Relation between quantum fluctuations and the performance enhancement of quantum annealing in a nonstoquastic Hamiltonian

Abstract

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.

Improving quantum annealing of the ferromagnetic p -spin model through pausing

“…The latter reduces the efficiency of adiabatic computations as the minimal gap ∆ closes exponentially as a function of n [42]. Several techniques are known to mitigate the detrimental effects of first-order QPTs in the quantum annealing of the p-spin model, such as non-stoquastic AQC [43,[51][52][53], inhomogeneous driving [44,54], or reverse annealing [29,45,[55][56][57]. We will not discuss these techniques here.…”

May a Dissipative Environment Be Beneficial for Quantum Annealing?

“…For p > 3, the introduction of a non-stoquastic potential in the Hamiltonian is expected to turn the first-order QPT into a second-order QPT, where the gap closes polynomially as a function of the system size. This procedure does not hold for p = 3 [6]. We measure energies in units E and times in units E −1 , and we set Γ = E. With this choice, the minimal gap is equal to ∆ ≈ 0.47E.…”

Exponential Enhancement of the Efficiency of Quantum Annealing by Non-Stoquastic Hamiltonians

“…We are now investigating this problem, and the result will soon be published (Susa et al, 2016). One of the hints may lie in the sign of coefficients of the ground-state wave function in the standard computational basis.…”

“…The answer is positive as far as the properties of phase transitions are concerned: Jörg et al (2010) used full quantum statistical-mechanical tools to reach the same conclusion as above. Quantum effects should be carefully taken into account if one wishes to fully understand the behavior of the energy gap for finite-size systems, as was done by Jörg et al (2010), and to describe more subtle properties of the system around the phase transition and within the ferromagnetic phase (Susa et al, 2016). However, the classical analysis is sufficient to predict the type of phase transitions in the thermodynamic limit.…”