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arXiv:1804.04251 · Physical review. B./Physical review. B

Optimal working point in dissipative quantum annealing

Luca Arceci, Simone Barbarino, Davide Rossini, Giuseppe E. Santoro

Published Aug 27, 2018• 27 citations
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Abstract

We study the effect of a thermal environment on the quantum annealing dynamics of a transverse-field Ising chain. The environment is modeled as a single Ohmic bath of quantum harmonic oscillators weakly interacting with the total transverse magnetization of the chain in a translationally invariant manner. We show that the density of defects generated at the end of the annealing process displays a minimum as a function of the annealing time, the so-called optimal working point, only in rather special regions of the bath temperature and coupling strength plane. We discuss the relevance of our results for current and future experimental implementations with quantum annealing hardware.

DOI
10.1103/physrevb.98.064307

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We study the effect of a thermal environment on the quantum annealing dynamics of a transverse-field Ising chain. The environment is modeled as a single Ohmic bath of quantum harmonic oscillators weakly interacting with the total transverse magnetization of the chain in a translationally invariant manner. We show that the density of defects generated at the end of the annealing process displays a minimum as a function of the annealing time, the so-called optimal working point, only in rather special regions of the bath temperature and coupling strength plane. We discuss the relevance of our results for current and future experimental implementations with quantum annealing hardware.
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In this thesis, the effect of dissipation is investigated in driven models of interest for quantum annealing and quantum topological pumping. Dissipation comes from coupling the system to a bosonic bath at thermal equilibrium, using the usual Caldeira-Leggett setting. The first original results presented deal with the dissipative Landau-Zener model, where we revisit the issue of whether dissipation can improve the final ground state probability. We shed light upon the importance of the coupling direction. Then, we move to dissipation effects in the quantum annealing of the Ising chain in transverse field: we study the conditions for the emergence or absence of an optimal annealing velocity, depending on system-bath interaction strength and bath temperature. Finally, we explore dissipation effects in topological pumping in the periodically-driven Rice-Mele model, where we find that a bath at low temperature can improve the pumping performance when adiabaticity is not perfectly fulfilled.
We show that quantum diffusion near a quantum critical point can provide an efficient mechanism of quantum annealing. It is based on the diffusion-mediated recombination of excitations in open systems far from thermal equilibrium. We find that, for an Ising spin chain coupled to a bosonic bath and driven by a monotonically decreasing transverse field, excitation diffusion sharply slows down below the quantum critical region. This leads to spatial correlations and effective freezing of the excitation density. Still, obtaining an approximate solution of an optimization problem via the diffusion-mediated quantum annealing can be faster than via closed-system quantum annealing or Glauber dynamics.
We present a study of the digitized quantum annealing protocol proposed by R. Barends et al. [Nature (London) 534, 222 (2016)]. Our analysis, performed on the benchmark case of a transverse Ising chain problem, shows that the algorithm has a well-defined optimal working point for the annealing time ${\ensuremath{\tau}}_{\mathrm{P}}^{\mathrm{opt}}$, scaling as ${\ensuremath{\tau}}_{\mathrm{P}}^{\mathrm{opt}}\ensuremath{\sim}\mathrm{P}$, where P\mathrm{P}P is the number of digital Trotter steps, beyond which the residual energy error shoots up toward the value characteristic of the maximally disordered state. We present an analytical analysis for the translationally invariant transverse Ising chain case, but our numerical evidence suggests that this scenario is more general, surviving, for instance, the presence of disorder.
Quantum annealing must be sufficiently slow to prevent defect formation. We study this problem in the random transverse-field Ising model via the adiabatic theorem to obtain the limiting rate at which defect formation can be avoided. Our approach based on the strong-disorder renormalization group (SDRG) technique views the annealing process as a gradual aggregation of strongly coupled clusters, giving access to the elementary events of error formation. We present a scaling theory of the statistics of these events in general dimensions and confront it to an extensive numerical analysis carried out in one, two, and three dimensions by an efficient implementation of the SDRG method. This approach is also validated by exact diagonalization in one dimension by means of the free-fermion technique. Our results consistently show a logarithmically slow decrease of the error density with the annealing time, characterized by the dimension-dependent critical exponents of the corresponding infinite-disorder fixed point of the model. Furthermore, we provide a map of individual events of potential defect formation in the excitation energy - control parameter plane, which may help in the design of optimal annealing paths.
We investigate the quantum annealing of the ferromagnetic ppp-spin model in a dissipative environment (p=5p=5p=5 and p=7p=7p=7). This model, in the large-ppp limit, codifies Grover's algorithm for searching in an unsorted database [L. K. Grover, Proceedings of the 28th Annual ACM Symposium on Theory of Computing (ACM, New York, 1996), pp. 212--219]. The dissipative environment is described by a phonon bath in thermal equilibrium at finite temperature. The dynamics is studied in the framework of a Lindblad master equation for the reduced density matrix describing only the spins. Exploiting the symmetries of our model Hamiltonian, we can describe many spins and extrapolate expected trends for large NNN and ppp. While at weak system-bath coupling the dissipative environment has detrimental effects on the annealing results, we show that in the intermediate-coupling regime, the phonon bath seems to speed up the annealing at low temperatures. This improvement in the performance is likely not due to thermal fluctuation but rather arises from a correlated spin-bath state and persists even at zero temperature. This result may pave the way to a new scenario in which, by appropriately engineering the system-bath coupling, one may optimize quantum annealing performances below either the purely quantum or the classical limit.