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arXiv:1705.05379 · Physical review. B./Physical review. B

Dissipative Landau-Zener problem and thermally assisted Quantum Annealing

Luca Arceci, Simone Barbarino, Rosario Fazio, Giuseppe E. Santoro

Published Aug 1, 2017• 44 citations
arXivHTMLPDF

Abstract

We revisit here the issue of thermally assisted Quantum Annealing by a detailed study of the dissipative Landau-Zener problem in the presence of a Caldeira-Leggett bath of harmonic oscillators, using both a weak-coupling quantum master equation and a quasiadiabatic path-integral approach. Building on the known zero-temperature exact results [Wubs et al., Phys. Rev. Lett. 97, 200404 (2006)], we show that a finite temperature bath can have a beneficial effect on the ground-state probability only if it couples also to a spin direction that is transverse with respect to the driving field, while no improvement is obtained for the more commonly studied purely longitudinal coupling. In particular, we also highlight that, for a transverse coupling, raising the bath temperature further improves the ground-state probability in the fast-driving regime. We discuss the relevance of these findings for the current quantum-annealing flux qubit chips.

DOI
10.1103/physrevb.96.054301

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We revisit here the issue of thermally assisted Quantum Annealing by a detailed study of the dissipative Landau-Zener problem in the presence of a Caldeira-Leggett bath of harmonic oscillators, using both a weak-coupling quantum master equation and a quasiadiabatic path-integral approach. Building on the known zero-temperature exact results [Wubs et al., Phys. Rev. Lett. 97, 200404 (2006)], we show that a finite temperature bath can have a beneficial effect on the ground-state probability only if it couples also to a spin direction that is transverse with respect to the driving field, while no improvement is obtained for the more commonly studied purely longitudinal coupling. In particular, we also highlight that, for a transverse coupling, raising the bath temperature further improves the ground-state probability in the fast-driving regime. We discuss the relevance of these findings for the current quantum-annealing flux qubit chips.
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 study Landau-Zener transitions in a qubit coupled to a bath at zero temperature. A general formula that is applicable to models with a nondegenerate ground state is derived. We calculate exact transition probabilities for a qubit coupled to either a bosonic or a spin bath. The nature of the baths and the qubit-bath coupling is reflected in the transition probabilities. For diagonal coupling, when the bath causes energy fluctuations of the diabatic qubit states but no transitions between them, the transition probability coincides with the standard Landau-Zener probability of an isolated qubit. This result is universal as it does not depend on the specific type of bath. For pure off-diagonal coupling, by contrast, the tunneling probability is sensitive to the coupling strength. We discuss the relevance of our results for experiments on molecular nanomagnets, in circuit QED, and for the fast-pulse readout of superconducting phase qubits.
We formulate and analyze a double-slit proposal for quantum annealing, which involves observing the probability of finding a two-level system (TLS) undergoing evolution from a transverse to a longitudinal field in the ground state at the final time tf. We demonstrate that for annealing schedules involving two consecutive diabatic transitions, an interference effect is generated akin to a double-slit experiment. The observation of oscillations in the ground state probability as a function of tf (before the adiabatic limit sets in) then constitutes a sensitive test of coherence between energy eigenstates. This is further illustrated by analyzing the effect of coupling the TLS to a thermal bath: increasing either the bath temperature or the coupling strength results in a damping of these oscillations. The theoretical tools we introduce significantly simplify the analysis of the generalized Landau-Zener problem. Furthermore, our analysis connects quantum annealing algorithms exhibiting speedups via the mechanism of coherent diabatic transitions to near-term experiments with quantum annealing hardware.
We discuss the quantum annealing of the fully-connected ferromagnetic p-spin model in a dissipative environment at low temperature. This model, in the large p limit, encodes in its ground state the solution to the Grover’s problem of searching in unsorted databases. In the framework of the quantum circuit model, a quantum algorithm is known for this task, providing a quadratic speed-up with respect to its best classical counterpart. This improvement is not recovered in adiabatic quantum computation for an isolated quantum processor. We analyze the same problem in the presence of a low-temperature reservoir, using a Markovian quantum master equation in Lindblad form, and we show that a thermal enhancement is achieved in the presence of a zero temperature environment moderately coupled to the quantum annealer.
We determine the Landau-Zener transition probability in a dissipative environment including both longitudinal as well as transversal quantum-mechanical noise originating from a single noise source. For this, we use the numerically exact quasiadiabatic path integral, as well as the approximative nonequilibrium Bloch equations. We find that transversal quantum noise in general influences the Landau-Zener probability much more strongly than longitudinal quantum noise does at a given temperature and system-bath coupling strength. In other words, transversal noise contributions become important even when the coupling strength of transversal noise is smaller than that of longitudinal noise. We furthermore reveal that transversal noise renormalizes the tunnel coupling independent of temperature. Finally, we show that the effect of mixed longitudinal and transversal noise originating from a single bath cannot be obtained from an incoherent sum of purely longitudinal and purely transversal noise.
Abstract We formulate and analyze a double-slit proposal for quantum annealing, which involves observing the probability of finding a two-level system (TLS) undergoing evolution from a transverse to a longitudinal field in the ground state at the final time tf. We demonstrate that for annealing schedules involving two consecutive diabatic transitions, an interference effect is generated akin to a double-slit experiment. The observation of oscillations in the ground state probability as a function of tf (before the adiabatic limit sets in) then constitutes a sensitive test of coherence between energy eigenstates. This is further illustrated by analyzing the effect of coupling the TLS to a thermal bath: increasing either the bath temperature or the coupling strength results in a damping of these oscillations. The theoretical tools we introduce significantly simplify the analysis of the generalized Landau-Zener problem. Furthermore, our analysis connects quantum annealing algorithms exhibiting speedups via the mechanism of coherent diabatic transitions to near-term experiments with quantum annealing hardware.