Limits of Computation

Reiter, Edna E.; Johnson, Clayton Matthew

doi:10.1201/b12992

Cited by 7 publications

(3 citation statements)

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“…C OMBINATORIAL optimization is used to solve many practical problems in various fields and involves finding the optimal solution for a given objective function subject to a set of constraints [1]. Combinatorial optimization problems are often NP-hard, meaning that finding the optimal solution requires an exponentially large amount of time with respect to the problem size [2]. One such approach is simulated annealing, which is a stochastic optimization method inspired by the physical annealing process in materials science [3], [4].…”

Section: Introductionmentioning

confidence: 99%

Local Energy Distribution Based Hyperparameter Determination for Stochastic Simulated Annealing

Onizawa,

Kuroki,

Shin

et al. 2023

IEEE Open J. Signal Process.

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This paper presents a local energy distribution based hyperparameter determination for stochastic simulated annealing (SSA). SSA is capable of solving combinatorial optimization problems faster than typical simulated annealing (SA), but requires a time-consuming hyperparameter search. The proposed method determines hyperparameters based on the local energy distributions of spins (probabilistic bits). The spin is a basic computing element of SSA and is graphically connected to other spins with its weights. The distribution of the local energy can be estimated based on the central limit theorem (CLT). The CLTbased normal distribution is used to determine the hyperparameters, which reduces the time complexity for hyperparameter search from O(n 3 ) of the conventional method to O(1). The performance of SSA with the determined hyperparameters is evaluated on the Gset and K2000 benchmarks for maximum-cut problems. The results show that the proposed method achieves mean cut values of approximately 98% of the best-known cut values.

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Section: Introductionmentioning

confidence: 99%

Local Energy Distribution Based Hyperparameter Determination for Stochastic Simulated Annealing

Onizawa,

Kuroki,

Shin

et al. 2023

IEEE Open J. Signal Process.

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“…Combinatorial optimization problems can often be represented by Ising models, which are mathematical representations of networks or graphs. These problems are often categorized as NP-hard 13 , meaning that the time required to find the optimal solution tends to grow exponentially with the size of the problem, making them computationally challenging to solve. The goal of simulated annealing in this context is to minimize the 'energy' of the Ising model, where 'energy' is a metaphor for the objective or cost function of the optimization problem.…”

Section: Introductionmentioning

confidence: 99%

Enhanced Convergence in p-bit Based Simulated Annealing with Partial Deactivation for Large-Scale Combinatorial Optimization Problems

Onizawa,

Hanyu

2023

Preprint

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This article critically investigates the limitations of the simulated annealing algorithm using probabilistic bits (pSA) in solving large-scale combinatorial optimization problems. The study begins with an in-depth analysis of the pSA process, focusing on the issues resulting from unexpected oscillations among p-bits. These oscillations hinder the energy reduction of the Ising model and thus obstruct the successful execution of pSA in complex tasks. Through detailed simulations, we unravel the root cause of this energy stagnation, identifying the feedback mechanism inherent to the pSA operation as the primary contributor to these disruptive oscillations. To address this challenge, we propose two novel algorithms, time average pSA (TApSA) and stalled pSA (SpSA). These algorithms are designed based on partial deactivation of p-bits and are thoroughly tested using Python simulations on maximum cut benchmarks that are typical combinatorial optimization problems. On the 16 benchmarks from 800 to 5,000 nodes, the proposed methods improve the normalized cut value from 0.8% to 98.4% on average in comparison with the conventional pSA.

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“…Third, the GI problem is mathematically interesting; though many sub-classes of the problem can be solved in polynomial time by specialized classical solvers, the run time of the best general solution is exponential and has remained at since 198367. The classical computational complexity of the problem is currently considered to be NP -intermediate8, and the quantum computational complexity of the problem is unknown. Graph isomorphism is a non-abelian hidden subgroup problem and is not known to be easy in the quantum regime910.…”

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confidence: 99%

Experimental quantum annealing: case study involving the graph isomorphism problem

Zick

Shehab

French

2015

Sci Rep

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Quantum annealing is a proposed combinatorial optimization technique meant to exploit quantum mechanical effects such as tunneling and entanglement. Real-world quantum annealing-based solvers require a combination of annealing and classical pre- and post-processing; at this early stage, little is known about how to partition and optimize the processing. This article presents an experimental case study of quantum annealing and some of the factors involved in real-world solvers, using a 504-qubit D-Wave Two machine and the graph isomorphism problem. To illustrate the role of classical pre-processing, a compact Hamiltonian is presented that enables a reduced Ising model for each problem instance. On random N-vertex graphs, the median number of variables is reduced from N2 to fewer than N log2 N and solvable graph sizes increase from N = 5 to N = 13. Additionally, error correction via classical post-processing majority voting is evaluated. While the solution times are not competitive with classical approaches to graph isomorphism, the enhanced solver ultimately classified correctly every problem that was mapped to the processor and demonstrated clear advantages over the baseline approach. The results shed some light on the nature of real-world quantum annealing and the associated hybrid classical-quantum solvers.

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Limits of Computation

Cited by 7 publications

References 0 publications

Local Energy Distribution Based Hyperparameter Determination for Stochastic Simulated Annealing

Local Energy Distribution Based Hyperparameter Determination for Stochastic Simulated Annealing

Enhanced Convergence in p-bit Based Simulated Annealing with Partial Deactivation for Large-Scale Combinatorial Optimization Problems

Experimental quantum annealing: case study involving the graph isomorphism problem

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