From near to eternity: Spin-glass planting, tiling puzzles, and constraint-satisfaction problems

Hamze, Firas; Jacob, Darryl C.; Ochoa, Andrew J.; Perera, Dilina; Wang, Wenlong; Katzgraber, Helmut G.

doi:10.1103/physreve.97.043303

Cited by 21 publications

(25 citation statements)

References 58 publications

(73 reference statements)

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“…By generating 3SAT instances with planted solutions (see, e.g., refs. 17,18 ) and transforming these to non-stoquastic Hamiltonians via the mapping prescribed by Theorems 1 (or 2), one would be able to generate 3-local (or 6-local) Hamiltonians that are stoquastic, but are computationally hard to transform into a stoquastic form.…”

Section: Methodsmentioning

confidence: 99%

On the computational complexity of curing non-stoquastic Hamiltonians

2019

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Quantum many-body systems whose Hamiltonians are non-stoquastic, i.e., have positive off-diagonal matrix elements in a given basis, are known to pose severe limitations on the efficiency of Quantum Monte Carlo algorithms designed to simulate them, due to the infamous sign problem. We study the computational complexity associated with ‘curing’ non-stoquastic Hamiltonians, i.e., transforming them into sign-problem-free ones. We prove that if such transformations are limited to single-qubit Clifford group elements or general single-qubit orthogonal matrices, finding the curing transformation is NP-complete. We discuss the implications of this result.

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Section: Methodsmentioning

confidence: 99%

On the computational complexity of curing non-stoquastic Hamiltonians

2019

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“…The tile planting method [44,45] decomposes the problem graph into edge-disjoint vertex-sharing subgraphs. It produces scalable problems with highly tunable complexity.…”

Section: Benchmark Problemsmentioning

confidence: 99%

Computational overhead of locality reduction in binary optimization problems

Valiante

Hernandez

Barzegar

et al. 2021

Computer Physics Communications

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This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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“…Computing the partial derivatives from the definition of F TAP in Eq. (14) we obtain the Hessian matrix elements…”

Section: Stability Of M =mentioning

confidence: 99%

“…On devices natively encoding problems consisting of binary variables, this can be problematic. Various techniques have recently been published [12][13][14][15] for constructing planted Ising instances on sparse graph topologies, which in some cases appear to yield quite difficult problems [14], but in common with short-ranged disordered models in general, most of their known properties are inferred from numerical simulation and much remains unexplored about which features make them amenable as benchmarks for given algorithms.…”

Section: Introductionmentioning

confidence: 99%

Wishart planted ensemble: A tunably rugged pairwise Ising model with a first-order phase transition

et al. 2020

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We propose the Wishart planted ensemble, a class of zero-field Ising models with tunable algorithmic hardness and specifiable (or planted) ground state. The problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a sign-inverted variant of the Hopfield model. The Hamiltonian contains only 2-spin interactions, with the coupler matrix following a type of Wishart distribution. The class exhibits a classical first-order phase transition in temperature. For some parameter settings the model has a locally stable paramagnetic state, a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape. We analytically probe the ensemble thermodynamic properties by deriving the Thouless-Anderson-Palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis; extensive Monte Carlo simulations confirm our predictions of the first-order transition temperature. The class exhibits a wide variation in algorithmic hardness as a generation parameter is varied, with a pronounced easy-hard-easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes. By deriving the ensemble-averaged energy distribution and taking into account finite-precision representation, we propose an analytical expression for the location of the hardness peak and show that at fixed precision, the number of constraints in the integer program must increase with system size to yield truly hard problems. The Wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms.

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From near to eternity: Spin-glass planting, tiling puzzles, and constraint-satisfaction problems

Cited by 21 publications

References 58 publications

On the computational complexity of curing non-stoquastic Hamiltonians

On the computational complexity of curing non-stoquastic Hamiltonians

Computational overhead of locality reduction in binary optimization problems

Wishart planted ensemble: A tunably rugged pairwise Ising model with a first-order phase transition

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