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Edited by: Itay Hen, University of Southern California, USA

Reviewed by: Faisal Shah Khan, Khalifa University, United Arab Emirates; Evgeny Andriyash, D-Wave Systems Inc., Canada; Layla Hormozi, Massachusetts Institute of Technology, USA

Specialty section: This article was submitted to Quantum Computing, a section of the journal Frontiers in ICT

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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

Quantum annealing is a metaheuristic for combinatorial optimization problems (Kadowaki,

To numerically test the performance of quantum annealing, one often uses quantum Monte Carlo simulation, which is a classical algorithm to sample the equilibrium distribution of the transverse-field Ising model. Although the quantum Monte Carlo simulation is designed to sample the equilibrium Boltzmann distribution, it has been found that some aspects of dynamics of quantum annealing can also be described by quantum Monte Carlo simulations (Isakov et al.,

Related to the above observation is the concept of stoquastic Hamiltonians (Bravyi et al., ^{1}

There exist several studies related to this idea. Farhi et al. (

The present article describes the findings in Seki and Nishimori (

The first problem to be discussed is the ferromagnetic

In the present section, we consider the case with

Simulated annealing is a classical heuristic to sample the Boltzmann distribution with the temperature decreasing from a very high value to zero (Kirkpatrick et al.,

As shown in Figure

What will happen if we apply quantum annealing to the same problem? The conventional choice of the transverse-field Ising model for quantum annealing has the Hamiltonian

For our problem Hamiltonian of equation (_{x}, m_{y}_{z}_{x}, m_{y}_{z}

As one sees in Figure _{0}. The authors of Jörg et al. (

One may wonder if the above analysis using a classical vector would properly describe the essential features of quantum annealing under the Hamiltonian equation (

It is also worth noticing that the performance of quantum annealing is comparable to that of simulated annealing discussed in the previous section, both of which should spend an exponentially long time to reach the ground state. In this sense, there is no ‘limited quantum speedup’ in the present case, according to the classification of Rønnow et al. (

We next study the non-stoquastic case with the Hamiltonian

Quantum annealing starts at

The analysis proceeds as before by the replacement of the normalized total spin operator with a classical unit vector. The energy per site is

Two typical examples of the behavior of this energy at

Comparison of Figures

^{+} (quantum paramagnetic) and F′ (ferromagnetic) phases is for second-order phase transitions, and all other boundaries represent first-order transitions. A new phase QP^{−} exists for ^{+} phase. The two ferromagnetic phases F and F′ are clearly separated for

The case of

The case of ^{+} and QP^{−}. The former has θ = 0 with magnetization vector ^{−} phase appears at the top left corner of the phase diagram, where the antiferromagnetic multiple-_{x}^{k}^{+} and F′ phases. The first-order transition line within the ferromagnetic phase between F and F′ extends toward QP^{−} at the upper left part of the phase diagram. For larger ^{−} phase as seen for

Although the above analyses use only classical variables, it has been shown that quantum statistical–mechanical computations reproduce those phase boundaries quantitatively very faithfully in the thermodynamic limit (Seki and Nishimori,

It is important to study how the environment affects the behavior of the system. One of the standard models to describe the interactions of the system with its environment is the following Hamiltonian (Leggett et al., _{l}_{l}

Following the previous analysis (see also Sinha and Dattagupta, _{α}_{c}

Equation (_{α}^{2}. If we consider for simplicity the case of

One may wonder if the above results for the

The Hopfield model with

It is impossible to apply the simple classical method used in the

The parameter ^{x}

The energy of equation (^{x}^{x}

When only

When

When ^{x}

Extremization conditions of ^{x}^{2}

We next discuss the case of ^{3}

^{3}

We have therefore established that the antiferromagnetic multiple-

We have shown that antiferromagnetic multiple-

We have also shown for the ferromagnetic

It should be remembered that antiferromagnetic multiple-

One may wonder if there is any other way to show that an antiferromagnetic multiple-

It is an interesting question how far the present results for the infinite-range fully connected models apply to more realistic problems with relatively sparse connections, e.g., a problem on a finite-dimensional lattice with short-range interactions. It is of course difficult to say something with confidence without explicit evidence. Nevertheless, our experience in the physics of phase transitions suggests that a mean-field analysis often provides reliable results also for finite-dimensional systems with a finite number of connections per site as far as qualitative descriptions are concerned (Nishimori and Ortiz,

HN played a major role in conducting the research described in this article. KT contributed to section

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Most of the technical results described in this article appeared in Seki and Nishimori (

^{1}It is to be noticed that, in some cases, it is non-trivial to efficiently simulate a stoquastic Hamiltonian. See, for example, Hastings and Freedman (

^{2}It has been pointed out in Knysh (

^{3}Notice that the replica symmetric ansatz (Nishimori,