The quantum annealing gap and quench dynamics in the exact cover problem

Bernhard Irsigler and Tobias Grass

ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

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Quenching and annealing are extreme opposites in the time evolution of a quantum system: Annealing explores equilibrium phases of a Hamiltonian with slowly changing parameters and can be exploited as a tool for solving complex optimization problems. In contrast, quenches are sudden changes of the Hamiltonian, producing a non-equilibrium situation. Here, we investigate the relation between the two cases. Specifically, we show that the minimum of the annealing gap, which is an important bottleneck of quantum annealing algorithms, can be revealed from a dynamical quench parameter which describes the dynamical quantum state after the quench. Combined with statistical tools including the training of a neural network, the relation between quench and annealing dynamics can be exploited to reproduce the full functional behavior of the annealing gap from the quench data. We show that the partial or full knowledge about the annealing gap which can be gained in this way can be used to design optimized quantum annealing protocols with a practical time-to-solution benefit. Our results are obtained from simulating random Ising Hamiltonians, representing hard-to-solve instances of the exact cover problem.

Quantum annealing is a method to solve optimization problems via Hamiltonian engineering and quantum state preparation. The method requires that, at any time during the preparation process, the annealing is slow compared to the energy gap above the ground state. Unfortunately, for complex problems small gaps are a common bottleneck of quantum annealing. In this paper, we present strategies to gain knowledge of the gap function for specific computational problems by performing quench experiments and applying machine learning techniques. This knowledge is then shown to facilitate the design of optimized annealing algorithms. As compared to the usual homogeneous protocol, these protocols show an enhanced performance with increased annealing fidelities (see Figure).

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[1] Tobias Grass, "Quantum Annealing Sampling with a Bias Field", Physical Review Applied 18 4, 044036 (2022).

[2] Jian Lin, Zhengfeng Zhang, Junping Zhang, and Xiaopeng Li, "Hard-instance learning for quantum adiabatic prime factorization", Physical Review A 105 6, 062455 (2022).

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