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arXiv:1203.2418 · Physical Review E

Quantum annealing with antiferromagnetic fluctuations

Yuya Seki, Hidetoshi Nishimori

Published May 10, 2012• 134 citations
arXivHTMLPDF

Abstract

We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties in finding the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of p. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of p for which the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of p efficiently, in contrast to the case with only simple transverse-field fluctuations.

DOI
10.1103/physreve.85.051112

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We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties in finding the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of p. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of p for which the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of p efficiently, in contrast to the case with only simple transverse-field fluctuations.
We investigate quantum annealing (QA) with antiferromagnetic transverse interactions for the generalized Hopfield model with k -body interactions. The goal is to study the effectiveness of antiferromagnetic interactions, which were shown to help us avoid problematic first-order quantum phase transitions in pure ferromagnetic systems, in random systems. We estimate the efficiency of QA by analyzing phase diagrams for two cases where the number of embedded patterns is finite or extensively large. The phase diagrams of the model with finite patterns show that there exist annealing paths that avoid first-order transitions at least for . The same is true for the extensive case with k = 4 and 5. In contrast, it is impossible to avoid first-order transitions for the case of finite patterns with k = 3 and the case of extensive number of patterns with k = 2 and 3. The spin-glass phase hampers the QA process in the case of k = 2 and extensive patterns. These results indicate that QA with antiferromagnetic transverse interactions is efficient also for certain random spin systems.
We study the relation between quantum fluctuations and the significant enhancement of the performance of quantum annealing in a mean-field Hamiltonian. First-order quantum phase transitions were shown to be reduced to second order by antiferromagnetic transverse interactions in a mean-field-type many-body-interacting Ising spin system in a transverse field, which means an exponential speedup of quantum annealing by adiabatic quantum computation. We investigate if and how quantum effects manifest themselves around these first- and second-order phase transitions to understand if the antiferromagnetic transverse interactions appended to the conventional transverse-field Ising model induce notable quantum effects. By measuring the proximity of the semiclassical spin-coherent state to the true ground state as well as the magnitude of the concurrence representing entanglement, we conclude that significant quantum fluctuations exist around second-order transitions, whereas quantum effects are much less prominent at first-order transitions. Although the location of the transition point can be predicted by the classical picture, system properties near the transition need quantum-mechanical descriptions for a second-order transition but not necessarily for first order. It is also found that quantum fluctuations are large within the ferromagnetic phase after a second-order transition from the paramagnetic phase. These results suggest that the antiferromagnetic transverse interactions induce marked quantum effects, and this fact would be related to closely to the significant enhancement of the performance of quantum annealing.
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 to
The performance of open-system quantum annealing is adversely affected by thermal excitations out of the ground state. While the presence of energy gaps between the ground and excited states suppresses such excitations, error correction techniques are required to ensure full scalability of quantum annealing. Quantum annealing correction (QAC) is a method that aims to improve the performance of quantum annealers when control over only the problem (final) Hamiltonian is possible, along with decoding. Building on our earlier work [S. Matsuura et al., Phys. Rev. Lett. 116, 220501 (2016)], we study QAC using analytical tools of statistical physics by considering the effects of temperature and a transverse field on the penalty qubits in the ferromagnetic ppp-body infinite-range transverse-field Ising model. We analyze the effect of QAC on second (p=2p=2p=2) and first ($p\ensuremath{\ge}3$) order phase transitions, and construct the phase diagram as a function of temperature and penalty strength. Our analysis reveals that for sufficiently low temperatures and in the absence of a transverse field on the penalty qubit, QAC breaks up a single, large free-energy barrier into multiple smaller ones. We find theoretical evidence for an optimal penalty strength in the case of a transverse field on the penalty qubit, a feature observed in QAC experiments. Our results provide further compelling evidence that QAC provides an advantage over unencoded quantum annealing.
Reverse annealing is a variant of quantum annealing that starts from a given classical configuration of spins (qubits). In contrast to the conventional formulation, where one starts from a uniform superposition of all possible states (classical configurations), quantum fluctuations are first increased and only then decreased. One then reads out the state as a proposed solution to the given combinatorial optimization problem. We formulate a mean-field theory of reverse annealing using the fully connected ferromagnetic ppp-spin model, with and without random longitudinal fields, and analyze it in order to understand how and when reverse annealing is effective at solving this problem. We find that the difficulty arising from the existence of a first-order quantum phase transition, which leads to an exponentially long computation time in conventional quantum annealing, is circumvented in the context of this particular problem by reverse annealing if the proximity of the initial state to the (known) solution exceeds a threshold. Even when a first-order transition is unavoidable, the difficulty is mitigated due to a smaller jump in the order parameter at a first-order transition, which implies a larger rate of quantum tunneling. Reverse annealing has not been studied analytically before, and this study paves the way toward a systematic understanding of this relatively unexplored protocol in a broader context.
This paper deals with fully-connected mean-field models of quantum spins with p-body ferromagnetic interactions and a transverse field. For p=2 this corresponds to the quantum Curie-Weiss model (a special case of the Lipkin-Meshkov-Glick model) which exhibits a second-order phase transition, while for p>2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the first-order transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the first-order transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.