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arXiv:1909.00817 · Physical review. B./Physical review. B

Optimal working point in digitized quantum annealing

Glen Bigan Mbeng, Luca Arceci, Giuseppe E. Santoro

Published Dec 2, 2019• 19 citations
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Abstract

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 $\mathrm{P}$ 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.

DOI
10.1103/physrevb.100.224201

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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 \ensuremathτPopt{\ensuremath{\tau}}_{\mathrm{P}}^{\mathrm{opt}}\ensuremathτPopt​, scaling as \ensuremathτPopt\ensuremath∼P{\ensuremath{\tau}}_{\mathrm{P}}^{\mathrm{opt}}\ensuremath{\sim}\mathrm{P}\ensuremathτPopt​\ensuremath∼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.
Digitizing an adiabatic evolution is a strategy able to combine the good performance of gate-based quantum processors with the advantages of adiabatic algorithms, thus providing a hybrid model for efficient quantum information processing. In this work we develop validity conditions for high-fidelity digital adiabatic tasks. To this end, we assume a digitizing process based on the Suzuki-Trotter decomposition, which allows us to introduce a digitized adiabatic theorem. As a consequence of this theorem, we show that the performance of such a hybrid model is limited by the fundamental constraints on the adiabatic theorem validity, even in ideal quantum processors. We argue how our approach predicts the existence of intrinsic nonadiabatic errors reported by Barends et al. [Nature (London) 534, 222 (2016)] through an empirical study of digital annealing. In addition, our approach allows us to explain the existence of a scaling of the number of Suzuki-Trotter blocks for the optimal digital circuit with respect to the optimal adiabatic total evolution time, as reported by Mbeng et al. [Phys. Rev. B 100, 224201 (2019)] through robust numerical analysis of digital annealing. We illustrate our results through two examples of digitized adiabatic algorithms, namely, the two-qubit exact-cover problem and the three-qubit adiabatic factorization of the number 21.
We establish and discuss a number of connections between a digitized version of Quantum Annealing (QA) with the Quantum Approximate Optimization Algorithm (QAOA) introduced by Farhi et al. (arXiv:1411.4028) as an alternative hybrid quantum-classical variational scheme for quantum-state preparation and optimization. We introduce a technique that allows to prove, for instance, a rigorous bound concerning the performance of QAOA for MaxCut on a 222-regular graph, equivalent to an unfrustrated antiferromagnetic Ising chain. The bound shows that the optimal variational error of a depth-P\mathrm{P}P quantum circuit has to satisfy εPres≥(2P+2)−1ε^\mathrm{res}_{\mathrm{P}}\ge (2\mathrm{P}+2)^{-1}εPres​≥(2P+2)−1. In a separate work (Mbeng et al., arXiv:1911.12259) we have explicitly shown, exploiting a Jordan-Wigner transformation, that among the 2P2^{\mathrm{P}}2P degenerate variational minima which can be found for this problem, all strictly satisfying the equality εPres=(2P+2)−1ε^\mathrm{res}_{\mathrm{P}}=(2\mathrm{P}+2)^{-1}εPres​=(2P+2)−1, one can construct a special {\em regular} optimal solution, which is computationally optimal and does not require any prior knowledge about the spectral gap. We explicitly demonstrate here that such a schedule is adiabatic, in a digitized sense, and can therefore be interpreted as an optimized digitized-QA protocol. We also discuss and compare our bound on the residual energy to well-known results on the Kibble-Zurek mechanism behind a continuous-time QA. These findings help elucidating the intimate relation between digitized-QA, QAOA, and optimal Quantum Control.
We show how a digitized version of Quantum Annealing can be made optimal, realizing the best possible solution allowed by quantum mechanics in the shortest time, without any prior knowledge on the location and properties of the spectral gap. Our findings elucidate the intimate relation between digitized-QA, optimal Quantum Control, and recently proposed hybrid quantum-classical variational algorithms for quantum-state preparation and optimization. We illustrate this on the simple benchmark problem of an unfrustrated antiferromagnetic Ising chain in a transverse field.
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.
Simulated quantum annealing (SQA) is a classical computational strategy that emulates a quantum annealing (QA) dynamics through a path-integral Monte Carlo whose parameters are changed during the simulation. Here we apply SQA to the one-dimensional transverse field Ising chain, where previous works have shown that, in the presence of disorder, a coherent QA provides a quadratic speedup with respect to classical simulated annealing, with a density of Kibble-Zurek defects decaying as ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}2}$ as opposed to ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{({log}_{10}\ensuremath{\tau})}^{\ensuremath{-}1}, \ensuremath{\tau}$ being the total annealing time, while for the ordered case both give the same power law ${\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{QA}}\ensuremath{\approx}{\ensuremath{\rho}}_{\mathrm{KZ}}^{\mathrm{SA}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$. We show that the dynamics of SQA, while correctly capturing the Kibble-Zurek scaling ${\ensuremath{\tau}}^{\ensuremath{-}1/2}$ for the ordered case, is unable to reproduce the QA dynamics in the disordered case at intermediate $\ensuremath{\tau}$. We analyze and discuss several issues related to the choice of the Monte Carlo moves (local or global in space), the time-continuum limit needed to eliminate the Trotter-discretization error, and the long autocorrelation times shown by a local-in-space Monte Carlo dynamics for large disordered samples.
Quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA) are two special cases of the following control problem: apply a combination of two Hamiltonians to minimize the energy of a quantum state. Which is more effective has remained unclear. Here we analytically apply the framework of optimal control theory to show that generically, given a fixed amount of time, the optimal procedure has the pulsed (or "bang-bang") structure of QAOA at the beginning and end but can have a smooth annealing structure in between. This is in contrast to previous works which have suggested that bang-bang (i.e., QAOA) protocols are ideal. To support this theoretical work, we carry out simulations of various transverse field Ising models, demonstrating that bang-anneal-bang protocols are more common. The general features identified here provide guideposts for the nascent experimental implementations of quantum optimization algorithms.