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 nonstoquastic 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 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 XX 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 XX interactions depending on whether the XX 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 XX interactions. We further analyze how inhomogeneous driving of the transverse field affects the performance of the system without XX interactions and find that inhomogeneity in the transverse field removes the first-order transition if appropriately implemented.

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We study the critical properties of finite-dimensional dissipative quantum spin systems with uniform ferromagnetic interactions. Starting from the transverse-field Ising model coupled to a bath of harmonic oscillators with Ohmic spectral density, we generalize its classical representation to classical spin systems with O(n) symmetry and then take the large-n limit to reduce the system to the spherical model. The exact solution to the resulting spherical model with long-range interactions along the imaginary-time axis shows a phase transition with static critical exponents coinciding with those of the conventional short-range spherical model in d + 2 dimensions, where d is the spatial dimensionality of the original quantum system. This implies the dynamical exponent to be z = 2. These conclusions are consistent with the results of Monte Carlo simulations and renormalization group calculations for dissipative transverse-field Ising and O(n) models in one and two dimensions. The present approach therefore serves as a useful tool to analytically investigate the properties of quantum phase transitions of the dissipative transverse-field Ising and related models. Our method may also offer a platform to study more complex phase transitions in dissipative finite-dimensional quantum spin systems, which recently receive renewed interest under the context of quantum annealing in a noisy environment.2