Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1504.07991 · Physical Review Letters

Heavy Tails in the Distribution of Time to Solution for Classical and Quantum Annealing

Damian S. Steiger, Troels F. Rønnow, Matthias Troyer

Published Dec 4, 2015• 39 citations
arXivHTMLPDF

Abstract

For many optimization algorithms the time to solution depends not only on the problem size but also on the specific problem instance and may vary by many orders of magnitude. It is then necessary to investigate the full distribution and especially its tail. Here, we analyze the distributions of annealing times for simulated annealing and simulated quantum annealing (by path integral quantum Monte Carlo simulation) for random Ising spin glass instances. We find power-law distributions with very heavy tails, corresponding to extremely hard instances, but far broader distributions-and thus worse performance for hard instances-for simulated quantum annealing than for simulated annealing. Fast, nonadiabatic, annealing schedules can improve the performance of simulated quantum annealing for very hard instances by many orders of magnitude.

DOI
10.1103/physrevlett.115.230501

Related papers

Optimization by quantum annealing: Lessons from hard satisfiability problems

Demian A. Battaglia, Giuseppe E. Santoro, Erio Tosatti

The path integral Monte Carlo simulated quantum annealing algorithm is applied to the optimization of a large hard instance of the random satisfiability problem (N = 10,000). The dynamical behavior of the quantum and the classical annealing are compared, showing important qualitative differences in the way of exploring the complex energy landscape of the combinatorial optimization problem. At variance with the results obtained for the Ising spin glass and for the traveling salesman problem, in the present case the linear-schedule quantum annealing performance is definitely worse than classical annealing. Nevertheless, a quantum cooling protocol based on field-cycling and able to outperform standard classical simulated annealing over short time scales is introduced.

arXiv:cond-mat/0502468•Physical Review E•Jun 29, 2005•107 citations

Degeneracy, degree, and heavy tails in quantum annealing

Andrew D. King, Emile Hoskinson, Trevor M. Lanting +2 more

Both simulated quantum annealing and physical quantum annealing have shown the emergence of ``heavy tails'' in their performance as optimizers: The total time needed to solve a set of random input instances is dominated by a small number of very hard instances. Classical simulated annealing, in contrast, does not show such heavy tails. Here we explore the origin of these heavy tails, which appear for inputs with high local degeneracy---large isoenergetic clusters of states in Hamming space. This category includes the low-precision Chimera-structured problems studied in recent benchmarking work comparing the D-Wave Two quantum annealing processor with simulated annealing. On similar inputs designed to suppress local degeneracy, performance of a quantum annealing processor on hard instances improves by orders of magnitude at the 512-qubit scale, while classical performance remains relatively unchanged. Simulations indicate that perturbative crossings are the primary factor contributing to these heavy tails, while sensitivity to Hamiltonian misspecification error plays a less significant role in this particular setting.

arXiv:1512.07325•Physical Review A•May 18, 2016•27 citations

Optimizing Schedules for Quantum Annealing

Daniel Herr, Ethan Brown, Bettina Heim +3 more

Classical and quantum annealing are two heuristic optimization methods that search for an optimal solution by slowly decreasing thermal or quantum fluctuations. Optimizing annealing schedules is important both for performance and fair comparisons between classical annealing, quantum annealing, and other algorithms. Here we present a heuristic approach for the optimization of annealing schedules for quantum annealing and apply it to 3D Ising spin glass problems. We find that if both classical and quantum annealing schedules are similarly optimized, classical annealing outperforms quantum annealing for these problems when considering the residual energy obtained in slow annealing. However, when performing many repetitions of fast annealing, simulated quantum annealing is seen to outperform classical annealing for our benchmark problems.

arXiv:1705.00420•May 1, 2017•6 citations

Sampling diverse near-optimal solutions via algorithmic quantum annealing

Masoud Mohseni, Marek M. Rams, Sergei V. Isakov +5 more

Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open problems is the lack of ergodicity, or mode collapse, for typical stochastic solvers based on Monte Carlo techniques leading to poor generalization or lack of robustness to uncertainties. Currently, there is no universal metric to quantify such performance deficiencies across various solvers. Here, we introduce a new diversity measure for quantifying the number of independent approximate solutions for NP-hard optimization problems. Among others, it allows benchmarking solver performance by a required time-to-diversity (TTD), a generalization of often used time-to-solution (TTS). We illustrate this metric by comparing the sampling power of various quantum annealing strategies. In particular, we show that the inhomogeneous quantum annealing schedules can redistribute and suppress the emergence of topological defects by controlling space-time separated critical fronts, leading to an advantage over standard quantum annealing schedules with respect to both TTS and TTD for finding rare solutions. Using path-integral Monte Carlo simulations for up to 1600 qubits, we demonstrate that nonequilibrium driving of quantum fluctuations, guided by efficient approximate tensor network contractions, can significantly reduce the fraction of hard instances for random frustrated 2D spin glasses with local fields. Specifically, we observe that by creating a class of algorithmic quantum phase transitions, the diversity of solutions can be enhanced by up to 40% with the fraction of hard-to-sample instances reducing by more than 25%.

arXiv:2110.10560•Physical review. E•Dec 11, 2023•8 citations

Quantum combinatorial optimization beyond the variational paradigm: Simple schedules for hard problems

Tim Bode, Krish Ramesh, Tobias Stollenwerk

Advances in quantum algorithms suggest a tentative scaling advantage on certain combinatorial optimization problems. Recent work, however, has also reinforced the idea that barren plateaus render variational algorithms ineffective on large Hilbert spaces. Hence, finding annealing protocols by variation ultimately appears to be difficult. Similarly, the adiabatic theorem fails on hard problem instances with first-order quantum phase transitions. Here we show how to use the spin coherent-state path integral to shape the geometry of quantum adiabatic evolution, leading to annealing protocols at polynomial overhead that provide orders-of-magnitude improvements in the probability to measure optimal solutions, relative to linear protocols. These improvements are not obtained on a controllable toy problem but on randomly generated hard instances (Sherrington-Kirkpatrick and maximum 2-satisfiability), making them generic and robust. Our method works for large systems and may thus be used to improve the performance of state-of-the-art quantum devices.

arXiv:2411.07646•Physical Review A•Mar 11, 2025•0 citations

Nonstoquastic Hamiltonians and quantum annealing of an Ising spin glass

Layla Hormozi, Ethan Brown, Giuseppe Carleo +1 more

Quantum annealers are computing machines that utilize quantum effects to solve hard optimization problems. Understanding the circumstances under which quantum annealers are more efficient than other optimization methods is a challenging open problem. A clue to this conundrum comes from studying the complexity of quantum annealers. In general, the complexity of a physical system relates to how efficiently the system can be simulated by classical algorithms. One then asks: if some class of quantum annealers are hard to simulate classically, are they also more powerful optimization machines? Here, the authors study quantum annealers belonging to two different complexity classes: the so-called ``stoquastic'' systems, which can be efficiently simulated with classical algorithms, and ``nonstoquastic'' systems, currently with no efficient classical treatment. Applied to a prototypical optimization problem, the authors observe that, on average, the stoquastic systems perform better on easier instances of the problem, while the more complex nonstoquastic annealers show significant advantage when applied to the hardest instances. The authors conjecture that the performance of the nonstoquastic Hamiltonians is closely related to their internal structure during the annealing process and elucidate how properties such as magnetic frustration can play a key role in devising more powerful quantum annealers.

arXiv:1609.06558•Physical review. B./Physical review. B•May 15, 2017•97 citations
For many optimization algorithms the time to solution depends not only on the problem size but also on the specific problem instance and may vary by many orders of magnitude. It is then necessary to investigate the full distribution and especially its tail. Here, we analyze the distributions of annealing times for simulated annealing and simulated quantum annealing (by path integral quantum Monte Carlo simulation) for random Ising spin glass instances. We find power-law distributions with very heavy tails, corresponding to extremely hard instances, but far broader distributions-and thus worse performance for hard instances-for simulated quantum annealing than for simulated annealing. Fast, nonadiabatic, annealing schedules can improve the performance of simulated quantum annealing for very hard instances by many orders of magnitude.
The path integral Monte Carlo simulated quantum annealing algorithm is applied to the optimization of a large hard instance of the random satisfiability problem (N = 10,000). The dynamical behavior of the quantum and the classical annealing are compared, showing important qualitative differences in the way of exploring the complex energy landscape of the combinatorial optimization problem. At variance with the results obtained for the Ising spin glass and for the traveling salesman problem, in the present case the linear-schedule quantum annealing performance is definitely worse than classical annealing. Nevertheless, a quantum cooling protocol based on field-cycling and able to outperform standard classical simulated annealing over short time scales is introduced.
Both simulated quantum annealing and physical quantum annealing have shown the emergence of ``heavy tails'' in their performance as optimizers: The total time needed to solve a set of random input instances is dominated by a small number of very hard instances. Classical simulated annealing, in contrast, does not show such heavy tails. Here we explore the origin of these heavy tails, which appear for inputs with high local degeneracy---large isoenergetic clusters of states in Hamming space. This category includes the low-precision Chimera-structured problems studied in recent benchmarking work comparing the D-Wave Two quantum annealing processor with simulated annealing. On similar inputs designed to suppress local degeneracy, performance of a quantum annealing processor on hard instances improves by orders of magnitude at the 512-qubit scale, while classical performance remains relatively unchanged. Simulations indicate that perturbative crossings are the primary factor contributing to these heavy tails, while sensitivity to Hamiltonian misspecification error plays a less significant role in this particular setting.
Classical and quantum annealing are two heuristic optimization methods that search for an optimal solution by slowly decreasing thermal or quantum fluctuations. Optimizing annealing schedules is important both for performance and fair comparisons between classical annealing, quantum annealing, and other algorithms. Here we present a heuristic approach for the optimization of annealing schedules for quantum annealing and apply it to 3D Ising spin glass problems. We find that if both classical and quantum annealing schedules are similarly optimized, classical annealing outperforms quantum annealing for these problems when considering the residual energy obtained in slow annealing. However, when performing many repetitions of fast annealing, simulated quantum annealing is seen to outperform classical annealing for our benchmark problems.
Sampling a diverse set of high-quality solutions for hard optimization problems is of great practical relevance in many scientific disciplines and applications, such as artificial intelligence and operations research. One of the main open problems is the lack of ergodicity, or mode collapse, for typical stochastic solvers based on Monte Carlo techniques leading to poor generalization or lack of robustness to uncertainties. Currently, there is no universal metric to quantify such performance deficiencies across various solvers. Here, we introduce a new diversity measure for quantifying the number of independent approximate solutions for NP-hard optimization problems. Among others, it allows benchmarking solver performance by a required time-to-diversity (TTD), a generalization of often used time-to-solution (TTS). We illustrate this metric by comparing the sampling power of various quantum annealing strategies. In particular, we show that the inhomogeneous quantum annealing schedules can redistribute and suppress the emergence of topological defects by controlling space-time separated critical fronts, leading to an advantage over standard quantum annealing schedules with respect to both TTS and TTD for finding rare solutions. Using path-integral Monte Carlo simulations for up to 1600 qubits, we demonstrate that nonequilibrium driving of quantum fluctuations, guided by efficient approximate tensor network contractions, can significantly reduce the fraction of hard instances for random frustrated 2D spin glasses with local fields. Specifically, we observe that by creating a class of algorithmic quantum phase transitions, the diversity of solutions can be enhanced by up to 40% with the fraction of hard-to-sample instances reducing by more than 25%.
Advances in quantum algorithms suggest a tentative scaling advantage on certain combinatorial optimization problems. Recent work, however, has also reinforced the idea that barren plateaus render variational algorithms ineffective on large Hilbert spaces. Hence, finding annealing protocols by variation ultimately appears to be difficult. Similarly, the adiabatic theorem fails on hard problem instances with first-order quantum phase transitions. Here we show how to use the spin coherent-state path integral to shape the geometry of quantum adiabatic evolution, leading to annealing protocols at polynomial overhead that provide orders-of-magnitude improvements in the probability to measure optimal solutions, relative to linear protocols. These improvements are not obtained on a controllable toy problem but on randomly generated hard instances (Sherrington-Kirkpatrick and maximum 2-satisfiability), making them generic and robust. Our method works for large systems and may thus be used to improve the performance of state-of-the-art quantum devices.
Quantum annealers are computing machines that utilize quantum effects to solve hard optimization problems. Understanding the circumstances under which quantum annealers are more efficient than other optimization methods is a challenging open problem. A clue to this conundrum comes from studying the complexity of quantum annealers. In general, the complexity of a physical system relates to how efficiently the system can be simulated by classical algorithms. One then asks: if some class of quantum annealers are hard to simulate classically, are they also more powerful optimization machines? Here, the authors study quantum annealers belonging to two different complexity classes: the so-called ``stoquastic'' systems, which can be efficiently simulated with classical algorithms, and ``nonstoquastic'' systems, currently with no efficient classical treatment. Applied to a prototypical optimization problem, the authors observe that, on average, the stoquastic systems perform better on easier instances of the problem, while the more complex nonstoquastic annealers show significant advantage when applied to the hardest instances. The authors conjecture that the performance of the nonstoquastic Hamiltonians is closely related to their internal structure during the annealing process and elucidate how properties such as magnetic frustration can play a key role in devising more powerful quantum annealers.