Scaling and Diabatic Effects in Quantum Annealing with a D-Wave Device

Abstract

We discuss quantum annealing of the two-dimensional transverse-field Ising model on a D-Wave device, encoded on L × L lattices with L ≤ 32. Analyzing the residual energy and deviation from maximal magnetization in the final classical state, we find an optimal L dependent annealing rate v for which the two quantities are minimized. The results are well described by a phenomenological model with two powers of v and L-dependent prefactors to describe the competing effects of reduced quantum fluctuations (for which we see evidence of the Kibble-Zurek mechanism) and increasing noise impact when v is lowered. The same scaling form also describes results of numerical solutions of a transverse-field Ising model with the spins coupled to noise sources. We explain why the optimal annealing time is much longer than the coherence time of the individual qubits.The prospect of simulating theoretical quantum manybody Hamiltonians with controllable engineered systems is now an important motivation for atomic and quantum device physics [1-3]. Systems explored for creating such "synthetic quantum matter" include ultracold gases [4][5][6][7][8][9], photonic devices [10][11][12][13][14], polaritons [15], and trapped ions [16][17][18][19][20][21][22]. Another emerging simulation platform is large arrays of superconducting qubits [23][24][25][26][27][28][29], which were originally envisioned in the context of quantum annealing (QA) as efficient solvers of classical optimization problems mapped to Ising like Hamiltonians [30][31][32][33][34][35][36][37][38][39][40][41][42][43]. To reach the classical ground state (the problem solution) in a QA process, strong quantum fluctuations are initially induced by applying a transverse field, which is quasi-adiabatically reduced to zero. QA devices operating according to this principle have entered industrial production and applications beyond the academic setting [23], motivated by the hope of more efficient solutions of NP-hard problems [33,44] and, more recently, quantum enhanced machine learning [45,46]. It is still unclear what systems (classes of optimization problems) are amenable to significant speedup, and to what extent QA can be realized in actual devices [41,[47][48][49][50][51][52][53][54][55].While the question of quantum speedups is essential, the potential of using QA devices as generic quantum many-body emulators motivates a broader range of investigations into the devices and how they can be exploited for probing various quantum phenomena. As an example, recently a QA device produced by D-Wave Systems was used in an impressive study of a quantum phase transition of a quantum spin glass [29]. An important question in applications of QA devices, for optimization or quantum simulation, is whether the desired adiabatic evolution is sufficiently realized in the presence of noise (the environment) and finite annealing time. This question motivates studies of the dependence of measured properties on the annealing time [27,[57][58][59], which also impacts the effects of noise. For this...

Probing the universality of topological defect formation in a quantum annealer: Kibble-Zurek mechanism and beyond

“…In the short-time region of the two-dimensional experiment of Ref. [51] where the KZM is likely to apply, the system seems to be much less susceptible to noise and the exponent α is close to the theoretical value of an ideal, isolated system as mentioned above. These observations suggest that how noise affects the system behavior strongly depends on the problem type as well as on the annealing time range.…”

“…4 and presented in Ref. [51] as well as in Refs. [46,76] show that the kink density can be nonmonotonic if the temperature is finite.…”

“…Other closely related work is the recent experiment probing the two-dimensional transverse-field Ising model on the square lattice by Weinberg et al [51], which demonstrated nonmonotonicity in the kink density as a function of the annealing time t a . In the short annealing time regime, the kink density decreases as a function of t a , as in our case.…”

“…Indeed, the KZM scaling is sensitive to, e.g., nonlinear driving schemes [38,39], disorder [40,41], inhomogeneities in the system [42][43][44][45] and decoherence [46][47][48][49]. The use of the KZM to assess the performance of a quantum annealer was studied by Gardas et al [50], focusing on the one-dimensional case on two previous generation D-Wave 2X devices, and by Weinberg et al [51] on a current generation D-Wave 2000Q (DW2KQ) device, focusing on the Ising Hamiltonian on the two-dimensional square lattice.…”

Coherent Many-Body Oscillations Induced by a Superposition of Broken Symmetry States in the Wake of a Quantum Phase Transition

“…More recently, KZM was adapted to quantum phase transitions [42][43][44][45][46][47]. Theoretical developments as well as experimental tests [80][81][82][83][84][85][86][87][88][89][90] of the quantum KZM (QKZM) followed. The classic experiment [88], where a quantum Ising chain in the transverse field is emulated by Rydberg atoms, is consistent with the predicted the density of defects and with is quench-rate dependent scaling [43][44][45].…”