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arXiv:1801.02005 · Journal of the Physical Society of Japan

Exponential Speedup of Quantum Annealing by Inhomogeneous Driving of the Transverse Field

Yuki Susa, Yu Yamashiro, Masayuki Yamamoto, Hidetoshi Nishimori

Published Jan 17, 2018• 75 citations
arXivHTMLPDF

Abstract

We show, for quantum annealing, that a certain type of inhomogeneous driving of the transverse field erases first-order quantum phase transitions in the p-body interacting mean-field-type model with and without longitudinal random field. Since a first-order phase transition poses a serious difficulty for quantum annealing (adiabatic quantum computing) due to the exponentially small energy gap, the removal of first-order transitions means an exponential speedup of the annealing process. The present method may serve as a simple protocol for the performance enhancement of quantum annealing, complementary to non-stoquastic Hamiltonians.

DOI
10.7566/jpsj.87.023002

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We show, for quantum annealing, that a certain type of inhomogeneous driving of the transverse field erases first-order quantum phase transitions in the p-body interacting mean-field-type model with and without longitudinal random field. Since a first-order phase transition poses a serious difficulty for quantum annealing (adiabatic quantum computing) due to the exponentially small energy gap, the removal of first-order transitions means an exponential speedup of the annealing process. The present method may serve as a simple protocol for the performance enhancement of quantum annealing, complementary to non-stoquastic Hamiltonians.
Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.
We solve the mean-field-like ppp-spin Ising model under a spatiotemporal inhomogeneous transverse field to study the effects of inhomogeneity on the performance of quantum annealing. We previously found that the problematic first-order quantum phase transition that arises under the conventional homogeneous field protocol can be avoided if the temperature is zero and the local field is completely turned off site by site after a finite time. We show in the present paper that, when these ideal conditions are not satisfied, another series of first-order transitions appear, which prevents us from driving the system while avoiding first-order transitions. Nevertheless, under these nonideal conditions, quantitative improvements can be obtained in terms of narrower tunneling barriers in the free-energy landscape. A comparison with classical simulated annealing establishes a limited quantum advantage in the ideal case, since inhomogeneous temperature driving in simulated annealing cannot remove a first-order transition, in contrast to the quantum case. The classical model of spin-vector Monte Carlo is also analyzed, and we find it to have the same thermodynamic phase diagram as the quantum model in the ideal case, with deviations arising at nonzero temperature.
We study the energy gap between the ground state and the first excited state of a mean-field-type non-stoquastic Hamiltonian by a semi-classical analysis. The fully connected mean-field model with ppp-body ferromagnetic interactions under a transverse field has a first-order quantum phase transition for p≥3p\ge 3p≥3. This first-order transition is known to be reduced to second order for p≥5p\ge 5p≥5 by an introduction of antiferromagnetic transverse interactions, which makes the Hamiltonian non-stoquastic. This reduction of the order of transition means an exponential speedup of quantum annealing by adiabatic processes because the first-order transition is shown to have an exponentially small energy gap whereas the second order case does not. We apply a semi-classical method to analytically derive the explicit expression of the rate of the exponential decay of the energy gap at first-order transitions. The result reveals how the property of first-order transition changes as a function of the system parameters. We also derive the exact closed-form expression for the critical point where the first-order transition line disappears within the ferromagnetic phase. These results help us understand how the antiferromagnetic transverse interactions affect the performance of quantum annealing by controlling the effects of non-stoquasticity in the Hamiltonian.
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 NNN. 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 NNN 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.
We study the weak-strong cluster problem for quantum annealing in its mean-field version as proposed by Albash [Phys. Rev. A 99 (2019) 042334] who showed by numerical diagonalization that non-stoquastic XXXXXX interactions (non-stoquastic catalysts) remove the problematic first-order phase transition. We solve the problem exactly in the thermodynamic limit by analytical methods and show that the removal of the first-order transition is successfully achieved either by stoquastic or non-stoquastic XXXXXX interactions depending on whether the XXXXXX interactions are introduced within the weak cluster, within the strong cluster, or between them. We also investigate the case where the interactions between the two clusters are sparse, i.e. not of the mean-field all-to-all type. The results again depend on where to introduce the XXXXXX interactions. We further analyze how inhomogeneous driving of the transverse field affects the performance of the system without XXXXXX interactions and find that inhomogeneity in the transverse field removes the first-order transition if appropriately implemented.
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.