© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Cambridge University Press eBooks

Adiabatic Quantum Computations and Transverse Field Models

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti, Uma Divakaran, T. F. Rosenbaum, Diptiman Sen

Published Jan 28, 2015• 0 citations

Abstract

Multivariable optimization problems belong to an interdisciplinary field of research connecting diverse fields like computer science, information theory and statistical physics. It involves finding an optimal solution out of the many available states or configurations, possibly satisfying a set of constraints. For physicists, the problem can be visualized as finding the global minima of the energy landscape of a given complex Hamiltonian, which is quite difficult to obtain by the obvious method of examining the energy of each state. This is due to the huge number of available states which grows exponentially with the system size (for N Ising spins, the configuration space grows as 2 N ). The search for such a global minimum becomes all the more complex when the landscape consists of several local minima separated by macroscopic barriers ( O ( N )) arising from frustrating constraints in the systems. To solve such an optimization problem in a stochastic case, the concept of simulated annealing (SA) was put forward by Kirkpatrick et al . [437]. In this method, a tunable noise is artificially induced (thermal-like fluctuations equivalent to that of a metallurgical annealing) which helps the system to escape from the local minima by hopping above the energy barriers, and staying longer in minimum energy valley(s) as the noise variable is slowly reduced to zero starting from a high value. At the same time, it is also known that in complex systems, due to many closely spaced low lying states, the annealing needs to be performed infinitely slowly to reach the true ground state; this is indeed a major drawback. We shall elaborate on this point later when we discuss the effect of system size on the minimum gap between the ground state and the first excited state of the quantum many body system under consideration. The numerical methods used in such studies are variants of Monte Carlo methods.

DOI: 10.1017/cbo9781107706057.016

Related papers

Hybrid annealing: Coupling a quantum simulator to a classical computer

Tobias Graß, Maciej Lewenstein

Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Annealing strategies, either classical or quantum, explore the configuration space by evolving the system under the influence of thermal or quantum fluctuations. The thermal annealing dynamics can rapidly freeze the system into a low-energy configuration, and it can be simulated well on a classical computer, but it easily gets stuck in local minima. Quantum annealing, on the other hand, can be guaranteed to find the true ground state and can be implemented in modern quantum simulators; however, quantum adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here, we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such a hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasidegenerate ground states.

arXiv:1611.09729•Physical Review A•May 3, 2017•11 citations

A Tour of Adiabatic Quantum Computing

Nicholas R. Allgood

Different types and approaches for using quantum mechanical events for computation have developed quickly with the quantum technology revolution. One of these, adiabatic quantum computing, uses the adiabatic theorem as a paradigm for computation. According to the adiabatic theorem, if conditions gradually change and a system is permitted to adjust its configuration (changing the probability density), the associated eigenstate will be of the final Hamiltonian if the system starts in an eigenstate of its initial Hamiltonian. Simply put, a quantum mechanical system that is subjected to rapidly changing and varying conditions, there isn’t enough time for a function to adapt which leaves the spatial probability density unchanged. A closely related optimization process that utilizes the adiabatic theorem known as quantum annealing, finds a global minimum over a candidate set of solutions by processing quantum fluctuations. Quantum annealing is used primarily in combinatorial optimization problems where the search space is large and discrete, containing multiple local minima.

IntechOpen eBooks•Sep 11, 2023•0 citations

Computational Bottlenecks of Quantum Annealing

Sergey Knysh

A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of $N$ qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by $O(\log N)$ bottlenecks inside the spin glass phase, where the gap scales as a stretched exponential. For smaller $N$, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield Model. Qualitative comparison with the Sherrington-Kirkpatrick Model leads to similar conclusions.

arXiv:1506.08608•Jun 29, 2015•3 citations

Adiabatic optimization without local minima

Michael Jarret, Stephen P. Jordan

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two ``lobes'' separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.

Quantum Information and Computation•Mar 1, 2015•3 citations

Adiabatic optimization without local minima

Michael Jarret, Stephen P. Jordan

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wave-function consists of two lobes separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.

Quantum Information and Computation•Mar 1, 2015•6 citations

Adiabatic optimization without local minima

Michael Jarret, Stephen P. Jordan

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two "lobes" separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.

arXiv:1405.7552•May 29, 2014•0 citations

Multivariable optimization problems belong to an interdisciplinary field of research connecting diverse fields like computer science, information theory and statistical physics. It involves finding an optimal solution out of the many available states or configurations, possibly satisfying a set of constraints. For physicists, the problem can be visualized as finding the global minima of the energy landscape of a given complex Hamiltonian, which is quite difficult to obtain by the obvious method of examining the energy of each state. This is due to the huge number of available states which grows exponentially with the system size (for N Ising spins, the configuration space grows as 2 N ). The search for such a global minimum becomes all the more complex when the landscape consists of several local minima separated by macroscopic barriers ( O ( N )) arising from frustrating constraints in the systems. To solve such an optimization problem in a stochastic case, the concept of simulated annealing (SA) was put forward by Kirkpatrick et al . [437]. In this method, a tunable noise is artificially induced (thermal-like fluctuations equivalent to that of a metallurgical annealing) which helps the system to escape from the local minima by hopping above the energy barriers, and staying longer in minimum energy valley(s) as the noise variable is slowly reduced to zero starting from a high value. At the same time, it is also known that in complex systems, due to many closely spaced low lying states, the annealing needs to be performed infinitely slowly to reach the true ground state; this is indeed a major drawback. We shall elaborate on this point later when we discuss the effect of system size on the minimum gap between the ground state and the first excited state of the quantum many body system under consideration. The numerical methods used in such studies are variants of Monte Carlo methods.

Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Annealing strategies, either classical or quantum, explore the configuration space by evolving the system under the influence of thermal or quantum fluctuations. The thermal annealing dynamics can rapidly freeze the system into a low-energy configuration, and it can be simulated well on a classical computer, but it easily gets stuck in local minima. Quantum annealing, on the other hand, can be guaranteed to find the true ground state and can be implemented in modern quantum simulators; however, quantum adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here, we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such a hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasidegenerate ground states.

Different types and approaches for using quantum mechanical events for computation have developed quickly with the quantum technology revolution. One of these, adiabatic quantum computing, uses the adiabatic theorem as a paradigm for computation. According to the adiabatic theorem, if conditions gradually change and a system is permitted to adjust its configuration (changing the probability density), the associated eigenstate will be of the final Hamiltonian if the system starts in an eigenstate of its initial Hamiltonian. Simply put, a quantum mechanical system that is subjected to rapidly changing and varying conditions, there isn’t enough time for a function to adapt which leaves the spatial probability density unchanged. A closely related optimization process that utilizes the adiabatic theorem known as quantum annealing, finds a global minimum over a candidate set of solutions by processing quantum fluctuations. Quantum annealing is used primarily in combinatorial optimization problems where the search space is large and discrete, containing multiple local minima.

A promising approach to solving hard binary optimisation problems is quantum adiabatic annealing (QA) in a transverse magnetic field. An instantaneous ground state --- initially a symmetric superposition of all possible assignments of

N

qubits --- is closely tracked as it becomes more and more localised near the global minimum of the classical energy. Regions where the energy gap to excited states is small (e.g. at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by

O(\log N)

bottlenecks inside the spin glass phase, where the gap scales as a stretched exponential. For smaller

N

, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield Model. Qualitative comparison with the Sherrington-Kirkpatrick Model leads to similar conclusions.

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two ``lobes'' separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wave-function consists of two lobes separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.

Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two "lobes" separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.