Shown is an example of a path system defined on a weighted graph that induces a cycle that cannot be broken by computing a minimum spanning tree without increasing the total weight of the path system. The initial path system has a total weight equal to 46 (the red path and the blue path each has a weight of 13, the orange path has a weight of 8, and the purple path has a weight of 12). The MST of the graph induced on the path system includes all the edges of the graph except for the edge of weight 10. The path system generated by computing all-pairs shortest paths on the MST and then computing a source-target matching has a total weight of 52=1×8+5×4+6×4, irrespective of the specific choice of the matching (among 4!=24 possibilities), as the edges of weight 5 will each be in two paths and the edge of weight 6 will be in four paths. A similar example can be constructed for the case of uniformly weighted graphs, where deleting some edge might increase the weight of the path system.

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    • Figure 3
      Figure 3

      Reducing the fraction of displaced atoms using the rerouting subroutine in the absence of loss. (a) Mean fraction of displaced atoms f¯ν=〈Naν/Na0〉 computed for various configuration sizes for the baseline (yellow circles), assignment-rerouting (orange squares), and 3-approx (purple inverted triangles) algorithms. (b) Distribution of the relative fraction of displaced atoms and its mean value for various configuration sizes computed as the ratio of the fraction of displaced atoms for the assignment-rerouting algorithm and the baseline reconfiguration algorithm.

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    • Figure 4
      Figure 4

      Reducing the number of transfer operations using the rerouting and ordering subroutines in the absence of loss for a target configuration of 32×32 atoms. (a) Distribution of the number of EDI cycles per displaced atom using the baseline (yellow), ARO (red), and 3-approx (red) algorithms. (b) Distribution of the number of transfer operations computed relative to the 3-approx for the baseline (yellow) and ARO (red) algorithms. (c) Mean relative number of transfer operations for the ARO algorithm computed relative to the baseline reconfiguration algorithm for various configuration sizes. (d) Distribution of the relative number of transfer operations for the ARO algorithm computed relative to the baseline reconfiguration algorithm.

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    • Figure 5
      Figure 5

      Reducing the total number of control operations using the ARO algorithm in the absence of loss for a target configuration of 32×32 atoms. (a) Distribution of the number of transfer and displacement operations for the baseline (yellow), ARO (red), and 3-approx (purple) algorithms. (b) Distribution of the number of control operations for the baseline (yellow), ARO (red), and 3-approx (purple) algorithms. (c) and (d) Distribution of (c) the relative number of control operations and (d) its mean value for various configuration sizes for the ARO algorithm computed relative to the baseline (yellow) and 3-approx algorithms (purple).

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    • Figure 6
      Figure 6

      Increasing the mean success probability using the ARO algorithm. The mean success probability p¯ for preparing a configuration of NaT=Ntx×Ntx atoms in an array of Nt=Ntx×Nty traps is shown for (a) Ntx=16, (b) Ntx=16 and Nty=36,38, (c) Ntx=32, and (d) Ntx=32 and Nty=86,88. The markers represent the 3-approx (purple inverted triangles), baseline (yellow disks), red-rec (blue triangles), and ARO (red squares) algorithms.

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    • Figure 7
      Figure 7

      Examples of atom reconfiguration problems on graphs. (a) Example problem for which ordering the paths of a path system reduces the number of transfer operations. Executing the move associated with either P2 or P3 (or both) before the move associated with P1 would force τ2 or τ3 (or both) to move twice. (b) Example problem for which a token (here τ3) can be isolated without increasing the weight of the path system. (c) Example problem for which the displacement distance and the number of displaced tokens cannot be simultaneously minimized. Two tokens need to move to minimize the displacement distance, whereas one token needs to move if minimizing displacements is not imposed.

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    • Figure 8
      Figure 8

      Merging the edge color for the case where exactly one of Vχb∖V(se) and Vχa∖V(se) is empty. (a) and (b) Starting from cycle C″ where the edge color (red) is noncontiguous, we construct a cycle C′′′ (blue line) in which the color of edge e (red) is contiguous via path Pχ (red solid line on the cycle to highlight the segments; red dashed line indicates edges in the path that are not shared with the cycle).

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    • Figure 9
      Figure 9

      Leveraging a source-target corner to reduce the number of paths that induce the special cycle. (a) Example of a sequence of three paths that induce a special cycle that contains the edge e★ (thick blue line). (b) Updated sequence of paths after swapping the target vertices of P2 and P3, which can then be removed from the sequence of paths, as it no longer contributes any edge to the special cycle.

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    • Figure 10
      Figure 10

      Breaking a cycle in a path system induced on a reduced set of paths without source-target corners. (a) Example of a path system with a special cycle formed by four paths. (b) Even reduced path system and (c) odd reduced path system obtained after breaking the cycle. Because both reduced path systems have the same overall weight and the odd path system does not contain the edge e★ (thick blue line), the odd reduced path system gets picked.

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