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arXiv:2209.01737 · Physical Review A

Nonstoquastic catalyst for bifurcation-based quantum annealing of the ferromagnetic p -spin model

Yuki Susa, Takashi Imoto, Yuichiro Matsuzaki

Published May 1, 2023• 4 citations
arXivHTMLPDF

Abstract

Introducing a nonstoquastic catalyst is a promising avenue to improve quantum annealing with the transverse field. In the present paper, we propose a nonstoquastic catalyst for bifurcation-based quantum annealing described by the spin-1 operators to improve the efficiency of a ground-state search. To investigate the effect of the nonstoquastic catalyst, we study the ferromagnetic $p$-spin model, which has difficulty with finding the ground state due to the first-order phase transition for quantum annealing. A semiclassical analysis shows that the problematic first-order phase transition can be eliminated by introducing the proposed nonstoquastic catalyst with the appropriate amplitude. We also numerically calculate the minimum energy gap for a finite-size system by diagonalizing the Hamiltonian. We find that while the energy gap decreases exponentially with increasing system size for the original Hamiltonian, it decreases polynomially against the system size for the Hamiltonian with the nonstoquastic catalyst. This result implies that the proposed nonstoquastic catalyst has the potential to improve the performance of bifurcation-based quantum annealing.

DOI
10.1103/physreva.107.052401

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Introducing a nonstoquastic catalyst is a promising avenue to improve quantum annealing with the transverse field. In the present paper, we propose a nonstoquastic catalyst for bifurcation-based quantum annealing described by the spin-1 operators to improve the efficiency of a ground-state search. To investigate the effect of the nonstoquastic catalyst, we study the ferromagnetic ppp-spin model, which has difficulty with finding the ground state due to the first-order phase transition for quantum annealing. A semiclassical analysis shows that the problematic first-order phase transition can be eliminated by introducing the proposed nonstoquastic catalyst with the appropriate amplitude. We also numerically calculate the minimum energy gap for a finite-size system by diagonalizing the Hamiltonian. We find that while the energy gap decreases exponentially with increasing system size for the original Hamiltonian, it decreases polynomially against the system size for the Hamiltonian with the nonstoquastic catalyst. This result implies that the proposed nonstoquastic catalyst has the potential to improve the performance of bifurcation-based quantum annealing.
Nonstoquastic drivers are known to improve the performance of quantum annealing by reducing first-order phase transitions into second-order ones in several mean-field-type model systems. Nevertheless, statistical-mechanical analysis shows that some target Hamiltonians still exhibit unavoidable first-order transitions even with nonstoquastic drivers, making them difficult for quantum annealing to solve. Recently, a mechanism called coherent catalysis was proposed by Durkin [Phys. Rev. A 99, 032315 (2019)], in which he showed the existence of a particular point on the line of first-order phase transitions where the energy gap scales polynomially as expected for a second-order transition. We show by extensive numerical computations that this phenomenon is observed in a few additional mean-field-type optimization problems where nonstoquastic drivers fail to change the order of phase transition in the thermodynamic limit. This opens up the possibility of using coherent catalysis to search for exponential speedups in systems previously thought to be exponentially slow for quantum annealing to solve.
We introduce antiferromagnetic quantum fluctuations into quantum annealing in addition to the conventional transverse-field term. We apply this method to the infinite-range ferromagnetic p-spin model, for which the conventional quantum annealing has been shown to have difficulties in finding the ground state efficiently due to a first-order transition. We study the phase diagram of this system both analytically and numerically. Using the static approximation, we find that there exists a quantum path to reach the final ground state from the trivial initial state that avoids first-order transitions for intermediate values of p. We also study numerically the energy gap between the ground state and the first excited state and find evidence for intermediate values of p for which the time complexity scales polynomially with the system size at a second-order transition point along the quantum path that avoids first-order transitions. These results suggest that quantum annealing would be able to solve this problem with intermediate values of p efficiently, in contrast to the case with only simple transverse-field fluctuations.
The viability of nonstoquastic catalyst Hamiltonians to deliver consistent quantum speedups in quantum adiabatic optimization remains an open question. The infinite-range ferromagnetic ppp-spin model is a rare example exhibiting an exponential advantage for nonstoquastic catalysts over its stoquastic counterpart. We revisit this model and note how the incremental changes in the ground-state wave function give an indication of how the nonstoquastic catalyst provides an advantage. We then construct two new examples that exhibit an advantage for nonstoquastic catalysts over stoquastic catalysts. The first is another infinite range model that is only 2-local, but also exhibits an exponential advantage, and the second is a geometrically local Ising example that exhibits a growing advantage up to the maximum system size we study.
We introduce a two-parameter approximate counter-diabatic term into the Hamiltonian of the transversefield Ising model for quantum annealing to accelerate convergence to the solution, generalizing an existing single-parameter approach.The protocol is equivalent to unconventional diabatic control of the longitudinal and transverse fields in the transverse-field Ising model and thus makes it more feasible for experimental realization than an introduction of new terms such as nonstoquastic catalysts toward the same goal of performance enhancement.We test the idea for the p-spin model with p = 3, which has a first-order quantum phase transition, and show that our two-parameter approach leads to significantly larger ground-state fidelity and lower residual energy than those by traditional quantum annealing and by the single-parameter method.We also find a scaling advantage in terms of the time-to-solution as a function of the system size in a certain range of parameters as compared to the traditional methods in the sense that an exponential time complexity is reduced to another exponential complexity with a smaller coefficient.Although the present method may not always lead to a drastic exponential speedup in difficult optimization problems, it is useful because of its versatility and applicability for any problem after a simple algebraic manipulation, in contrast to some other powerful prescriptions for acceleration such as nonstoquastic catalysts in which one should carefully study in advance if it works in a given problem and should identify a proper way to meticulously control the system parameters to achieve the goal, which is generally highly nontrivial.
In adiabatic quantum computation the required run-time to reach a given ground-state fidelity is dictated by the size of the minimum gap that appears between the ground and first-excited state in the annealing spectrum. In general the presence of avoided level crossings demands an exponential increase in the annealing time with the system size which has consequences both for the efficiency of the algorithm and the required qubit coherence times. One promising avenue being explored to produce more favorable gap scaling is the introduction of nonstoquastic X X couplings in the form of a catalyst—of particular interest are catalysts which utilize accessible information about the optimization problem in their construction. Here we show extreme sensitivity of the effect of an X X catalyst to subtle changes in the encoding of the optimization problem. In particular, we observe that a targeted catalyst containing a single coupling can significantly reduce the gap closing with system size at an avoided level crossing. For slightly different encodings of the same problems however, these same catalysts result in closing gaps in the annealing spectrum. To understand the origin of these closing gaps, we study how the evolution of the ground-state vector is altered by the presence of the catalyst and find that the negative components of the ground-state vector are key to understanding the response of the gap spectrum. We also consider how and when these closing gaps could be utilized in diabatic quantum annealing protocols—a promising alternative to adiabatic quantum annealing in which transitions to higher energy levels are exploited to reduce the run time of the algorithm. Published by the American Physical Society 2024
Quantum annealing is a promising algorithm for solving combinatorial optimization problems. It searches for the ground state of the Ising model, which corresponds to the optimal solution of a given combinatorial optimization problem. The guiding principle of quantum annealing is the adiabatic theorem in quantum mechanics, which guarantees that a system remains in the ground state of its Hamiltonian if the time evolution is sufficiently slow. According to the adiabatic theorem, the annealing time required for quantum annealing to satisfy the adiabaticity scales inversely proportional to the square of the minimum energy gap between the ground state and the first excited state during time evolution. As a result, finding the ground state becomes significantly more difficult when the energy gap is small, creating a major bottleneck in quantum annealing. Expanding the energy gap is one strategy for improving the performance of quantum annealing; however, its implementation in actual hardware remains difficult. This study proposes a method for efficiently solving instances with small energy gaps by introducing additional local terms to the Hamiltonian and exploiting the diabatic transition remaining in the small energy gap. The proposed method achieves an approximate quadratic speedup of the exponential scaling exponent in time to solution compared to the conventional quantum annealing. In addition, we investigate the transferability of the parameters obtained with the proposed method.