Energy Gap at First-Order Quantum Phase Transitions: An Anomalous Case

Tsuda, Junichi; Yamanaka, Yuuki; Nishimori, Hidetoshi

doi:10.7566/jpsj.82.114004

Cited by 26 publications

(29 citation statements)

References 18 publications

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“…As a side remark, we mention that in the {Γ, κ} parameter space, for very small κ ∼ 0 the gap closes very fast to be factorially or exponentially small, without any associated phase transition -this anomalous behaviour has been documented previously in Ref. 24 and relates to the fact a finite number of spins n = 2j cannot exactly represent an irrational value of the variable z = m/j. (In the current context this is not an interesting limit because the problem Hamiltonian vanishes when κ = 0.)…”

Section: Adiabatic Evolution Across the Softened Phase Transitionsupporting

confidence: 62%

“…From FIG.10 it's seen that reducing κ also lowers the barrier, it becomes completely suppressed only for κ = 0. This limit, however, brings us back to the anomalous case discussed earlier 24 . How does the optimal non-stoquastic driver contribution κ c scale with the system size n = 2j?…”

Section: Adiabatic Evolution Across the Softened Phase Transitionmentioning

confidence: 79%

“…The optimal gap (black dots) eventually converges on the asymptote (cyan line) predicted by the continuous model, eqn. (24). The rate of gap closure is always faster than that of the Lipkin Meshkov Glick (2-spin) model 12,13 ∆c ∝ j −4/3 , a continuous phase transition.…”

Section: T * : Time To Solution Via Optimal Pathmentioning

confidence: 92%

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Quantum speedup at zero temperature via coherent catalysis

Durkin

2019

Phys. Rev. A

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It is known that secondary non-stoquastic drivers may offer speed-ups or catalysis in some models of adiabatic quantum computation accompanying the more typical transverse field driver. Their combined intent is to raze potential barriers to zero during adiabatic evolution from a false vacuum to a true minimum; first order phase transitions are softened into second order transitions. We move beyond mean-field analysis to a fully quantum model of a spin ensemble undergoing adiabatic evolution in which the spins are mapped to a variable mass particle in a continuous one-dimensional potential. We demonstrate the necessary criteria for enhanced mobility or 'speed-up' across potential barriers is actually a quantum form of the Rayleigh criterion. Quantum catalysis is exhibited in models where previously thought not possible, when barriers cannot be eliminated. For the 3-spin model with secondary anti-ferromagnetic driver, catalysed time complexity scales between linear and quadratically with the number of qubits. As a corollary, we identify a useful resonance criterion for quantum phase transition that differs from the classical one, but converges on it, in the thermodynamic limit.

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Section: Adiabatic Evolution Across the Softened Phase Transitionsupporting

confidence: 62%

Section: Adiabatic Evolution Across the Softened Phase Transitionmentioning

confidence: 79%

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Quantum speedup at zero temperature via coherent catalysis

Durkin

2019

Phys. Rev. A

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“…It was shown explicitly that reverse annealing can turn first order quantum phase transitions into second order transitions in a simple problem, the p-spin model, by choosing an appropriate annealing path. It is generally the case that a first-order quantum phase transition is associated with an exponentially closing energy gap ∆ between the ground state and the first excited state as a function of system size, whereas a second order transition is associated with a polynomially closing gap (though exceptions are known [17][18][19]). On the other hand, according to the adiabatic theorem of quantum mechanics, the computation time τ is inversely proportional to a small power of the energy gap ∆ [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of reverse annealing for the fully connected p -spin model

Yamashiro

Ohkuwa²,

Nishimori

et al. 2019

Phys. Rev. A

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Reverse annealing is a relatively new variant of quantum annealing, in which one starts from a classical state and increases and then decreases the amplitude of the transverse field, in the hope of finding a better classical state than the initial state for a given optimization problem. We numerically study the unitary quantum dynamics of reverse annealing for the mean-field-type p-spin model and show that the results are consistent with the predictions of equilibrium statistical mechanics. In particular, we corroborate the equilibrium analysis prediction that reverse annealing provides an exponential speedup over conventional quantum annealing in terms of solving the p-spin model. This lends support to the expectation that equilibrium analyses are effective at revealing essential aspects of the dynamics of quantum annealing. We also compare the results of quantum dynamics with the corresponding classical dynamics, to reveal their similarities and differences. We distinguish between two reverse annealing protocols we call adiabatic and iterated reverse annealing. We further show that iterated reverse annealing, as has been realized in the D-Wave device, is ineffective in the case of the p-spin model, but note that a recently-introduced protocol ("h-gain"), which implements adiabatic reverse annealing, may lead to improved performance.

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“…This implies that QA can follow the instantaneous ground state and find the desired ground state in a polynomial time. In contrast, if a system undergoes a first-order quantum phase transition, the gap often decays exponentially at the transition point [11][12][13], and QA cannot solve the problem efficiently although an anomalous exception is known to exist [14]. Thus, we can estimate the efficiency of QA by analyzing the existence and order of quantum phase transitions, which is reflected in the phase diagram.…”

Section: Introductionmentioning

confidence: 99%

Quantum annealing with antiferromagnetic fluctuations

2012

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We investigate quantum annealing 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 quantum annealing 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 5 ≤ k ≤ 21. 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 quantum annealing process in the case of k = 2 and extensive patterns. These results indicate that quantum annealing with antiferromagnetic transverse interactions is efficient also for certain random spin systems.

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Energy Gap at First-Order Quantum Phase Transitions: An Anomalous Case

Cited by 26 publications

References 18 publications

Quantum speedup at zero temperature via coherent catalysis

Quantum speedup at zero temperature via coherent catalysis

Dynamics of reverse annealing for the fully connected p -spin model

Quantum annealing with antiferromagnetic fluctuations

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