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arXiv:1811.09980 · Physical Review A

Role of nonstoquastic catalysts in quantum adiabatic optimization

Tameem Albash

Published Apr 26, 2019• 52 citations
arXivHTMLPDF

Abstract

The viability of nonstoquastic catalyst Hamiltonians to deliver consistent quantum speedups in quantum adiabatic optimization remains an open question. The infinite-range ferromagnetic $p$-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.

DOI
10.1103/physreva.99.042334

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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.
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.
Quantum annealing promises to solve combinatorial optimization problems by preparing the ground state of a target Hamiltonian. Standard annealing protocols are, however, stoquastic and can thus be simulated by sign-problem-free quantum Monte-Carlo methods. To obtain a true quantum advantage, it has been proposed to use non-stoquastic catalyst Hamiltonians. Active only at intermediate stages of the protocol, these can, for certain problems, convert first-order into second-order quantum phase transitions and thus permit an exponential speedup over the stoquastic protocol. At the same time, the non-stoquastic catalyst renders quantum Monte-Carlo methods inefficient. It remains, however, an open question how other classical computation methods are affected by the non-stoquastic terms. We address this question by computing quantum resources -- entanglement entropy and stabilizer Rényi entropy -- whose presence makes classical computations based on tensor networks and stabilizer-tableau methods exponentially hard. We compare these with the spectral gap along the annealing path for two paradigmatic benchmark models, the fully connected ppp-spin model and a geometrically local Ising model. While the exact behavior shows a subtle dependency on the underlying model and the annealing path, our numerics suggest consistently that the scaling of entanglement and non-stabilizerness is at least maintained in the deeply non-stoquastic regime and in some cases even significantly enhanced. Our results thus suggest that improvements of quantum performance in non-stoquastic annealing coincide with significant presence of quantum computational resources.
Quantum annealers are computing machines that utilize quantum effects to solve hard optimization problems. Understanding the circumstances under which quantum annealers are more efficient than other optimization methods is a challenging open problem. A clue to this conundrum comes from studying the complexity of quantum annealers. In general, the complexity of a physical system relates to how efficiently the system can be simulated by classical algorithms. One then asks: if some class of quantum annealers are hard to simulate classically, are they also more powerful optimization machines? Here, the authors study quantum annealers belonging to two different complexity classes: the so-called ``stoquastic'' systems, which can be efficiently simulated with classical algorithms, and ``nonstoquastic'' systems, currently with no efficient classical treatment. Applied to a prototypical optimization problem, the authors observe that, on average, the stoquastic systems perform better on easier instances of the problem, while the more complex nonstoquastic annealers show significant advantage when applied to the hardest instances. The authors conjecture that the performance of the nonstoquastic Hamiltonians is closely related to their internal structure during the annealing process and elucidate how properties such as magnetic frustration can play a key role in devising more powerful quantum annealers.
Nonstoquastic Hamiltonians are hard to simulate due to the sign problem in quantum Monte Carlo simulation. It is, however, unclear whether nonstoquasticity can lead to advantage in quantum annealing. Here we show that YYYYYY interactions between the qubits make the adiabatic path during quantum annealing, and therefore the performance, dependent on spin-reversal transformations. With the right choice of spin-reversal transformation, a nonstoquastic Hamiltonian with YYYYYY interaction can outperform stoquastic Hamiltonians with similar parameters. We introduce an optimization protocol to determine the optimal transformation and discuss the effect of suboptimality.
The role of nonstoquasticity in the field of quantum annealing and adiabatic quantum computing is an actively debated topic. We study a strongly-frustrated quasi-one-dimensional quantum Ising model on a two-leg ladder to elucidate how a first-order phase transition with a topological origin is affected by interactions of the $\ifmmode\pm\else\textpm\fi{}XX$-type. Such interactions are sometimes known as stoquastic (negative sign) and nonstoquastic (positive sign) ``catalysts''. Carrying out a symmetry-preserving real-space renormalization group analysis and extensive density-matrix renormalization group computations, we show that the phase diagrams obtained by these two methods are in qualitative agreement with each other and reveal that the first-order quantum phase transition of a topological nature remains stable against the introduction of both XXXXXX-type catalysts. This is the first study of the effects of nonstoquasticity on a first-order phase transition between topologically distinct phases. Our results indicate that nonstoquastic catalysts are generally insufficient for removing topological obstacles in quantum annealing and adiabatic quantum computing.
The potential quantum speedup in solving optimization problems via adiabatic quantum annealing is often hindered by the closing of the energy gap during the anneal, especially when this gap scales exponentially with system size. In this work, we alleviate this bottleneck by demonstrating that for the NP-complete Maximum Weighted Independent Set (MWIS) problem, an informed choice of n -local catalysts (operators involving n qubits) can reopen the gap during the annealing process. By analyzing first-order phase transitions in toy instances of the MWIS problem, we first identify direct-tunneling catalysts that effectively eliminate the transition and provide an analytical discussion on when the sign of the catalyst influences its impact. We then reveal that n -local catalysts exponentially improve gap scaling and in certain scenarios are as effective as direct-tunnel coupling between two minima. Utilizing this understanding, we show that they also increase the efficiency of ground-state preparation via adiabatic quantum annealing in random graphs and analytically demonstrate the necessity of their placement across unfrustrated loops in the graph for effective performance in MWIS problems. Additionally, using a circuit implementation of the n -local catalyst (requiring 2 n nearest-neighbor gates), we demonstrate that both the circuit depth and the total number of gates required to solve the problem are reduced by several orders of magnitude when compared to the discrete-time version of traditional quantum annealing using local drivers. Our analysis suggests that nonlocal quantum fluctuations entangling multiple qubits as a catalyst are key to achieving the desired quantum advantage.