Control of quantum phase transitions in mean-field models for quantum annealing

Inhomogeneous quantum annealing (IQA), in which transverse fields are turned off one by one rather than simultaneously, has been proposed as an effective way to avoid the first-order phase transitions that impede conventional quantum annealing (QA). Here we explicitly study the dynamics of IQA, rather than merely the thermodynamics, and find that it is appreciably slower than the phase diagram would suggest. Interestingly, this slowdown manifests both when IQA succeeds in circumventing phase transitions and when it fails. Even in the absence of transitions, such as for the mean-field models that have been analyzed previously, IQA is slower than expected by a factor of the number of spins $N$ . More significantly, we show that in non-mean-field models, first-order transitions are likely to be quite common, and the gap at such transitions is not merely exponential in $N$ but exactly zero. Thus IQA cannot reach the ground state on any timescale. Both of these results can be understood through the simple observation that a spin's magnetization becomes conserved once its field is turned off during the IQA protocol.Inhomogeneous quantum annealing (IQA), in which transverse fields are turned off one by one rather than simultaneously, has been proposed as an effective way to avoid the first-order phase transitions that impede conventional quantum annealing (QA). Here we explicitly study the dynamics of IQA, rather than merely the thermodynamics, and find that it is appreciably slower than the phase diagram would suggest. Interestingly, this slowdown manifests both when IQA succeeds in circumventing phase transitions and when it fails. Even in the absence of transitions, such as for the mean-field models that have been analyzed previously, IQA is slower than expected by a factor of the number of spins $N$. More significantly, we show that in non-mean-field models, first-order transitions are likely to be quite common, and the gap at such transitions is not merely exponential in $N$ but exactly zero. Thus IQA cannot reach the ground state on any timescale. Both of these results can be understood through the simple observation that a spin's magnetization becomes conserved once its field is turned off during the IQA protocol.

arXiv:2502.03535Physical Review AJun 2, 20251 citation

平均場模型の量子アニーリングにおける量子相転移の制御

優也関, Yuya Seki

The present thesis studies statistical-mechanical properties of mean-field models to reveal an effect of various types of quantum fluctuations against the efficiency of quantum annealing.Quantum annealing is a quantum computation model to search the ground state of Ising spin systems by taking advantage of quantum fluctuations.The quantum fluctuations are usually induced by the transverse-field term.However, it has been reported that the quantum annealing with the transverse-field term has a difficulty of exponentially long running time for certain problems.For example, quantum annealing cannot find even the ground state of a simple ferromagnetic system.The difficulty is closely related to the phenomena of quantum phase transitions.Quantum annealing typically requires exponentially long time to find the ground state of a system, if the system undergoes a first-order quantum phase transition in the thermodynamics limit.Hence, avoiding firstorder quantum phase transitions is a critical issue for quantum annealing.A solution to avoid the difficulty mentioned above is to use a degree of freedom of quantum annealing.Although many studies adopt the transverse-field term to induce quantum fluctuations, there is no restriction to use the term.The present thesis shows that the difficulty for the simple ferromagnetic system can be avoided by using transverse antiferromagnetic interactions.Furthermore, the antiferromagnetic term turns out to be also effective for a random spin system.We next focus on the macroscopic degeneracy of the ground states of the antiferromagnetic term.In order to reveal the effect of the macroscopic degeneracy, the Wajnflasz-Pick model is analyzed.The Wajnflasz-Pick model is interesting also in terms of providing an alternative way to control the order of quantum phase transitions.iii I would like to thank my supervisor Prof. Hidetoshi Nishimori for his guidance and revisions of the thesis and the papers we co-authored.The studies in the present thesis could not have been accomplished without his insightful advice.I am also grateful to Dr. Kazutaka Takahashi for fruitful discussions.I would like toThe present thesis studies statistical-mechanical properties of mean-field models to reveal an effect of various types of quantum fluctuations against the efficiency of quantum annealing.Quantum annealing is a quantum computation model to search the ground state of Ising spin systems by taking advantage of quantum fluctuations.The quantum fluctuations are usually induced by the transverse-field term.However, it has been reported that the quantum annealing with the transverse-field term has a difficulty of exponentially long running time for certain problems.For example, quantum annealing cannot find even the ground state of a simple ferromagnetic system.The difficulty is closely related to the phenomena of quantum phase transitions.Quantum annealing typically requires exponentially long time to find the ground state of a system, if the system undergoes a first-order quantum phase transition in the thermodynamics limit.Hence, avoiding firstorder quantum phase transitions is a critical issue for quantum annealing.A solution to avoid the difficulty mentioned above is to use a degree of freedom of quantum annealing.Although many studies adopt the transverse-field term to induce quantum fluctuations, there is no restriction to use the term.The present thesis shows that the difficulty for the simple ferromagnetic system can be avoided by using transverse antiferromagnetic interactions.Furthermore, the antiferromagnetic term turns out to be also effective for a random spin system.We next focus on the macroscopic degeneracy of the ground states of the antiferromagnetic term.In order to reveal the effect of the macroscopic degeneracy, the Wajnflasz-Pick model is analyzed.The Wajnflasz-Pick model is interesting also in terms of providing an alternative way to control the order of quantum phase transitions.iii I would like to thank my supervisor Prof. Hidetoshi Nishimori for his guidance and revisions of the thesis and the papers we co-authored.The studies in the present thesis could not have been accomplished without his insightful advice.I am also grateful to Dr. Kazutaka Takahashi for fruitful discussions.I would like to

Mar 1, 20160 citations

QUANTUM ANNEALING WITH ANTIFERROMAGNETIC FLUCTUATIONS FOR MEAN-FIELD MODELS

Yuya Seki, Hidetoshi Nishimori

Apr 1, 20141 citation

Mean-field analysis of quantum annealing with XX-type terms

Hidetoshi Nishimori

Bulletin of the American Physical SocietyMar 14, 20160 citations

N

. More significantly, we show that in non-mean-field models, first-order transitions are likely to be quite common, and the gap at such transitions is not merely exponential in

N

but exactly zero. Thus IQA cannot reach the ground state on any timescale. Both of these results can be understood through the simple observation that a spin's magnetization becomes conserved once its field is turned off during the IQA protocol.